Maths dimensions 7

Maths Dimensions 7 essential learning

Pearson Education Australia A division of Pearson Australia Group Pty Ltd Level 9, 5 Queens Road Melbourne 3004 Australia www.pearsoned.com.au/schools Offices in Sydney, Brisbane and Perth, and associated companies throughout the world. Copyright © Pearson Education Australia 2006 First published 2006 Reproduction and Communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10 per cent of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given remuneration notice(s) to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact Copyright Agency Limited (www.copyright.com.au). Reproduction and Communication for other purposes Except as permitted under the Act (for example a fair dealing for the purposes of study, research, criticism or review) no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above. Edited by Marta Veroni Designed by Nicole Melbourne and Kim Ferguson-Somerton Cover designed by Kim Ferguson-Somerton Typeset by Nicole Melbourne Technical illustrations by Margaret Hastie and Wendy Gorton Cartoons by Connah Brecon Cover images by Getty Images Prepress work by The Type Factory Produced by Pearson Education Australia Printed in China National Library of Australia Cataloguing-in-Publication data Maths dimensions 7: essential learning. For secondary school students. ISBN 0 7339 7633 6. 1. Mathematics - Textbooks. I. Bull, Ian. 510

Every effort has been made to trace and acknowledge copyright. However, should any infringement have occurred, the publishers tender their apologies and invite copyright holders to contact them.

Ian Bull, Bob Howes, Karen Kimber, Caroline Nolan, Kimm Noonan

Maths Dimensions 7 essential learning

Sydney, Melbourne, Brisbane, Perth and associated companies around the world

Contributing Authors Ron Barassi (Animations) David Barton (Homework Book) Peter Fox (Technology Activities) Greg Hardham (Companion Website)

Acknowledgements The authors would like to acknowledge the support and encouragement of their families over the many hours spent completing the original Maths for Vic series and now this Maths Dimensions series. In particular, grateful thanks are afforded to Jenny, William and Lucinda for making available their home on numerous occasions when the author team originally met to develop the series. Our initial aim was to produce a sequence of Maths resources that would make the discipline of Mathematics come alive in the twenty-first century and be relevant to the new millennium and the host of new technology available. With the release of the Maths Dimensions series that process has been further refined in the light of the Victorian Essential Learning Standards (VELS) to produce a course of mathematics that is rich, relevant and rewarding in a whole-curriculum sense. The strength of the material contained in the series is a by-product of the energy and commitment shown by the author team whose years of experience are too numerous to mention. Through healthy debate, we sought to produce a series true to our aims. Students of mathematics need to be challenged and to challenge us as educators, and the enrichment section for each chapter enables that challenge to be presented to all students in context. For the students who want to achieve their best, studying hard will produce results. We would like to thank the following for permission to reproduce copyright material. The following abbreviations are used in this list: b = bottom, t = top, c = centre, l = left, r = right. Adric Macht/Stockxchng Website: p. 25. Australian Picture Library: Philip de Bay, pp. 8, 9, 10–37 (border); MPTV, p. 33; Thierry Orban, p. 181. Australian Sports Commission: p. 399t. Austral International: pp. 121, 191. Bull, Ian: pp. 17, 19, 74, 79, 97, 105, 126l, 126r, 127cl, 127bc, 127bl, 128tl, 128tr, 128cl, 128cr, 128bl, 128br, 141, 197, 200, 201, 206tl, 206tr, 206bl, 206br, 208tl, 208tr, 208cl, 208cr, 208bl, 208br, 209tl, 209tr, 209cl, 209cr, 209bl, 209br, 211, 233, 252l, 252r, 259 (backs of houses, bridge, house, kitchen, pylon, sculpture), 260l, 260r, 277, 315, 332, 333, 334–355 (border), 344, 373, 382, 383, 384–423 (border), 385, 402, 409, 417b, 418.

Matt Groening Productions, Inc.: Reprinted from Bart Simpson’s Guide to Life, published by Harper Collins, © 1993 Matt Groening Productions, Inc. All rights reserved. The Simpsons © and TM Twentieth Century Fox Film Corporation. All rights reserved, p. 207. Melbourne Sports and Aquatic Centre: Provided by Melbourne Sports and Aquatic Centre, p. 305. Melway Publishing Pty Ltd: Copyright Melway Publishing Pty Ltd, reproduced with permission, p. 297. NASA: pp. 58, 59, 60–93 (border), 72, 88, 188, 189, 190–219 (border), 242, 356, 357, 358–381 (border). Pearson Education Australia: Ben Killingsworth, p. 204; Nada Jolic, p. 339.

Crown Limited: pp. 280, 281, 282–305 (border), 296.

Photolibrary: D. Roberts/SPL, p. 41.

The M.C. Escher Company: M.C. Escher’s Horsemen © 2005 The M.C. Escher Company – Holland. All rights reserved. <www.mcescher.com>, p. 275t; M.C. Escher’s Symmetry Drawing E70: Butterflies © 2005 The M.C. Escher Company – Holland. All rights reserved. <www.mcescher.com>, p. 275b.

Reserve Bank of Australia: p. 11 (notes). Royal Australian Mint: pp. 11 (coins), 16, 18, 136t, 136b, 148tl, 148tr, 148tcl, 148tcr, 362. Texas Instruments: pp. 83, 319, 411. West Australian Newspapers: p. 393.

Every effort has been made to trace and acknowledge copyright. However, should any infringement have occurred, the publishers tender their apologies and invite copyright owners to contact them.

Contents Contributing authors

iv

Acknowledgements

iv

The Maths Dimensions Package 4 How to use this book

6

Chapter 1 Whole Number Revision

8

1A

2G

Squares and square roots

47

2H

Prime factors

48

2I

Odds and evens

49

Puzzles

50

Applications

52

Enrichment

54

Revision

56

Chapter 3 Fractions

58

Place value

10

3A

Shaded diagrams

60

1B

Addition

12

3B

1C

Subtraction

14

Mixed numbers and improper fractions

62

3C

Adding with the same denominator

64

3D

Adding with different denominators

66

3E

Tetral fractions

69

3F

Subtracting with the same denominator

71

3G

Subtracting with different denominators

73

3H

Exploring multiplication of fractions

75

1D

Multiplication

16

1E

Division

18

1F

Doubling and halving

20

1G

Order of operations

22

1H

Estimation

24

1I

Four by four

25

1J

Number systems of the past

28

Puzzles

30

Applications

32

Enrichment

34

Revision

36

Chapter 2 Number Patterns 2A

Exploring number patterns

38

3I

Multiplying fractions

76

3J

Dividing fractions

78

3K

Fraction of quantities

80

3L

Squares and square roots of fractions

82

Order of operations

84

40

2B

Multiples

42

3M

2C

Factors

43

Puzzles

86 88

2D

Divisibility tests

44

Applications

2E

Exploring primes and composites

45

Enrichment

90

46

Revision

92

2F

Index notation

1

Chapter 4 Decimals and Percentages 4A

Place value and notation

96

4B

Estimation of decimals

98

4C

Rounding decimals

4D

Adding decimals

99 100

4E

Subtracting decimals

102

4F

Multiplying decimals

104

4G

Dividing by whole numbers

106

4H

Dividing by decimals

107

4I

Exploring powers of 10

108

4J

Decimal Defender

110

4K

Fractions and decimals

111

4L

Converting decimals to fractions

112

4M

Percentages

113

4N

Finding percentages of quantities

114

4O

Calculating percentages

115

Puzzles

116

Applications

118

Enrichment

120

Revision

122

Chapter 5 Length and Perimeter 5A

Units used to measure lengths

124 126

5B

Estimating by using known values

127

5C

Estimating lengths

129

5D

Reading scales when measuring

131

5E

Measuring lengths accurately

133

5F

Converting length units

135

5G

Adding and subtracting lengths

137

5H

Perimeter of shapes with straight sides 139

5I

Exploring measurement in the past

143

5J

Rhino

144

Puzzles

146

Applications

148

Enrichment

150

Revision

152

Chapter 6 Area and Volume

2

94

154

6A

Finding and comparing areas

156

6B

Using grids to find areas

158

6C

Exploring areas of faces

160

6D

Area of rectangles

161

6E

Area of parallelograms

164

6F

Cabri triangles

166

6G

Area of triangles

168

6H

Volume as a measure of space

171

6I

Exploring volume

173

6J

Volume of rectangular prisms

174

6K

Volume of prisms

Puzzles

176 178

Applications

180

Enrichment

182

Revision

184

Chapter 7 Time and Mass

188

7A

Timelines

190

7B

Time conversions

193

7C

Time differences and the calendar

195

7D

Time differences in hours and minutes 198

7E

Using timetables

200

7F

Time zones

202

7G

Ordering events and flow charts

204

7H

Mass and conversion of units of mass 210

Puzzles

212

Applications

214

Enrichment

216

Revision

218

Chapter 8 Angles

220

8A

Naming angles

8B

Types of angles

222 223

8C

Measuring angles

225

8D

Using a protractor to draw angles

229

8E

Complementary angles

230

8F

Supplementary angles

232

8G

Angles in a circle

234

8H

What’s your angle?

Puzzles

236 238

Applications

240

Enrichment

242

Revision

244

Chapter 9 Polygons

246

9A

Triangles: Side properties

248

9B

Triangles: Angle properties

251

9C

Finding the third angle in a triangle

253

9D

Exterior angle properties of triangles

255

12C

Solving equations by inverse operation 338

9E

Triangle sums

256

12D

Solving two- and three-step equations 340

9F

Quadrilaterals

258

12E

Inequations

342

9G

Angle sum of a quadrilateral

260

12F

Exploring the coordinate plane

345

9H

Polygons

263

12G

Just CAS

346

9I

Angle sum of polygons

265

Puzzles

348

9J

Exploring polygon constructions

268

Applications

350

9K

Exploring geometric designs

270

Enrichment

352

Puzzles

272

Revision

354

Applications

274

Enrichment

276

Revision

278

Chapter 10 Location

Chapter 13 Probability

356

13A

The language of chance

358

13B

Theoretical probability

360

280

13C

Exploring simple experiments

362

10A

Directions in two dimensions

282

13D

Exploring spinners

364

10B

The coordinate number plane

284

13E

Exploring games of chance

365

10C

Handy reflections

288

13F

Exploring card games

366

10D

Scale diagrams and maps

292

13G

Spinners

368

10E

Maps and bearings

294

13H

Spinner simulations

370

298

13I

Using statistics to find probabilities

372

Applications

300

Puzzles

374

Enrichment

302

Applications

376

Revision

304

Enrichment

378

Revision

380

Puzzles

Chapter 11 Algebra Symbols 11A

Writing expressions

308

Chapter 14 Statistics

11B

Pronumerals

310

14A

Collecting data

11C

Multiplying and dividing pronumerals

312

14B

Creating and interpreting tables

388

11D

The distributive law: Expanding brackets

14C

Column and bar graphs

390

314

14D

Interpreting line graphs

394

11E

Substituting into expressions

316

14E

Pie graphs

396

11F

Calculator subs

318

14F

Dot plots and the mode

398

11G

Exploring dot patterns

320

14G

The mean

401

11H

Exploring match patterns

321

14H

The median and the range

403

11I

Rules and formulas

322

14I

Stem-and-leaf plots

405

Puzzles

324

14J

Venn diagrams

406

Applications

326

14K

Two-way tables

409

Enrichment

328

14L

Bad spell of wheather

Revision

330

Puzzles

414

Applications

416

Enrichment

418

Revision Questions

420

Chapter 12 Equations and Inequations

306

332

12A

Solving equations by inspection

334

12B

Solving equations with flow charts

336

Answers

382 384

411

424

3

The Maths Dimensions package The Maths Dimensions series has been written to cover the learning statements and standards for the Victorian Essential Learning Standards (VELS) at level 5 in the discipline of Mathematics.

Maths Dimensions 7 essential learning

Score:

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76 33 6

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Maths Dimensions Coursebook

1

includes Student CD A full-colour student coursebook with free interactive Student CD containing: •

electronic copy of the coursebook with printable pages

•

Companion Website on CD—no Internet access required

•

live link to the Companion Website

•

free software downloads such as TIConnectVI.6™ and Cabri®Jr.

4

Maths Dimensions Homework Book Each homework book provides a structured homework program for students which: •

links directly to the coursebook

•

includes technology-free and technologyactive questions

•

includes multiple-choice and extendedanswer questions

•

provides topic-based and cumulative revision.

Maths Dimensions 7 essential learning

Teacher’s Edition CD CD should launch automatically. If not, double click on ‘Setup’. Consult the ‘ReadMe’ file on this CD for further information.

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Maths Dimensions 7 essential learning

Student CD CD should launch automatically. If not, double click on ‘Setup’. Consult the ‘ReadMe’ file on this CD for further information.

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Maths Dimensions Teacher’s Edition includes Student CD and Teacher’s Edition CD Each teacher’s edition has reduced pages of the coursebook with ‘wrap-around’ teacher’s notes, and includes: •

additional examples and worked solutions

•

teaching notes for the Technology Activities in the coursebook, including graphics calculators (TI-84 family) and CAS technology (TI-89)

•

information on the technology in the Companion Website

•

VELS activities for Interdisciplinary Learning and Physical, Personal and Social Learning

•

Teacher’s Edition CD including fully worked solutions to coursebook exercises, revision sheets and chapter tests, Foundation and Challenge worksheets, correlation grids and teaching plans, answers to the Homework Book and worksheets

•

Student CD with an electronic copy of the coursebook and an archived copy of the Companion Website.

For more information on the Maths Dimensions series, visit www.pearsoned.com.au/schools

Maths Dimensions Companion Website www.pearsoned.com.au/schools The Companion Website contains a wealth of support material for teachers and students and has been written to enhance the coursebook content. It includes: •

Review Questions such as auto-correcting multiple-choice, pattern matching and fill-inthe-gap questions for each chapter

•

Technology Applications including activities using Excel spreadsheets and The Geometer’s Sketchpad to assist students to develop and apply understanding of concepts and skills

•

Animations for students to explore concepts in a creative and visually stimulating way

•

Drag and Drop Interactives that can be used to revise basic skills and key terms used in each chapter

•

Teacher’s Resource Centre which is password protected and includes revision sheets, chapter tests, correlation grids, teaching plans and lots of other teacher’s resource material.

5

How to use this book The Maths Dimensions series has been written to cover the learning statements and standards for the Victorian Essential Learning Standards (VELS) at level 5 in the discipline of Mathematics.

Chapter opening pages Chapter opening pages include a contemporary or historical context for the content and provide students with a list of the skills covered in the chapter.

These features are found at the end of each chapter: Puzzles Puzzles are included for extra skill practice.

Theory and exercise sections Theory and exercise sections contain explanations, examples and exercises designed to develop understanding of concepts and provide opportunities for students to practise new skills. Technologyfree questions are included for students to strengthen 'by hand' skills.

Applications Applications investigate and apply mathematical ideas in a creative way and provide activities for a range of student abilities.

Enrichment Explorations Explorations are scattered throughout the chapters for students to work independently on nonstandard problems and construct their own understanding.

6

Enrichments contain challenging tasks for students to apply and extend their understanding of concepts.

Revision Questions Revision Questions provide opportunities for students to consolidate understanding of basic skills.

Icons used in the coursebook Technology Activities Technology Activities include interesting tasks for students using both graphics calculator and CAS technology. •

Graphics calculator examples use TI-84 family outputs.

These icons are used throughout the coursebook to link together the different components of the series: Companion Website These icons indicate where Companion Website resource activities can be used to enhance the teaching and learning of mathematical concepts. Activities include: •

review questions

•

technology applications

•

animations

•

drag and drop interactives.

Teacher’s Edition

•

CAS technology examples use TI-89 outputs.

•

Free software downloads such as TIConnectV1.6™ and Cabri®Jr are available from the Maths Dimensions Student CD and Companion Website.

Use the live Companion Website or the Companion Website on the CD at the back of this book to access the software downloads from the drop-down menu.

These icons indicate where Teacher’s Edition resource material can be used to assist in the development and assessment of mathematical understanding. Material includes: •

revision sheets

•

chapter tests

•

Challenge worksheets

•

Foundation worksheets

Homework Book These icons indicate where a homework task from the Homework Book can be set for students. Puzzle These icons identify skills that are covered in the Puzzle section at the end of the chapter.

7

CHAPTER

1 Whole Number Revision

Whole Number Revision From the beginning of human existence, people have needed numbers to count and record information. Early civilisations such as the Egyptians, Romans, Indians, Greeks and Arabians developed their own number systems and methods of calculation. The Hindus developed today’s decimal system in 600–900 AD. Every whole number can be written by using only ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

This chapter covers the following skills: • Understanding place value • Adding, subtracting, multiplying and dividing whole numbers • Doubling and halving mentally • Estimating values • Converting numbers from different counting systems

• Simplifying using order of operations B Work out the calculations inside the brackets first. If there is more than one operation inside the brackets, then they must also follow the rules of BODMAS O If the question contains fractions of or powers of, then these are calculated next D Work out the division and M multiplication calculations, working across the page from left to right A Work out the addition and S subtraction calculations, working across the page from left to right

1A

Place value The Hindu–Arabic or whole number system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit represents a different value depending on its position in the number. The numbers 3482 and 2 497 501 are shown in place value columns below. Millions

Hundred thousands

2

4

Ten thousands

Thousands

Hundreds

Tens

Units

3

4

8

2

7

5

0

1

9

In numbers 10 000 and greater, a space is left after every three digits to the left of the last digit, to make the number easier to read.

Example

Solution

1 Write the number twelve thousand, seven hundred and four using digits.

12 704

2 Write 3 002 195 in words.

3 002 195 is three million, two thousand, one hundred and ninety-five

Exercise 1A 1 Write the following numbers in figures: a twenty-four b six hundred and twelve c two hundred and four d six thousand, four hundred and fifty-eight e thirty-seven thousand f five million, six thousand and one g six hundred thousand h forty million and sixty-seven i two hundred and fifteen thousand, and ninety-nine j one hundred and forty thousand, five hundred and thirty k eight hundred and nine million, eighty-seven thousand, nine hundred and twenty 2 Write each of these numbers in words: a 48 b 291 e 2063 f 9 745 100 i 224 000 000 j 872 004

c 125 909 g 67 405 k 4 000 016

3 What is the place value of the 3 in each of the following numbers? a 13 b 23 002 c 3 090 765 e 65 834 f 2 435 769 g 24 530 000 i 308 457 888 j 4538 k 1 306 501

10

Maths Dimensions 7

d 3490 h 9003 l 213 500 d 356 h 557 213 l 10 301

1A 4 What is the value of the 3 in each of the following numbers? a 123 b 34 c 3054 e 309 642 f 452 135 222 g 23 986 457

d 329 h 3 566 421

5 Write each of these sums as a single number: a 30 000 + 2000 + 300 + 40 + 3 b 400 000 + 20 000 + 500 + 20 c 900 000 + 40 000 + 300 + 10 + 1 d 45 000 + 400 + 80 + 1 e 3 000 000 + 40 000 + 300 f 12 000 + 500 + 70 + 8 6 Write the numbers formed from the following: a 8 hundreds and 6 tens b 3 ten thousands, 5 tens and 6 units c 4 thousands, 8 hundreds and 3 tens d 80 million, 5 ten thousands and 3 tens e 1 ten thousand, 6 hundreds, 8 tens and 7 units f 5 millions, 4 hundred thousands, 6 thousands and 7 units 7 Write the largest number possible in each case by using each digit once: a 1, 4 and 8 b 3, 5, 9 and 8 c 6 and 1 d 2, 4, 9, 6 and 1 e 4, 6, 7, 1, 9 and 0 f 9, 8, 7, 4, 3, 2, 1 and 0 8 Write the smallest number possible in each case by using each digit once: a 5 and 8 b 5, 7, 6, 3, 1 and 0 c 5, 9, 3 and 6 d 2, 4, 3, 9, 7, 0 and 5 e 4, 8 and 7 f 6, 7, 4, 3 and 2 9 Write the following lists of numbers in order from smallest to largest: a 123, 321, 145, 16, 35, 245, 12 b 306, 316, 360, 366, 603, 660, 303 c 4007, 4070, 4707, 4770, 4777 d 55 976, 55 980, 55 809, 55 796, 55 789 e 345, 467, 34, 56, 721, 3, 5005 f 12 405, 52 401, 42 510, 20 451 10 Write down the total amount of money in each of these photographs: a

b

c

d

Chapter 1 Whole Number Revision

11

1B

Addition Addition is used to combine two or more whole numbers to find their sum or total value. When adding numbers, be sure to keep the digits in their correct place value position.

Example

Solution

Find the sum of each of the following:

a 235 + 42

235 + 42 277 11

b 358 + 469

358 + 469 827

Exercise 1B 1 Find the sum of each of the following: a 123 b 457 + 34 + 681

c +

2987 853

d

45 + 347

e

235 89 + 1246

f

89 234 + 9359

g

3 456 12 438 + 931

h

5 34 + 6781

i

254 976 + 3591

j

5781 43 + 9

k

532 5681 + 2855

l

90 7852 + 1666

2 Place the following addition questions in columns and then work out the answers: a What is the sum of 2265, 8642 and 2451? b Add 45 671 and 89 065. c What do you get when you add 45 651 and 3459? d What is 492 plus 571 plus 3490? e Find the result of 3561 + 56 + 8900. f What is the answer to 346 plus 45 698 plus 445 621? 3 Sarah went to the shopping centre to buy some new clothes. She bought a pair of shoes for $45, jeans for $82, two T-shirts for $16 each and a windcheater for $28. How much did she spend altogether? 4 Matthew needed new skateboard gear. He bought a new skateboard for $436, a helmet for $134, and some elbow and knee guards for $75. How much did Matthew spend? 5 Mr Young goes to a cattle sale to buy some cows. He buys five cows which cost him $785, $862, $804, $930 and $912. How much does Mr Young spend on his purchases?

12

Maths Dimensions 7

1B 6 A Toyota sales person makes five new car sales in one week. He sells a Land Cruiser for $45 955, a Camry for $27 350, an Echo for $16 975, a Corolla for $20 890 and an Avalon for $35 420. What was the total of his sales for the week? 7 Find the shortest distance between Melbourne and Sydney by using the map opposite.

Bathurst 214 Sydney

8 Three cousins live in different 487 193 731 suburbs of Melbourne. They decide to meet in Melbourne 374 Goulburn and see a film at the cinema. Albury 203 Refer to the map below to answer the following questions: Seymour a How far does Alice have 103 to travel if she lives in 1037 Melbourne Ringwood? b What distance does Bonnie cover to get to Melbourne if she lives in Keilor? c How far from Melbourne is Celeste’s house in Clayton? d Which cousin travels the least distance? Keilor e How far in total do the girls travel to get to Melbourne? 20 km f Bonnie is having a party. How far do Alice and Celeste have to travel to get to Bonnie’s house? Melbourne 9 A bicycle company has launched its Mountaineer bicycle into New South Wales, Queensland and Western Australia. The company wishes to analyse its sales for October, November and December in each of these three regions. The sales for each region over the three months are listed in the table below.

Ringwood 26 km 22 km Clayton

Region

October

November

December

New South Wales

4235

6231

2451

Queensland

6854

2585

3567

Western Australia

2400

3476

3296

Use the table to answer the following questions: a What are the total sales for each region over the three-month period? b Which region had the highest sales? c What are the total sales for each month? d Which month produced the highest sales? e What are the total sales for the company over the three-month period?

Chapter 1 Whole Number Revision

13

1C

Subtraction Subtraction is used to find the difference between two numbers where one quantity is taken away from another quantity. The order in which numbers are subtracted is important here: the larger number must be placed first to find a positive difference. When subtracting numbers be sure to keep the digits in their correct place value position.

Example

Solution

Find the answer to each of the following subtraction questions:

a

489 − 213

489 − 213 276

b

4 11 14

5248 − 1362

5248 − 1362 3886

Exercise 1C 1 Find the answer to each of the following subtractions: a 875 b 829 c 921 − 51 − 517 − 810

d

86 − 42

e

235 − 89

f

5632 − 689

g

7632 − 1999

h

3412 − 621

i

967 − 383

j

5288 − 3598

k

8561 − 985

l

8572 − 5895

2 Place the following subtractions in columns and then work out the answers: a What is 8642 minus 3851? b Subtract 54 739 from 98 642. c What is the difference between 5651 and 999? d Take away 876 from 30 500. e Find the result of 56 000 − 32 879. f What number is 659 less than 34 232? 3 Emily spends a day shopping and buys clothes to the value of $438. If she started the day with $750 in her purse and hasn’t bought anything else, how much money does she have left? 4 The Jones family buys a house and land package valued at $225 000. If they have $23 750 to put towards the package, how much more money must they borrow from the bank?

14

Maths Dimensions 7

1C 5 Adam would like to know how many more days are left in the year. If it is not a leap year and it is the 10th of July today, find the answer for Adam. Use the rhyme ‘30 days have September, April, June and November. All the rest have 31, except February, which has 28 days and 29 days in a leap year’. 6 A silo, which is a container for storing grain, can hold 20 000 kilograms of wheat. If it currently has 8745 kilograms in it, how much more wheat can be added to the silo? 7 The Matterhorn and Jungfrau are two very popular tourist mountains in Switzerland. If the Matterhorn is 4477 metres high and the Jungfrau is 4158 metres high, how much higher is the Matterhorn than the Jungfrau? 8 The distance between Melbourne and Sydney via the Hume Highway is 873 kilometres. The map indicates the distances between some major towns along the Hume Highway. Use this map to answer the following questions: a How far is it from Euroa to Sydney? b How far is it from Melbourne to Albury? c Calculate the distance from Albury to Sydney. d Find the distance between Holbrook and Yass. e How many kilometres is it between Melbourne and Goulburn?

Sydney 193 km Yass

Goulburn 87 km

Holbrook 68 km Albury 75 km Wangaratta 87 km Euroa 148 km Melbourne

9 During a drought many water storage reservoirs become very low. a Lake Eppalock can hold 312 000 ML of water but currently only has 23 880 ML. Calculate how many megalitres are required to fill the lake. b Lake Nillahcootie has 36 981 ML currently. How many megalitres are required to reach its capcity of 40 000 ML? c The Hume Weir holds 3 038 000 ML when full. How many ML are needed to take its drought level of 928 330 ML up to capacity? d How many ML are needed to fill Waranga Basin from its current level of 182 738 ML to capacity of 411 000 ML? e Glenmaggie has 129 048 ML currently. How many megalitres are required to reach its capacity of 190 230 ML? f The Thomson dam can hold 1 068 000 ML of water but currently only has 496 382 ML. Calculate how many megalitres are required to fill the reservoir. Chapter 1 Whole Number Revision

15

1D

Multiplication Multiplication is the short way of adding up numbers in a pattern. DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

Three groups of five coins = 5 + 5 + 5 = 3 × 5 = 15 coins When multiplying numbers, or finding the product, write the smaller number under the larger number.

Example

Solution

Find the answer to each of the following multiplications: 15

a 428 × 7

428 × 7 2996 12 24

b 248 × 36

248 × 36 1488 7440 8928

Exercise 1D 1 How many coins are shown in each of the following? a b DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

2 Find the answer to each of the following: a 142 b 95 × 4 × 9 e i

16

218 × 47

f

424 × 99

j

Maths Dimensions 7

c

DOLLAR

c

310 × 53

g

309 × 102

k

208 × 8

d

1267 × 5

1219 67

h

×

2397 × 123

4578 89

l

×

3905 × 394

1D 3 Place the following multiplication questions in columns and then work out the answers: a What is 625 times 92? b What are 6 lots of 238? c What is the answer when 24 is multiplied by 287? d What is the product of 410 and 162? e What are 47 groups of 742? 4 Michael filled his car with 55 litres of unleaded fuel which cost 120 cents per litre. How much did it cost Michael to fill his car? 5 Sasha earns $68 each week working part time at a fast food outlet. How much does she earn for the year if she works for 45 weeks? 6 For the start of the winter term, Mia’s parents purchase three shirts for $44 each, a pair of pants for $86 and three windcheaters for $35 each. How much did Mia’s clothes cost? 7 A farmer sells 565 prime lambs for $45 each. How much did the farmer receive in total for the sale of his lambs? 8 The ferry between Victoria and Tasmania leaves from Port Melbourne and takes 14 hours to travel across Bass Strait to Devonport. a If the ferry makes 326 crossings per year, for how many hours is the ferry at sea? b If the ferry can carry 1300 people at an average cost of $230 per adult, how much money can be collected in fares? c The ferry can transport 350 cars at $40 each. What is the income from carrying cars? d The ferry can transport 35 semitrailers at $310 each. What is the income from carrying semitrailers? e What is the total income per trip from carrying 1300 adult passengers, 350 cars and 35 semitrailers? 9 The winner of the mountain bike endurance competition takes 8 minutes to complete a circuit. How long does it take him to complete the competition which involves 24 laps of the course? 10 Shen does a newspaper round each morning at 6 am on his bike. He travels 3 kilometres each morning, delivering 94 newspapers. At the end of the week Shen is paid $42 for his 7 delivery days. a How far does Shen travel in 1 week? b How far does he travel in 10 weeks? c How many newspapers does he deliver in 1 week? d How many newspapers are delivered by Shen in a year (52 weeks)? e How much money would Shen earn in 8 weeks? f How much money would Shen earn in a year?

Chapter 1 Whole Number Revision

17

1E

Division Division is used to find out how one number can be shared equally between groups. If 12 coins are shared or divided equally between four students, how many coins will each student receive? There are 4 groups of 3 in 12 and 12 ÷ 4 = 3, so each student receives three coins. Division is the opposite operation of multiplication. 56 ÷ 8 = 7

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

DOLLAR

Thus 7 groups of 8 = 56.

Or

8 + 8 + 8 + 8 + 8 + 8 + 8 = 56

Or

7 × 8 = 56

Example

Solution

Find the answer to each of the following division questions:

a 427 ÷ 7

061 7)427

b 22 368 ÷ 6

03 728 414 6)22 368

Exercise 1E 1 Find the answer to each of the following: a 2)4286 b 6)1266 d 4)1732 e 9)1872 g 7)46 123 h 12)6120 j 3)1686 k 5)4570

c f i l

8)2480 5)265 11)2937 7)6874

2 Set the following questions out correctly and then work out the answers: a 645 ÷ 3 b 1468 ÷ 4 c 3005 ÷ 5 d 594 ÷ 6 e 2037 ÷ 7 f 10 160 ÷ 8 g 8991 ÷ 9 h 4800 ÷ 10 i 29 161 ÷ 11 j 74 964 ÷ 12 k 4064 ÷ 8 l 144 291 ÷ 7 3 The answer to a division sum is also called a quotient. Set the following division questions out correctly before working out the quotients: a What is the quotient of 4575 ÷ 3? b What is 19 704 divided by 6? c How many times does 11 go into 6644? d How many groups of 8 are in 544? 4 Maria works in a department store and earns $480 per week. How much does she earn per day if she works from Monday to Friday? 5 John counts 47 520 legs in his flock of sheep, how many sheep does John have?

18

Maths Dimensions 7

1E 6 A hiker walked 192 kilometres over 12 days. How far did she walk per day? 7 John has 11 880 sheep which need to be shorn by his six shearers.

a If all the shearers shear at the same rate, how many sheep does each shearer shear? b If shearing lasts 9 days, how many sheep does each shearer shear in a day? c If the sheep are grazed in one of 20 flocks or groups, how many sheep are in a flock? 8 A piggery produced 14 166 piglets in a year. If each sow (mother pig) produces one litter of nine piglets on average, how many sows does the piggery have? 9 A car dealership sells 288 new cars per year. a How many cars are sold on average per month? b If a month consists of 4 weeks, how many cars are sold per week? 10 The Chan family travelled 1920 kilometres from Melbourne to holiday at Noosa. a How many kilometres per day did they travel, if it took them 4 days to arrive at Noosa? b If they travelled at an average speed of 80 kilometres per hour, how many hours of driving did they do each day? 11 A factory produces 1 440 000 bottles of soft drink per 5-day week. a If the factory has 10 identical machines, how many bottles of soft drink does each machine produce per week? b How many bottles of soft drink are produced by each machine in a day? c If the machines run for 8 hours each day, how many bottles of soft drink are produced by a machine each minute? d If each machine bottles 120 litres of soft drink in 1 minute, how many litres are in each bottle? 12 The Findlay family of four spent $8124·48 for the year on groceries. Calculate the following: a How much did they spend per person, in dollars and cents? b How much did they spend on groceries each month on average? c How much did they spend on groceries each week on average? 13 The Chesworths spent $2029·50 on 30 tonne of fertiliser for their farm. How much did the fertiliser cost per tonne?

Chapter 1 Whole Number Revision

19

1F

Doubling and halving Estimation gives us an approximate answer to a question, but there are other techniques which can give us the exact answer without doing detailed working out. When two different numbers are multiplied by the same number, the calculation can be simplified.

Example 1 Simplify the following to find the answers: a 36 × 8 + 44 × 8

b 23 × 5 − 18 × 5

Solution 36 × 8 + 44 × 8 = 36 lots of 8 plus 44 lots of 8 = (36 + 44) × 8 = 80 × 8 = 640 23 × 5 − 18 × 5 = (23 − 18) × 5 =5×5 = 25

In multiplication the order can be rearranged: 2 × 3 × 4 = 4 × 2 × 3. This can be used to simplify multiplication questions.

Example 2 Simplify the following to find the answers: a 20 × 18 × 5

b 61 × 30

Solution 20 × 18 × 5 = 20 × 5 × 18 = 100 × 18 = 1800 61 × 30 = 61 × 3 × 10 = 183 × 10 = 1830

The doubling and halving technique can also be used to make questions simpler without altering the value of the question.

Example

Solution

3 Use the doubling and halving technique to find the answers:

20

a 36 × 50

36 × 50 = 36 × 100 ÷ 2 = 3600 ÷ 2 = 1800

b 88 × 500

88 × 500 = 88 × 1000 ÷ 2 = 88 000 ÷ 2 = 44 000

Maths Dimensions 7

1F Exercise 1F 1 Rearrange the numbers to make it easier to find the answers: a 20 × 4 × 5 b 25 × 9 × 4 d 37 × 50 × 2 e 10 × 10 × 321 g 50 × 66 × 2 h 5 × 19 × 20 j 2 × 33 × 50 k 20 × 5 × 71

c f i l

10 × 56 × 10 34 × 4 × 25 99 × 10 × 10 4 × 15 × 25

2 Find an easy way to evaluate the following: a 8×3+2×3 b 8 × 55 + 2 × 55 d 5×6+7×6 e 19 × 12 + 31 × 12 g 14 × 8 − 6 × 8 h 34 × 21 − 24 × 21 j 98 × 7 − 18 × 7 k 63 × 12 − 13 × 12

c f i l

15 × 14 + 5 × 14 18 × 14 + 2 × 14 65 × 3 − 3 × 15 85 × 11 − 11 × 15

3 Evaluate mentally: a 56 × 30 e 200 × 34 i 19 × 30

b 21 × 20 f 500 × 12 j 50 × 23

c 80 × 14 g 20 × 45 k 90 × 18

4 Use the doubling and halving method to find the answers: a 28 × 50 b 50 × 91 d 52 × 500 e 5 × 387

d 60 × 15 h 85 × 40 l 66 × 20 c 36 × 500 f 46 × 5

5 Mario buys 28 computer disks at $4 each, while Eliza buys 32 CDs at $4 each. How much do they spend in total? 6 During the week of the swimming trials, Malcolm buys 5 pies at $1·40 each while Scott buys 5 pizza slices for $1·60 each. How much do the boys spend in total at the canteen for the week? 7 At their school swimming carnival, Kelvin and Glen competed in the 400-metre freestyle event, which is 8 laps of the pool. Glen took 39 seconds per lap, while Kelvin took 41 seconds. What was the boys’ combined time in the pool?

Chapter 1 Whole Number Revision

21

1G

Order of operations When a question contains more than one operation we need a system or set of rules to follow. BODMAS represents the order of operations that allows us to calculate questions correctly.

•••••••••••••

B Work out the calculations inside the brackets first. If there is more than one operation inside the brackets, then they must also follow the rules of BODMAS. O If the question contains fractions of or powers of, then these are calculated next. D Work out the division and multiplication calculations, working across the page M from left to right. A Work out the addition and subtraction calculations, working across the page S from left to right.

Example

Solution

Use BODMAS to evaluate each of the following:

a 2 + (3 × 6) − 5

2 + (3 × 6) − 5 = 2 + 18 − 5 = 20 − 5 = 15

b 5 × 4 + (18 ÷ 3) × 2

5 × 4 + (18 ÷ 3) × 2 =5×4+6×2 = 20 + 12 = 32

c 33 ÷ 3 + 4 × 9 − 6

33 ÷ 3 + 4 × 9 − 6 = 11 + 36 − 6 = 47 − 6 = 41

d

1 --2

of 12 × (3 + 1)

of 12 × (3 + 1) =6×4 = 24 1 --2

Exercise 1G 1 Evaluate each of the following: a 14 ÷ 2 + 3 × 4 b d 5 × ( 4 − 3) + 9 e g 3 × 15 − 6 × 5 h j 55 − 3 × 9 k m 5 + 24 − 13 n p 6×2÷3 q s 3 × 3 × 3 − (4 × 5) t v 2×9+7 w

22

Maths Dimensions 7

60 ÷ 3 ÷ 4 + 2 6 × ( 2 + 7) − 2 × 7 18 ÷ 2 × 10 − (7 × 8) (34 + 6) × 6 (144 ÷ 6) − 3 × 7 ( 9 − 4) × (6 + 3) ÷ 15 (1 + 2 + 3) × 10 + 40 16 ÷ 4 + 5

c f i l o r u x

(16 + 4) ÷ 5 × 3 5+7−3×2 6×9+4 8 + 4 × (26 − 15) (15 + 6) ÷ 7 × 2 (8 + 7) ÷ 5 × 3 − 1 (6 + 2 × 8) ÷ 2 + 32 16 − 3 × 2

1G 2 Evaluate the following: a

1 --2

of 16 × 5 – 2 × (10 – 7)

d 13 + 1--- of 18 + 4

1 --2

−2×6

b 6 + 34 ÷ 17 − 66 ÷ 11

c 9 − 8 × 7 ÷ 14 + 10

e 11 − 64 ÷ 8 + 5 × 11

f

(4 − 24 ÷ 8) × 55 − 9 × 5

3 Insert brackets where necessary to make the following equations true: a 6 + 8 − 1 = 13 b 54 ÷ 6 + 3 + 2 = 8 c 6 + 34 ÷ 2 + 3 ÷ 2 = 13 d 42 ÷ 5 + 2 × 5 = 30 e 15 + 4 × 5 − 5 = 15 f 81 ÷ 9 ÷ 2 + 1 = 3 g 12 − 2 × 5 = 50 h 120 ÷ 10 + 2 + 5 = 15 i 8 + 4 − 3 × 2 = 10 j 45 ÷ 3 × 2 + 16 = 46 k 18 ÷ 9 × 5 + 7 = 24 l 4 + 3 × 3 + 4 = 25 m 4 − 2 × 100 ÷ 25 = 8 n 100 ÷ 25 × 5 + 6 = 26 o 48 ÷ 16 + 5 × 2 = 16 4 Insert the symbols ×, ÷, +, − to make the following equations true: a (2 8) 5 = 2 b (14 5) 9 = 81 c 6 (8 3) = 30 d (21 5) 2 = 13 e 12 8 5 = 9 f 40 (14 7) = 38 g (10 8) (5 4) = 18 h (34 12) (12 4) = 66 i (36 9) (10 4) = 10 j (64 16) (16 8) = 8 k 20 100 25 = 24 l (9 3) (8 3) = 60 5 True or false? a 16 ÷ (3 + 1) × 3 + 6 = 36 c 5 × 10 + 3 × 8 ÷ 4 = 56 e (16 + 3 × 2) ÷ 11 − 1 = 1 g 15 × 3 ÷ (6 + 3) + 1 = 6 i 100 ÷ 25 × (6 + 2) = 26

b d f h j

(2 + 3 × 5) − (3 + 16 ÷ 4) = 18 44 ÷ 4 + 7 × 2 − 3 = 5 (48 + 2) ÷ 5 − 18 ÷ 3 = 4 (14 + 5 − 1) ÷ (2 × 5 − 1) = 3 64 ÷ 16 + 16 − 8 = 12

6 For each pair of equations below, write down which is the correct equation, i or ii: a i (2 + 4) × 3 = 14 ii 2 + 4 × 3 = 14 b i 3 + 36 × 2 = 75 ii 3 + 36 × 2 = 78 c i 4 × 8 + 2 = 16 ii 4 × 8 + 2 = 34 d i 16 ÷ 4 + 4 = 8 ii 16 ÷ (4 + 4) = 8 e i (13 + 8) ÷ (3 + 4) = 3 ii 13 + 8 ÷ 3 + 4 = 3 f i 81 ÷ 9 + 1 = 9 ii 81 ÷ 9 + 1 = 10

•••••••••••••••••••••

BODMAS The operations are completed in the order in which you go up these stairs. ■

Brackets first.

■

Fractions or powers of are on the second step, so calculate these next.

■

Multiplication and division are on the third step.

■

Addition and subtraction are on the top step. Calculate these last.

Remember that for division and multiplication, and for addition and subtraction, which are on the same steps, work across the page from left to right. Chapter 1 Whole Number Revision

23

1H

Estimation In many situations we do not need to calculate the exact answer to a question and we can do an approximate calculation in our head. This is a very useful technique to check calculations and ensure that we have not made any careless mistakes. To make estimates of an approximate answer we can round the numbers to the nearest multiple of 10 or 100, so each number in the approximation will have a non-zero digit at the start followed by zeroes. The symbol ≈ is used to represent ‘approximately equal’.

Example

Solution

Estimate the following:

a If there are 39 houses in a street and every household donates $5 in a doorknock appeal, about how much will be collected, $20, $200, $2000? b

i 23 × 506

ii 645 + 398

39 is approximately equal to 40 39 ≈ 40 39 × 5 ≈ 40 × 5 = 200 So approximately $200 will be collected.

i 23 × 506 ≈ 20 × 500 = 10 000

ii 645 + 398 ≈ 600 + 400 = 1000

Exercise 1H 1 Round each number to the nearest multiple of 10, 100 or 1000 and then estimate the answers to the following: a 54 + 234 b 1249 + 90 c 651 + 16 + 270 d 1239 + 854 + 45 e 789 − 88 f 1467 − 674 g 358 − 104 − 55 h 2478 − 865 i 45 × 89 j 478 × 12 k 49 × 82 × 105 l 372 × 34 × 2 m 83 ÷ 12 n 57 ÷ 6 o 804 ÷ 37 2 Michelle wishes to purchase the following items when shopping: jeans $84, shoes $156, jumper $42 and lunch $11. Estimate how much money she needs to have in her wallet to cover the cost of her purchases. 3 By air, Perth is 2120 kilometres from Adelaide, 2871 kilometres from Melbourne, 3278 kilometres from Sydney, 3091 kilometres from Canberra and 2656 kilometres from Darwin. Approximate each of these distances and then estimate the sum of these distances. 4 a Estimate how many runners are in this photograph.

24

Maths Dimensions 7

b Estimate how many paper clips are in this photograph.

Four by four 1I Introduction The aim of this activity is to use order of operations and other mathematical operations on the calculator repeatedly to solve a ‘series’ of problems. Most computing devices are programmed to perform calculations in a specific order. For example, 5 + 2 × 3 = 5 + (2 × 3). Multiplication is done before addition, so the brackets are not necessary in this case. This activity is designed to improve your awareness of the order of operations.

Equipment TI-83 Plus There are multiple answers to each question. For example, for zero you could use 4 + 4 – 4 – 4 or 44 – 44 or 4 ÷ 4 – 4 ÷ 4 and so on.

Technology activity 1I The order in which mathematical operations are carried out is important. For example: 4 ÷ 4 + 4 × 4 = 17 4+4×4÷4=8 In both cases the division is carried out first: 4 ÷ 4 = 1 4 ÷ 4 + 4 × 4 = 17 therefore becomes 1 + 4 × 4 = 17 4 + 4 × 4 ÷ 4 = 8 therefore becomes 4 + 4 × 1 = 8 The next operation is multiplication: 1 + 4 × 4 = 17 becomes 1 + 16 = 17 4 + 4 × 1 = 8 becomes 4 + 4 = 8 The two answers become clear. The four 4s have been combined using mathematical operations but the two answers are different. What is the result of 4 + 4 ÷ 4 + 4? Brackets can also be used to change the result: (4 + 4) ÷ 4 + 4 Can you use four 4s to produce a result of 1?

Chapter 1 Whole Number Revision

25

1I Problem Can you use four 4s to produce all the numbers between 0 and 100? The basic operations of +, –, ÷, ×, ( ) can be used; however, there are additional operations on the calculator that may help you solve this problem.

Calculator instructions Squared From the Home screen press 4

x2

ENTER .

Square root From the Home screen press 2nd

√

)

4

ENTER .

Note: √ is located on the x2 button.

Cubed From the Home screen press 4

MATH

ENTER .

3

Factorial From the Home screen press 4 and select option 4.

MATH

Press ENTER to execute the calculation. Permutations From the Home screen enter (4 + 4) then press MATH

, select option 3 then 4

ENTER .

Combining operations It is possible to combine operations to produce results numbers that only use one 4. From the Home screen press 2nd

√

4

)

MATH ,

select option 3, ENTER .

Use the last two sets of operations to produce 70 using only two 4s.

26

Maths Dimensions 7

1I Rules ■

You are only allowed to use the number 4; however, the calculator has built-in operations that use a 2 in the case of x2 and a 3 in the case of x3. These two operations are allowable as they are built into the calculator. For example, (42 + 43) ÷ 4 is acceptable but (45 + 4) ÷ 4 is not, because you would need to press 5 .

■

You must use all four 4s for a solution to be valid. It is therefore not always desirable to use operations such as x2. If you arrive at a solution using only three 4s you may be able to correct the solution by replacing 42 with 4 × 4.

Starting the problem Work in teams of four to produce as many different results as possible. Use a list of the first 100 integers, and 0, i.e. {0, 1, 2, 3 … 100}. Each time you obtain a new result, write down the equation next to the number and write your name next to it.

Scoring At the end of the lesson the teacher will collect all of your equations. A score will be determined to see which team is the winner. A point is earned for each number you discover. An additional 10 points is awarded for a sequence of 10 consecutive numbers. For example, if you have equations for all the numbers {6, 7, 8 … 15} then you score a point for each of these numbers plus a bonus 10 points.

Warning If one of your equations is incorrect you lose 5 points. It is therefore worthwhile making sure someone in your team checks your answer before you put it on your answer sheet.

Strategies There is a range of strategies your team could use in order to gain the highest score: ■

The highest score is not always obtained by having the most ‘answers’ filled in. If you have a couple of incorrect answers this can cause your score to go backwards very quickly! Make sure your answers have been thoroughly checked before you enter them on your sheet.

■

Incorrect answers can also be generated by not writing your answers using the correct notation. The notation used on the calculator is not perfect. You are permitted to check with your teacher to see how some of these expressions should be written. For example, ncr(4, 4) is not correct mathematical notation. The correct notation for this mathematical expression is 4C4.

■

Some numbers such as 2 can be generated using just one 4, i.e. 4. Try using just three 4s to generate some numbers. For example, 4 + 4 + 4 = 12. Now include the final 4 in the form 4 to generate 10 and 14. Can you use a similar approach to find expressions for 62 and 66?

■

Working as a team to explore numbers that can be obtained using either one, two or three 4s can make finding other solutions a lot quicker.

Chapter 1 Whole Number Revision

27

1J

Number systems of the past We are most familiar with the Hindu–Arabic number system because we use it every day. However, many early civilisations developed their own number system, including the Egyptians, Romans, Indians, Greeks and Arabians. The following tables show the Hindu–Arabic system and several others.

The Hindu–Arabic number system 1

2

3

4

5

6

7

8

9

10

It is uncertain if it was the Arabs or the Indians who first introduced the zero.

The Roman number system 0

1

2

3

4

5

I

II

III

IV

V

6

7

8

9

10

VI VII VIII IX

X

15

20

XV XX

50

100

L

C

The Romans wrote their numbers from left to right, as we do; however, the Roman system does not rely on the use of place value so there was no need for a symbol to represent zero. In the Roman number system 463 is written as CCCCLXIII.

The Egyptian number system Examples of this were found on stones, wood and pottery from about 3500 BC. Numbers were written from right to left. 1

2

100

3

1000

4

10 000

5

6

100 000

7

1 000 000

8

10 000 000

9

10

e.g. 256

The Egyptian number system does not rely on place value so there is no need for a zero.

The Greek number system The Greek number system was similar to the Egyptian one, except that the symbol for 5 was often combined with the symbol for 10, 100, 1000 and 10 000.

28

1

2

3

4

5

10

5 × 10 = 50

100

1000

10 000

I

II

III

IIII

Γ

Δ

ΓΔ

H

X

M

Maths Dimensions 7

1J The Chinese–Japanese number system The traditional Chinese–Japanese numerals are demonstrated below. Unlike the other systems this one was based on multiplication and was written down the page rather than across it. 1

2

3

4

5

6

7

8

9

10

100

1000

e.g. 3276

Exercise 1J 1 Represent each of the following numbers using the ancient Hindu–Arabic number system: a 46 b 97 c 257 d 308 e 1248 f 1004 2 Represent each of the following numbers using Roman numerals: a 59 b 137 c 73 d 14 e 482 f 124 3 Rewrite each of these Roman numerals into modern Hindu–Arabic: a XXIV b CCLV c CCCXI d CCCIX e XLV f XCIX 4 Rewrite each of the following numbers using the Egyptian hieroglyphics (remember to write them right to left): a 11 b 1245 c 8 d 10 000 e 1006 f 10 205 5 Decode the following Greek numbers: a ΔΔΓΙ b ΓΙΙΙ c ΓΔΔΔΓΙΙΙΙ d ΗΗΗΔΔΓΙΙ e ΧΔΙΙ f ΓΗΓΔΓΙ 6 Rewrite each of the following numbers using the traditional Chinese–Japanese number system (remember to write them down the page): a 93 b 84 c 51 d 238 e 782 f 1258 7 Explain in a sentence or two some of the advantages of the traditional Chinese–Japanese system. 8 Explain in a sentence or two some of the disadvantages of the Roman number system.

Chapter 1 Whole Number Revision

29

Puzzles 1 Place the numbers listed into the cross-number grid. The numbers can only be written downwards or left to right.

2 digits

3 digits

14 27 29 32 43

222 224 266 291 384 418 523

52 64 73 99

534 592 685 692 713 833 836

4 digits

5 digits

1331 2851 2885 3285 4256

35 636 36 817 41 678 94 367

7533 8453 9225 9238 9634

3

6

8

1

7

6 digits 143 667 154 927 692 586 842 283

2 In each of the following diagrams, the numbers in the circles at the end of the line must add to equal the number on the line. Find the number in each of the circles. a

b 17

14

c 29

22

16 13

20

27

15

17

d

e

40

31

15

f

19

22 13

19 26

31

33

29

13

10 11

24

30

Maths Dimensions 7

Ch1 3 Find the answer to the following questions and match the letter to the correct number below to solve the riddles: a What famous event occurred in 1616? A 4000 + 700 + 1

B 2 hundreds and 3 units D One hundred and thirteen E Twenty-six G Eight million, sixty-seven thousand H 3 tens and 6 units I 5 hundreds, 4 tens and 6 units K 300 000 + 2000 + 50 + 3 _____ 113

_____ 546

_____ _____ 746 302 053

_____ 36

_____ 76

_____ 4701

_____ 2100

_____ 113

_____ 32 310

_____ 36

_____ 4701

_____ _____ 746 302 053

_____ 26

L Seventy-six N 21 hundreds O 500 + 60 + 6 R Seven hundred and forty-six S 30 000 + 2000 + 300 + 10 T 900 000 + 400 + 6 Y 4 ten thousands, 5 tens and 3 units _____ 4701

_____ 113

_____ _____ 4701 900 406 _____ 203

b What is the definition of a stick? A 246 B 342 + 14 + 123

_____ _____ _____ _____ 746 900 406 566 8067000

C

_____ 4701

_____ 40 053

134 + 865

D +

1203 92

E

268 + 90

G

518 + 79

H

527 + 63

K

981 + 77

M

978 − 152

N

685 − 243

O

867 − 155

R

1298 − 367

T

802 − 35

V

4002 − 3185

W

1146 − 957

Z

6000 − 812

_____ 260

_____ 465

_____ 712

_____ 712

_____ 826

_____ 358

_____ 931

_____ 260

_____ 767

_____ 590

_____ 260

_____ 767

_____ 189

_____ 712

_____ 442

_____ 767

_____ 999

_____ 712

_____ 826

_____ 358

_____ 465

_____ 260

_____ 999

_____ 1058

_____ 442

_____ 597

Chapter 1 Whole Number Revision

31

Applications Number systems 1 Design your own base 10 counting system using unique symbols. Copy and complete the table showing your symbols: 0

1

2

3

4

5

6

7

Use the symbols to write the following numbers: a 43 b 102 c 269 d 2310 e 12 436 Can you design a counting system that is not base 10?

f

8

5841

9

g 998

Today’s date In this activity you will be using today’s date to write some maths questions that will need your knowledge of BODMAS. If the date is 13/2/03 (13 February 2003), then you use the digits 13203 to make a true statement. The last digit in the year will always be the answer to your question. Here are some examples: 1×3−2×0=3 (2 − 1 + 0) × 3 = 3

1 Use today’s date to make up five equations. 2 Use your birthday to make up five equations.

Number puzzle Arrange the numbers from 1 to 10 in the circles provided so that the difference between any two numbers next to each other is either 1 or 2.

Magic squares In a magic square, every row, column and diagonal adds up to the same magic number. Discover the magic numbers below and then use them to complete each of the missing boxes. 27

1

90

48

20

119

2

83

167

55

143

13

32

118

Maths Dimensions 7

34

173

131

137

149 125

107

3 107

118

217

116

125 128 183

106 152

Ch1 Basketball ladder The scores for the first round of a basketball competition are shown in the table below.

a Calculate the total score for each team using the following conversions: ■ foul shot = 1 point ■ goal = 2 points ■ 3 pointer = 3 points b Determine whether each team won or lost, or if the match was a draw. c Use a calculator to find the percentage for each team using the rule: points for percentage = --------------------------------- × 100 points against d Draw up the ladder showing the teams in order of percentage. Include all details in the table. Team

Foul shot

Goal

3 pointer

Springfield Shelbyville

5 1

30 23

5 1

Melb Midgits Vic Vipers

3 5

15 17

2 1

SA Sharks NT Newts

6 2

22 15

2 3

Qld Quokkas Sydney Stars

0 4

26 28

4 1

Tas Tigers WA Wallabies

2 1

22 15

0 5

Total score

Win loss draw

Percentage

Word sums In the following addition sums each digit has been replaced by a certain letter. Find the digit corresponding to each letter, given that each letter represents a different digit.

a +

T H R E E F O U R S E V E N

b

F O U R + F I V E N I N E

Chapter 1 Whole Number Revision

33

Enrichment 1 Replace the a

with the missing number in the following addition sums:

b

2 6 + 1 8

+

4 2 9

2 Replace the a

b

2 4 − 1 8

3 Replace the

0 2 7

9 1 1

c

1 6 8 − 4 8

b

1 8

d

9 − 1 9 9 4

c

2 4 × 1

4 Replace the

5

4 2 4 3 − 1 5

3

2

2

9 2

4 2 6 × 3

d

3 × 6

1 7 0 4 1 2 7

1 1 6

8 1 0

4 8 4

8

6 1

6 2

7

with the missing number in the following division sums: 4

)

+ 4 9 5

2 1 2 + 4

with the missing number in the following multiplication sums:

1

7

9 9

1 8

1 2 8 × 4

a

d

with the missing number in the following subtraction sums:

1 1 0

a

c

6

3 1 5

Example Use long division to solve: 12 098 ÷ 23

b

c

5 11

)

7 9 7

)

4

3 6

Solution 526 23) 12098 − 115

23 into 120 goes 5 times: 5 × 23 = 115.

59 − 46

This leaves 5. Bring the 9 down to the 5 to make 59. 23 into 59 goes twice: 2 × 23 = 46.

138 −138

This leaves 13. Bring the 8 down to make 138. 138 ÷ 23 = 6

000

5 Study the long division technique shown in the example and follow the process to find the answers to the questions below. a 6939 ÷ 27 b 8917 ÷ 37 c 62 453 ÷ 19 d 138 327 ÷ 21 e 7137 ÷ 13 f 46 980 ÷ 29 g 8128 ÷ 32 h 164 568 ÷ 24 i 72 437 ÷ 17

34

Maths Dimensions 7

Ch1 Binary numbers When you began to count you probably used your fingers, as our decimal system has ten digits. A computer does not have ten fingers, it has switches that are either on or off. Computers therefore use a system with only two numbers, 0 and 1, the binary number system. The place value for the binary number system is: 20 = 1, 21 = 2, 22 = 2 × 2 = 4, 23 = 2 × 2 × 2 = 8, 24 = 2 × 2 × 2 × 2 = 16 … For example, 10 1012 = 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 16 + 0 + 4 + 0 + 1 = 21 So 10 1012 = 2110, where the subscript refers to the base.

6 Convert the following binary numbers to decimal numbers: a 101 b 1000 d 11100 e 1010101 g 111000 h 1000000000

c 1101 f 111010101 i 101101101101

To convert a decimal number to a binary number, simply divide by 2 and record the remainder of each division. For example: 49 ÷ 2 = 24 remainder 1, 24 ÷ 2 = 12 remainder 0, 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0, 3 ÷ 2 = 1 remainder 1, 1 ÷ 2 = no whole remainder 1 49 = 110 0012

7 Write the binary representation for the following decimal numbers: a 22 b 79 c 35 d 56 f 100 g 164 h 256 i 599 The four rules for adding binary numbers are simple: 0+0=0 0 + 1 = 1 and 1 + 0 = 1 1 + 1 = 10 (put down a 0 and carry a 1) 1 + 1 + 1 = 11 (put down a 1 and carry a 1)

e 64 j 1024

Example 1 101 = 1310 + 1 001 = 910 10 110 = 2210

8 Convert each number to a binary number and add using binary addition: a 2+8 b 7+9 c 3 + 15 d 8 + 26 e 16 + 64 f 128 + 256 g 31 + 46 h 8 + 99 i 15 + 32 + 101 9 Does binary multiplication work in the same way as conventional multiplication? a Calculate 9 × 4, then convert each number to a binary number and multiply to check your answer. b Calculate 14 × 6 in binary multiplication. c Multiply 1 000 101 × 10 111, convert to a decimal number and multiply to check your answer. 10 Explore number systems other than base 2 and base 10, such as base 3: 30 = 1, 31 = 3, 32 = 9 …

Chapter 1 Whole Number Revision

35

Revision Exercise 1A 1 Write the following numbers in descending order, that is going from largest to smallest: a 2348, 65, 90 876, 567, 3258, 543 b 267, 627, 726, 266, 276, 626 c 404, 40, 440, 444, 400, 4, 44 2 Write the following numbers in words: a 230 b 123 562 c 1872

d 21

e 8

3 Write the following numbers: a four thousand, six hundred and twenty b one million, two hundred and forty-nine thousand and sixteen c nineteen thousand, four hundred and sixty-five d fifty thousand, nine hundred, two tens and five units

Exercise 1B 4 a Find the answers to the following: i 2349 ii 458 409 4892 + 4812 + 9827

iii

23 10 764 + 522

iii

23 000 − 8 796

iii

6035 × 307

b Add 308, 6670 and 201. c What is the answer to 568 plus 631?

Exercise 1C 5 a Find the answers to the following: i 498 ii 12 902 − 34 − 9 734 b What is 876 minus 235? c Find the difference between 7387 and 4649.

Exercise 1D 6 a Calculate the following: i 459 ×8

ii

b What is 740 multiplied by 12? c What is 39 groups of 489 equal to?

36

Maths Dimensions 7

1275 × 26

Ch1 Exercise 1E 7 a Calculate the following: i 6)948 ii 12)16 824 b What is the answer when 1416 is divided by 3? c How many groups of 11 are in 10 835?

iii 5)22 015

Exercise 1F 8 Evaluate mentally: a 44 × 500 d 5 × 78 × 20 g 6 × 67 + 67 × 4 j 471 × 50

b e h k

34 × 7 + 16 × 7 345 × 20 4806 × 50 34 × 200

c f i l

25 × 7 × 4 50 × 95 × 2 18 × 471 − 8 × 471 432 × 20

Exercise 1G 9 Find the answers to the following questions: a 56 ÷ 8 × 2 b 15 + 5 × 6 − 4 d 5 × ( 11 − 3) ÷ 4 e (6 − 6) × 7 + 3 g (25 − 8 × 2) ÷ 3 + 10 h 44 ÷ 2 ÷ 11 + 5

c (8 + 9) × 2 + 6 f 8 + 2 × 8 ÷ 16 i 3 × 6 − 18 ÷ 9

Exercise 1H 10 The Mount Buffalo National Park in Victoria contains the peaks of Mt Buffalo 1721 metres, The Horn 1723 metres, Eagle Point 1477 metres, and The Monolith 1419 metres. Find the approximate total height of the four peaks. 11 Rosetta travelled to Europe. It cost her $2399 for her plane ticket, and $875 for her tour of France, Spain, Portugal, Italy, Austria and Switzerland. She planned separate tours of Venice for $120 and the Greek islands for $340. Meals and accommodation were included in all the tours. Rosetta also took $2500 in spending money and travellers cheques. If Rosetta spent all her money and travellers cheques, estimate how much her holiday to Europe cost.

Exercise 1J 12 Represent each of the following numbers using Roman numerals: a 19 b 127 c 63 d 34 e 401 f 1234 13 Rewrite each of these Roman numerals as modern Hindu–Arabic numbers: a XIV b CCLVI c CCXXII d CVI e XLV f CL

Chapter 1 Whole Number Revision

37

CHAPTER

2 Number Patterns

Number Patterns The first European to discover the decimal system was Leonardo Fibonacci of Pisa, who translated an Arabic manuscript into Latin in 1202. Fibonacci’s book called Liber Abbaci, which means ‘Book of Calculating’, described the decimal system and all the rules of arithmetic for addition, subtraction, multiplication and division. Fibonacci is best known for a series of numbers created by adding the last two numbers: 1, 1, 2, 3, 5, 8, 13 . . . Fibonacci numbers appear in nature and art. Examples include flowers, shells and pine cones.

This chapter covers the following skills: • Exploring and comparing number patterns • Finding factors, common factors and highest common factors of given numbers • Finding multiples, common multiples and lowest common multiples of given numbers • Expressing a number as a product of its prime factors • Revising divisibility tests • Exploring prime and composite numbers • Expressing products of factors in index form • Finding squares and square roots • Investigating odd and even numbers

2A Exploring number patterns Whole numbers produce some interesting patterns when basic operations are applied. Patterns exist all around us in the real world if we know where to look. Number patterns can be simple like these: 1

3 2

1

5 2

2 1

7 2

4 2

9…

Each number is formed by adding 2 to the previous number.

2 7

3

11 …

Each number is formed by adding 1 to the difference between the previous two numbers.

4

Adding and multiplying numbers can produce some interesting patterns such as these: 1×7+1=8 12 × 7 + 2 = 86 123 × 7 + 3 = 864 1234 × 7 + 4 = 8642

3 × 67 = 201 6 × 67 = 402 9 × 67 = 603 12 × 67 = 804

In about 1202, Leonardo Fibonacci of Pisa discovered a sequence of numbers based on the problem ‘How many pairs of rabbits can be produced from a single pair in one year if it is assumed that every month each pair produces another pair and each new pair breeds from the second month?’ 1, 1, 2, 3, 5, 8, 13 … Each number is obtained by adding the two previous numbers.

Learning task 2A 1 Write the next five numbers in each of the number patterns below: a 4, 5, 6, 7, 8, 9 … b 2, 4, 6, 8, 10 … c 1, 3, 6, 10, 15 … d 1, 2, 3, 5, 7, 10, 13 … e 3, 5, 9, 15, 23 … f 1, 2, 4, 8, 16 … g 70, 59, 49, 40, 32 … h 2, 3, 5, 9, 17 … 2 Copy and complete the following number patterns: a 1×1= b 1×5+1= 11 × 11 = 2×5+2= 111 × 111 = 3×5+3= 1111 × 1111 = 4×5+4=

40

Maths Dimensions 7

c

1×9+2= 11 × 9 + 2 = 111 × 9 + 2 = 1111 × 9 + 2 =

2A 3 Use Fibonacci’s number pattern 1, 1, 2, 3, 5, 8, 13 … a Write down the first 20 numbers in the sequence. b Find the sum of the first 3 terms and compare it to the fifth term. c Find the sum of the first 4 terms and compare it to the sixth term. d Find the sum of the first 5 terms and compare it to the seventh term. e Find the sum of the first 6 terms and compare it to the eighth term. f Find the sum of the first 18 terms. 4 Use your library or the Internet to explore the Fibonacci number patterns that exist in nature and art. 5 When dots are arranged in the shape of a rectangle, the dots are counted to form rectangular numbers. The first three terms have been drawn for you:

3

8

15

a Draw the next two terms in the pattern. b Write down the first 10 rectangular numbers. c Find the difference between consecutive rectangular numbers for the first 10 terms. Is there a pattern? Explain your answer. 6 a For each of the patterns below, count the number of dots in each shape to give you the first two numbers in the pattern. i ii

b Find the difference between the number of dots in the first two terms. Is there a pattern? Explain your answer. c Draw the next shape in each pattern. d Write down the first 10 terms for each pattern. 7 The following number patterns are all formed by using the same process. Describe the pattern and write down the next three numbers in each pattern: a 2, 5, 11, 23 … b 3, 5, 9, 17 … c 1, 9, 25, 57 … d 5, 9, 17, 33 … e 2, 7, 17, 37 … f 1, 7, 19, 43 … g 3, 10, 24, 52 … h 1, 10, 28, 64 … Chapter 2 Number Patterns

41

2B

Multiples Multiples can be found by multiplying a number by all the whole numbers.

Example

Solution

1 List the multiples of 6 less than 65. 2 List the multiples of 9 less than 65. 3 Find the common multiples of 6 and 9 which are less than 65.

The common multiples of 6 and 9 which are less than 65 are 18, 36, 54.

4 Find the lowest common multiple of 6 and 9.

The lowest common multiple of 6 and 9 is 18.

6, 12, 18, 24, 30, 36, 42, 48, 54, 60 9, 18, 27, 36, 45, 54, 63

Exercise 2B 1 For each number below, list all multiples which are less than 100: a 6 b 7 c 8 d 9

e 10

2 For each number below, list all multiples which are less than 50: a 2 b 3 c 4 d 5 3 List the multiples of 11 between 40 and 80. 4 List the multiples of 12 between 140 and 200. 5 List the common multiples and then state the lowest common multiple of the following: a 2 and 5 b 3 and 4 c 7 and 9 d 6 and 7 e 6 and 8 f 6 and 10 6 Find the lowest common multiple of the following: a 2, 5 and 10 b 3, 4 and 5

c 6, 8 and 9

7 Sarah and Emily ride their bicycles around a track. Sarah completes one circuit in 42 seconds, while Emily takes 60 seconds. If they start together how long will it be before they are together at the start again? How many circuits will each girl do? 8 A lolly factory has four machines which produce snakes, mint leaves, bananas and musk sticks at different rates. A snake is produced every 12 seconds, a musk stick every 8 seconds, a banana every 6 seconds and a mint leaf every 3 seconds. If the machines start at the same time, how long will it take before all four produce a lolly at the same time? 9 Three Melbourne restaurants change their menus to give variety to their customers. The first restaurant changes its menu every 4 weeks, the second restaurant every 9 weeks and the third restaurant every 12 weeks. If they all change their menu today, how many weeks will it be before they all change their menu on the same day? If all the restaurants change their menus on 1 January, how many times will they change the menu on the same day during the year? 10 Lulu and Lillian are training for a swimming carnival by swimming laps of the pool. Lulu takes 24 seconds to swim a lap while Lillian takes 34 seconds. If they start together, how long will it be before they touch the wall at the same time? How many laps will each girl do?

42

Maths Dimensions 7

Factors

2C

The factors of a certain number are all the numbers which divide exactly into that number. They can be paired together except for factors of perfect squares.

Example 1 List all the factors of 24. 2 List all the factors of 36. 3 Find the common factors of 24 and 36. 4 Find the highest common factor of 24 and 36.

Solution 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 3, 4, 6, 9, 12, 18, 36 The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12. The highest common factor of 24 and 36 is 12.

Exercise 2C 1 List the factors of the following numbers: a 48 b 30 e 52 f 15 i 50 j 27

c 14 g 32 k 60

d 40 h 13 l 144

2 Find the highest common factor of the following pairs of numbers: a 48 and 30 b 15 and 30 c 14 and 27 d 13 and 52 e 40 and 32 f 40 and 50 g 30 and 60 h 48 and 144 i 32, 48 and 144 3 Two pieces of rope, 40 metres and 64 metres long, are to be cut into smaller lengths. What is the largest length into which the ropes can be cut so that every piece is the same length? 4 Two pieces of wood, 150 centimetres and 180 centimetres long, are to be cut into smaller lengths to make shelves that must all be the same length. What is the largest shelf length into which each piece of wood can be cut so that there is no wastage? 5 Keiko has three sections of hose to use in her garden’s automatic watering system. The hoses are 4 metres, 6 metres and 10 metres in length. She wishes to cut the hoses into equal lengths each as long as possible without having any offcuts. a How long would each piece of hose be? b How many pieces of hose will Keiko have for her garden? 6 Mr Chan has 24 students in his class whom he wishes to work with in groups. How many students would be in each group if all groups have the same number of students and no students are left out? In each case how many groups would Mr Chan have? 7 The scouts gather for their annual jamboree with representatives from Victoria, South Australia, Western Australia and Queensland. There are 84 scouts from Victoria, 126 from Queensland, 294 from South Australia and 462 from Western Australia. The scouts are distributed equally into groups. Use a calculator to find the highest common factor of 84, 126, 294 and 462, and state the maximum number of groups possible.

Chapter 2 Number Patterns

43

2D

Divisibility tests Knowing which numbers are more likely to divide into a certain number can make it much quicker to find all the factors, particularly prime factors, which we will look at next. Here are some easy divisibility tests or rules for numbers, which can help you find factors. Divisor

Divisibility test

Example

2

The number must be even. This means it can end in either 0, 2, 4, 6 or 8.

3568 is divisible by 2. This can be divided by 2 as it ends in an even number.

3

Add up all the digits. The result must be divisible by 3.

4086 is divisible by 3. 4 + 0 + 8 + 6 = 18 18 can be divided by 3 exactly.

4

The last two digits of the number must be divisible by 4.

6312 is divisible by 4. The last two digits are 12, which can be divided by 4 exactly.

5

The last digit must be either a 0 or a 5.

67 945 is divisible by 5. The last digit is a 5.

6

The number must pass the test for divisibility by 2 and 3.

4566 is divisible by 6. The number is even and the sum of the digits is 21. The answer is divisible by 2 and 3.

8

The last three digits of the number must be divisible by 8.

23 464 is divisible by 8. 464 ÷ 8 = 58

9

Add up all the digits. The result must be divisible by 9.

134 820 is divisible by 9. 1 + 3 + 4 + 8 + 2 + 0 = 18 18 ÷ 9 = 2

10

The number must end in a zero.

4 576 260 is divisible by 10 as the last digit is a zero.

Exercise 2D 1 Which of these numbers are divisible by: 45 55 724 8761 6210

a 5? 546

2 Which of these numbers are divisible by: 64 346 6780 545 87 452

a 3? 234

3 Which of these numbers are divisible by: a 2? 32 540 346 884 5672 65 321 54 984

b 10? b 6?

c 9?

b 4?

c 8?

4 Use the divisibility test to state whether the following numbers are divisible by 2, 3, 4, 5, 8 or 9: a 720 b 375 c 486 d 273 e 728 f 909 g 9843 h 10 520 i 22 959 5 Write down three four-digit numbers that are divisible by: a 2 b 3 c 6 d 5

44

Maths Dimensions 7

e 8

Exploring primes and composites 2E A prime number is a positive whole number that has exactly two factors: itself and one. Example: 37 = 1 × 37. The number 1 is not prime as it has only one factor. A composite number is a positive whole number that has more than two factors, i.e. it is not a prime. Example: 36 = 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.

Learning task 2E Copy the table below and follow the instructions to sieve out all the numbers which are not prime. ■

Cross out the number 1 because it is not a prime.

■

Circle the number 2 and cross out all of the multiples of 2.

■

Circle the number 3 and cross out all of the multiples of 3.

■

Circle the number 5 and cross out all the multiples of 5.

■

Circle the number 7 and cross out all the multiples of 7.

The numbers remaining are the prime numbers less than 100. 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

1 From the table above, write down all the prime numbers which are less than 100. 2 List all the composite numbers less than 20. 3 Christian Goldbach suggested that every even number greater than 2 can be written as the sum of two prime numbers. Examples: 4 = 2 + 2 6=3+3 8=3+5 Write the even numbers from 10 to 30 as a sum of two prime numbers. Chapter 2 Number Patterns

45

2F

Index notation Index notation is used when a number is multiplied by itself several times. The index or power tells us how many times the number must be multiplied by itself. 5 is the base

3 is the index or power

53 Example

Solution

1 Write 53 in expanded form. 2 Evaluate 35.

53 = 5 × 5 × 5

3 Evaluate 43 − 52.

43 − 52 = 4 × 4 × 4 − 5 × 5 = 64 − 25 = 39

35 = 3 × 3 × 3 × 3 × 3 = 243

Exercise 2F 1 Write each of the following in index form: a 2×2×2×2×2×2×2 c 17 × 17 × 17 e 23 × 23 × 23 × 23 g 1×1×1×1×1×1×1

b d f h

6×6×6 5×5×5×5×5×5×5×5×5×5 3×3×3×3×3 19 × 19

2 Write each of the following in expanded form: a 92 b 231 c 37 9 3 e 8 f 19 g 610

d 45 h 28

3 Evaluate: a 52 e 201

c 013 g 44

d 28 h 132

c 4 2 + 23 g 5 × 62 k 5 × 22

d 52 + 16 h 50 − 25 + 32 l 22 × 52

b 103 f 34

4 Evaluate (remember BODMAS): a 92 − 42 b 82 − 25 e 24 + 32 + 43 f 102 × 5 i 2 × 52 j (5 × 2)2

5 Arrange these index expressions in order from smallest to largest: 25 110 34 53 105 44 252 6 Four cat breeders each have four Persian cats, which each have four kittens. If the breeders take all their cats and kittens to the cat show, how many cages are needed if they must all be caged separately? 7 A bakery has three cake display cupboards. Each cupboard has six shelves and each shelf can hold six trays of cakes. How many cakes are on display when the cupboard is full, if each tray has six cakes across the width and six cakes along the length?

46

Maths Dimensions 7

Squares and square roots

2G

Square numbers When a number is multiplied by itself or raised to the power of 2, the number is said to be squared.

Example 1 Evaluate: a 42 b 72 2 Use a calculator to find 242.

Solution 42 = 4 × 4 = 16 72 = 7 × 7 = 49 242 = 576

Square roots The square root of a number is the opposite of squaring, you are finding the number that is multiplied by itself to give the number under the square root sign.

Example

Solution

3 Evaluate: a 64 b 49

64 = 8 since 8 × 8 = 64

4 Use a calculator to find 3136.

3136 = 56

49 = 7 since 7 × 7 = 49

Exercise 2G 1 Evaluate the following without using a calculator: a 52 b 22 c 102 f 122 g 132 h 112

d 92 i 202

e 82 j 402

2 Evaluate the following without using a calculator: a 36 b 9 c 25 100 81 225 f g h

d i

e j

3 Use your calculator to evaluate the following: a 172 b 262 c 302 f 522 g 472 h 332

d 1252 i 812

e 1002 j 632

4 Use your calculator to evaluate the following: a 324 b 2916 c 676 7921 4761 19·36 f g h

d i

e j

121 144

10 201 1·44

49 900

12 321 2·25

Chapter 2 Number Patterns

47

2H

Prime factors A prime factor is a factor of a positive whole number that is a prime number. Every composite number can be written as a product of its prime factors. This can be done using a ladder system or factor tree. Writing a number as a product of its prime factors is also called prime decomposition.

Example

Solution

1 Write 36 as a product of its prime factors by using a ladder system. 2

36

2

18

3

9

3

3

Place the 36 on the right-hand side of the ladder. Place the prime factors on the lefthand side of the ladder. Use the prime factors to divide the number down until you end up with 1. 36 = 2 × 2 × 3 × 3 = 22 × 32

1

2 Write 36 as a product of its prime factors by using a factor tree. 36 12

3

4

3

2

2

Find two numbers which multiply together to give 36. Let’s choose 3 × 12. Circle the number 3 as it is prime. 12 is composite so find two numbers which multiply to give 12. Let’s choose 3 × 4. Circle 3 as it is prime. 4 is composite so break it up into 2 × 2. Circle both of the 2s as they are prime. 36 = 2 × 2 × 3 × 3 = 22 × 32

Exercise 2H 1 Copy and complete these factor trees and ladders: a b c 105 182 21 3

7

51

d

5

17

23

1

1

2

2 Use a ladder system to write each of the following numbers as products of its prime factors: a 20 b 49 c 100 d 36 e 16 f 48 g 30 h 24 i 70 j 54 k 84 l 63 m 81 n 66 o 120 3 Use a factor tree to write each of the following numbers as products of its prime factors: a 42 b 65 c 90 d 28 e 72 f 196 g 400 h 224 i 420 j 560 k 1000 l 603 m 110 n 525 o 13 860

48

Maths Dimensions 7

Odds and evens

2I

Even numbers An even number can be divided exactly by 2 with no remainder. Every even number ends in an even digit. Examples: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 …

Odd numbers An odd number cannot be divided exactly by 2 and, therefore, is not even. Examples: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 … Zero is neither odd nor even.

Exercise 2I 1 State whether each number is odd, even or neither: a 34 b 1001 c 976 f 33 g 871 h 3902 k 45 l 100 003 m 542

d 432 i 64 n 63

e 0 j 28 o 22 680

2 List all the even numbers between 50 and 60 inclusive. (Inclusive means including the numbers mentioned.) 3 List all the even numbers between, but not including, 86 and 108. 4 List all the odd numbers between, but not including, 27 and 41. 5 List every second odd number between 1 and 21 inclusive. 6 List three consecutive (that is, in order) even numbers that add to 24. 7 List four consecutive even numbers that add to 52. 8 List three consecutive odd numbers that add to 33. 9 List four consecutive odd numbers that add to 24. 10 Add the following examples and use the results to answer part e: a 2+4= b 6 + 10 = c 12 + 4 = d 4 + 16 = e When an even number is added to an even number the result is always 11 Add the following examples and use the results to answer part e: a 3+5= b 11 + 7 = c 17 + 5 d 5 + 19 = e When an odd number is added to an odd number the result is always 12 Add the following examples and use the results to answer part e: a 3+8= b 4 + 11 = c 10 + 7 = d 12 + 11 = e When an odd number is added to an even number the answer is always

.

.

.

13 State whether the following are odd or even and give an example to support your answer: a the product of two even numbers b the product of two odd numbers c the product of an odd and an even number d the difference between two odd numbers e the difference between two even numbers f the difference between an odd number and an even number Chapter 2 Number Patterns

49

Puzzles 1 Match the numbers with their lowest common multiple to answer the following question:

Which mountain in Western Australia is the largest exposed rocky outcrop in the world?

A 6, 9 M 4, 6 T 3, 8

G 12, 14 N 3, 9 U 15, 25

H 12, 15 O 10, 16 W 10, 15

_____ 12

_____ 80

_____ 75

_____ 9

_____ 24

_____ 18

_____ 75

_____ 84

_____ 75

_____ 15

L S Y

_____ 24

_____ 75

7, 9 5, 15 6, 7

_____ 15

2 Match the numbers with their highest common factor to answer the following riddle:

What does a frog say when it is excited?

A 15, 100 G 35, 70 S 40, 72

50

B L T

D 108, 144 M 15, 18 W 121, 132

32, 64 48, 64 30, 40

_____ 10

_____ 7

_____ 5

_____ 36

_____ 16

_____ 24

_____ 5

_____ 11

_____ 12

_____ 8

_____ 7

_____ 3

Maths Dimensions 7

E 24, 36 O 21, 56 Y 72, 120

_____ 12

Ch2 3 Complete the cross-number puzzle using indices. 1.

4.

2.

3.

5.

6.

7.

9. 10.

11.

12.

13.

14.

Across 2. 5. 6. 9. 10. 13. 14.

Down

123

1. 2. 3. 4. 7. 11. 12. 13.

52 22 + 32 23 42 − 22 33 − 22 114

153 53 92 292 73 52 − 22 412 53 − 102 − 22

4 Complete the following number patterns to find the answer to the riddle:

What do you get when you cross a hedgehog with a worm?

A 4, 7, 10, 13, ____ D 1, 3, 6, 10, ____ I 2, 4, 8, 16, ____ Q 2, 6, 13, 23, ____ U 5, 25, 125, 625, ____

B 17, 34, 51, ____ E −9, −1, 8, 18, ____ L −1, −5, −12, −22, ____ R 1, 8, 27, 64, ____ W 4, 16, 64, 256, ____

_____ 68

_____ 16

_____ 125

_____ 68

_____ 1024

_____ 32

_____ 125

_____ 29

_____ 29

_____ 15

Chapter 2 Number Patterns

51

Applications Pascal’s Triangle Blaise Pascal was a French mathematician who investigated an arrangement of numbers that is now known as Pascal’s Triangle. It is a triangular array of numbers that dates back to Ancient China. This arrangement of numbers contains many different number patterns that relate to numerous areas of mathematics. 1 1 1 1 1 1 1

2 3

4 5

6

1 1 3 6

10 15

1 4

10 20

1 5

15

1 6

1

a Pascal’s Triangle contains many patterns. Work with a partner and try to find as many patterns as you can. To add more rows to the triangle, work out how each row is made from the numbers just above it. Look at the third and fourth rows. Row 1 1 Row 2

1

Row 3 Row 4

1 1

1 2

3

1 3

1

b How is 3 formed using the numbers above it? Check your pattern with other rows. Write this pattern in your book. c Copy rows 1–4 of the triangle into your book, then add six more rows to it. d What is the sum of the numbers in each row? Is there a pattern? If so, describe the pattern in your own words. Pascal’s Triangle also contains many patterns that form interesting shapes within the triangle.

e With a red pencil, shade in all the multiples of 5. Describe the shape and pattern of the shaded region. f

With a blue pencil, shade in all the multiples of 7. Describe the shape and pattern of the shaded region.

g With a green pencil, shade in all the even numbers. Describe the shape and pattern of the shaded region. Try to find some other patterns that form interesting shapes. Write down your instructions to form these shapes and swap them with your partner. Form the patterns.

52

Maths Dimensions 7

Ch2 Factor puzzle Find the smallest number with: a 1 factor b 2 factors c 3 factors d 4 factors e 5 factors f 6 factors Is there a pattern?

Brick walls You have been given bricks with which to build a brick wall. The bricks are twice as long as they are high. You are to investigate possible wall patterns if the wall is to be only two units high. If the wall is one unit wide, the wall will be made with only one brick on its end. If the wall is two units wide, the wall can be made in two ways: two bricks side by side on their ends or one brick lying long-ways and the other brick on top. If the wall is three units wide, the wall can be made in three ways as shown: 1 unit 2 units

2 units high

3 units wide

a How many wall patterns are possible for a wall four units wide? b How many different patterns are there for a wall of length 5 units? c Draw all the patterns. You may need dot paper to help you. d How many different patterns are there for a wall of length 6 units? e Draw all the patterns. f

Look at the number of wall patterns found for a wall of length 1, 2, 3, 4, 5 and 6 units. i Does this pattern look familiar? ii What name do we give to this pattern of numbers?

g Using your number pattern, how many different brick walls could be made for walls of the following lengths? i 8 units ii 11 units iii 30 units Chapter 2 Number Patterns

53

Enrichment 1 A city hotel has three different flashing neon signs. The top one flashes every 9 seconds, the side one flashes every 15 seconds and the sign at the bottom flashes every 21 seconds. At regular intervals all three signs flash at the same time. a At what interval do the top and side signs flash together? b At what interval do the top and the bottom signs flash together? c At what interval do the side and bottom signs flash together? d At 2 am the top and bottom signs flash together. Nine seconds later the top and side signs flash together. At what time will all three signs flash together? 2 At Toulton Post Office there are 200 post office boxes. The boxes are numbered and line a wall of the post office. At the start of the day all the boxes are empty. A post office employee places a brochure in each of them. He then places a letter in every second post office box. Next he places a letter in every third box. He continues in this way, placing a letter in every fourth post office box until he has only one letter left. This letter goes in the 200th box. Which boxes will have an even number of articles in them and which boxes have an odd number of articles in them when he has finished? Begin by looking at 20 post office boxes first and try to discover a pattern for odds and evens. Extend your theory to 50 or 100 post office boxes. 3 In a tennis tournament there are 128 players at the start. In the first round there are 64 matches as each person plays another person. Only the winners advance to the next round. This process continues until there are only two players left to play in the grand final. a If there are 64 players to advance to the second round, how many matches need to be played in the second round? b Express this number as a product of its prime factors in both expanded and index form. c How many matches will be played in the third round? d How many matches will be played in the fourth round? e How many matches will be played in total for the tournament? 4 Sometimes when you reverse the digits in a prime number the result is also a prime number. These are called reverse primes. Find all the reverse primes between 1 and 100. 5 A perfect number is a natural number that is equal to the sum of its integer factors not including the original number. For example, 6 is a perfect number, as 6 = 1 + 2 + 3. The prime factors of 6 are {1, 2, 3}. a What are the next three perfect numbers? b What are happy numbers? c Find the first six happy numbers.

54

Maths Dimensions 7

Ch2 The prime factor tree or the prime decomposition of numbers can be used to find the highest common factor (HCF) and lowest common multiple (LCM) without finding all the factors or multiples. They can also be used to find all the factors of a number.

Example

Solution

1 Find the HCF of 320 and 360.

320 = 2 × 2 × 2 × 2 × 2 × 2 × 5 360 = 2 × 2 × 2 × 3 × 3 × 5 The highest common factor will have prime factors in both expressions. HCF = 2 × 2 × 2 × 5 = 40 The factors of 320 will be 1, 2, 22, 23, 24, 25, 26, 5, 5 × 2, 5 × 22, 5 × 23, 5 × 24, 5 × 25, 5 × 26

2 Find the LCM of 40 and 36.

40 = 2 × 2 × 2 × 5 = 23 × 5 36 = 2 × 2 × 3 × 3 = 22 × 32 The lowest common multiple must have the highest power of a prime occurring in either number. LCM = 23 × 5 × 32 = 360

6 a b c d e f g h i

Find the lowest common multiple of 24, 30 and 40. Find the lowest common multiple of 693 and 9317. Find the highest common factor of 1178 and 1444. Find the highest common factor of 204 and 1380. List all the factors of 204 and 1380. Find the smallest whole number which has the factors 2, 3, 5, 10, 16, 22, 24 and 32. Find the largest three-digit number which has exactly three factors including 1 and itself. Find the largest three-digit number which has exactly six factors. A lolly manufacturer makes red, green, yellow and orange sugar-covered jubes. He has made 245 700 red jubes, 132 300 green jubes, 114 660 yellow jubes and 207 900 orange jubes. The manufacturer wishes to distribute the lollies equally among the packers so that they have equal numbers of each different-coloured jube. What is the greatest number of lollies that can be packaged?

7 The original Fibonacci sequence (1, 1, 2, 3, 5, 8, 13 …) is formed by adding the last two numbers together to form the next number in the pattern. a By using 2 and 2 as the first two terms and Fibonacci’s pattern, write down the first 10 terms. Find the sum of the first 10 terms. b Write down the first 10 terms if the starting numbers are 1 and 4. Find the sum of the first 10 terms. c Using the original Fibonacci sequence, find the sum of the first 10 terms. Try to find a connection between the sum and the seventh term in the sequence. Test your finding on the two sequences formed in part a and part b. Chapter 2 Number Patterns

55

Revision Learning task 2A 1 For each of the number shapes or patterns below, count the number of dots in each shape to find the first four numbers of the pattern. Write down the first 10 numbers in each pattern. a

b

2 For each of the number shapes below, count the number of dots to find the first three numbers of the pattern. Write down the first 10 numbers in each pattern. a

b

Exercise 2B 3 a b c d e f

List the multiples of 7 which are less than 40. List the multiples of 3 and the multiples of 5 which are less than 40. What are the common multiples of 4 and 6 which are less than 50? Find the lowest common multiple of 9 and 12. Find the lowest common multiple of 2, 6 and 7. Find the lowest common multiple of 3, 5 and 7.

4 Susan and Ngai are having races running up a flight of stairs. Susan can take the steps three at a time while Ngai can only take them two at a time. If both students reach the top with no stairs left over, give four possible answers for the number of steps. What is the least number of steps possible? 5 Dean can swim a lap of the pool in 36 seconds, Aiden takes 44 seconds and Hudson takes 66 seconds. If they all start together, how long does it take for them to all touch the wall at the same time?

Exercise 2C 6 a b c d e

List all the factors of 15 and 20. State which is the highest common factor of 15 and 20. Find the highest common factor of 14 and 21. Find the highest common factor of 36, 42 and 54. Find the highest common factor of 18, 51 and 63.

7 The Elliott family wishes to put new carpet squares in the lounge room of their holiday house. The rectangular room is 650 centimetres long and 475 centimetres wide. What is the largest square tile which could be used so that no tiles have to be cut? How many tiles will be placed along each wall? 8 A canoeing club has 126 members. If no members are left out, could they canoe in groups of: a 2? b 5? c 6? d 8?

56

Maths Dimensions 7

Ch2 Exercise 2D 9 Use your divisibility tests to fill in the table below. Use a tick for yes (✓), and a cross for no (✗). 2

3

4

5

6

8

9

10

3696 580 4752 4599

Learning task 2E 10 State whether the number is prime or composite: a 23 b 9 c 14 d 56

e 1001

f

61

Exercise 2F 11 Write each of the following in index form: a 4×4×4×4×4×4 c 10 × 10 × 10 × 10 × 10 × 10 × 10

b 9×9×9 d 2×2

12 Write each in expanded form: a 62 b 45

c 83

d 104

e 36

13 Evaluate: a 3 × 23

c (3 × 2)2

d (3 × 2)3

e 32 + 2 3

b 32 × 23

Exercise 2G 14 Evaluate without using a calculator: a 62

b 152

c

64

d

169

c

2116

d

10 609

15 Evaluate using a calculator: a 282

b 1072

Exercise 2H 16 Express the following numbers as a product of their prime factors by using a factor tree or a ladder system: a 18 b 64 c 26 d 32 e 242 f 128

Exercise 2I 17 a b c d

List all the even numbers between 44 and 60 not inclusive. List all the odd numbers between 17 and 33 not inclusive. List every second odd number between 3 and 17 inclusive. List three consecutive even numbers that add to 18. Chapter 2 Number Patterns

57

CHAPTER

3 Fractions

Fractions It takes the Moon about 29 1--- days to orbit the Earth, starting from a new moon and 2 continuing through the phases of first quarter, full moon, last quarter and then back again to another new moon. The Moon spins on its axis as it orbits the Earth, so we only ever see about 4-- of its surface. 7 In the 1960s, Soviet astronauts were the first to see the dark side of the Moon.

This chapter covers the following skills: • Using the language of fractions • Simplifying and finding equivalent fractions • Converting improper fractions to mixed numbers and vice versa • Adding and subtracting with like and unlike denominators • Investigating, with a calculator, multiplication of fractions • Finding fractions of whole quantities • Dividing fractions • Finding squares and square roots of fractions

• Simplifying using order of operations B Work out the calculations inside the brackets first. If there is more than one operation inside the brackets, then they must also follow the rules of BODMAS O If the question contains fractions of or powers of, then these are calculated next D Work out the division and M multiplication calculations, working across the page from left to right A Work out the addition and S subtraction calculations, working across the page from left to right

3A

Shaded diagrams A fraction involves a part of a whole. The top number is called the numerator and tells you how many pieces you have of the whole. The bottom number is called the denominator and tells you into how many pieces the whole was divided.

Example

Solution

1 What fraction of the diagram is shaded? a

b

The number of parts in the diagram = denominator The number of parts shaded = numerator numerator 3 ------------------------------ = --denominator 8 numerator 7 ------------------------------ = -----denominator 10

A proper fraction has a numerator which is smaller than the denominator. Examples: 2--- , 3

3 19 ------ , -----10 20

An improper fraction has a numerator which is larger than the denominator. Examples:

10 5 58 ------ , --- , -----9 3 23

A mixed number has a whole number and a proper fraction part. 9 Examples: 2 1--- , 10 4--- , 100 ----2

7

20

A fraction can be cancelled down if the numerator and denominator can be divided exactly by the same number. All fractions should be given in their simplest form.

Example

Solution

2 Simplify these fractions:

60

8 a ----10

8÷2 8 ------ = --------------10 10 ÷ 2 4 = --5

-----b 10 25

10 10 ÷ 5 ------ = --------------25 25 ÷ 5 2 = --5

-----c 12 36

12 12 ÷ 12 ------ = -----------------36 36 ÷ 12 1 = --3

Maths Dimensions 7

3A Exercise 3A 1 What fraction of each diagram is shaded? a

b

c

d

e

f

g

h

i

j

k

l

2 Label each of the following as proper (P), improper (I) or mixed number (M): a

2 --3

b 2 2---

c

7 --9

d

9 --5

e 1 4---

f

1 -----11

g 6 1---

h

5 --2

i

3 --4

j

4 1---

k

7 --3

l

m 2 1---

n

3 --5

o

15 -----4

5

2

25 -----7

4

5

3

3 Simplify the following fractions: a

6 --8

b

5 -----20

c

8 -----16

d

6 --9

e

10 -----40

f

14 -----21

g

35 -----42

h

12 -----30

i

13 -----26

j

56 -----64

k

12 -----60

l

6 -----18

m

99 --------108

n

12 -----40

o

35 -----56

p

16 -----28

q

56 -----96

r

45 -----72

s

33 --------121

t

78 --------169

u

8 -----10

v

12 -----14

w

21 -----36

x

15 -----25

y

96 --------120

Chapter 3 Fractions

61

3B

Mixed numbers and improper fractions To change a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. This answer becomes the new numerator. The denominator stays the same. (Remember BODMAS for the numerator—multiplication comes before the addition.)

Example

Solution

1 Change the mixed numbers to improper fractions: 2 4--- =

a 2 4--5

5

= 3 7--- =

b 3 7--9

9

= 2 1--- =

c 2 1--2

2

=

2×5+4 --------------------5 14 -----5 3×9+7 --------------------9 34 -----9 2×2+1 --------------------2 5 --2

To change improper fractions to mixed numbers, divide the denominator into the numerator. This answer becomes the whole number. Place the remainder in the numerator and the denominator remains the same.

Example

Solution

2 Change the improper fractions to mixed numbers: a

34 -----5

34 -----5

= 6 4---

b

24 -----7

24 -----7

= 3 3---

5

7

Finding equivalent fractions is the opposite of simplifying. It is the process of making the fraction larger. You need this skill for addition and subtraction of fractions with different denominators. To find an equivalent fraction the numerator and denominator are both multiplied by the same number.

Example

Solution

3 Find the equivalent fractions: a

b

62

1 --3

2 --5

=

=

--6

-----25

=

=

3 ---

=

6 ---

Maths Dimensions 7

-----21

1 --3 1 --3 1 --3

×

2 --5 2 --5

×

× ×

×

2 --2 3 --3 7 --7

=

5 --5 3 --3

=

= =

=

2 --6 3 --9 7 -----21 10 -----25 6 -----15

3B To compare two or more fractions, change the fractions to equivalent fractions with the same denominator.

Example

Solution

4 Which is larger

3 --4

or 4--- ?

3 --4

5

×

So

4 --5

5 --5

=

15 -----20

×

4 --5

=

4 --4

16 -----20

is bigger than 3--- . 4

Exercise 3B 1 Convert the following mixed numbers to improper fractions: a 2 1---

b 3 2---

2

f

5

2 5---

d 4 1---

h 3 4---

i

4

g 10 1---

6

c 1 1--5

e 1 3---

3

7

5

5 1 -----

2 7---

j

12

9

k 3 1---

l

5 3---

m 4 3---

-----n 1 11

o 10 4---

3 p 9 -----

q 3 2---

r 7 4---

s 4 3---

t 12 5---

6

11

4

8

3

12

5

9

7

8

2 Change the following improper fractions to mixed numbers: a

8 --3

b

7 --2

c

9 --5

d

29 -----6

e

19 -----8

f

9 --4

g

27 -----8

h

16 -----7

i

31 -----7

j

21 -----5

k

41 -----9

l

22 -----3

m

83 -----10

n

85 -----11

o

13 -----6

p

19 -----4

q

33 -----8

r

46 -----7

s

49 -----5

t

77 -----10

3 Fill in the gaps below: a

1 --2

=

-----12

=

4 ---

b

2 --3

=

10 ------

=

-----21

c

3 --5

=

-----20

=

18 ------

d

3 --4

=

12 ------

e

5 --6

=

-----12

=

25 ------

f

2 --7

=

18 ------

=

-----77

g

4 --5

=

-----15

=

36 ------

h

1 --4

=

8 ---

i

4 --9

=

-----18

=

20 ------

j

3 -----10

k

1 --8

=

2 ------

=

-----32

l

5 --9

=

20 ------

=

18 ------

=

--------110

= =

-----28

-----20

=

-----63

4 Change each of the fractions in the following pairs to equivalent fractions and state which is larger: a

5 --7

or

6 --7

b

10 -----11

d

5 --6

or

8 -----11

e

4 --9

or or

2 --5

8 -----11

c

2 --3

or

3 --5

f

2 --7

or

3 -----10

Chapter 3 Fractions

63

3C

Adding with the same denominator Only fractions with the same denominator can be added. Always check the solution and simplify the fraction if required. Improper fractions need to be converted to mixed numbers.

Example

Solution

1 Add these fractions: 1 --7

a

5 --9

b

+

+

3 --7 1 --7

+

3 --7

=

4 --7

5 --9

+

7 --9

=

12 -----9 1 3--9 1 1--3

7 --9

= =

When adding mixed numbers, add the whole number parts and add the fraction parts.

2 2 1--- + 1 3--8

8

2 1--- + 1 3--- = 3 4--8

8

8

= 3 1--2

Exercise 3C 1 Add these fractions (draw diagrams if necessary). Remember to fully simplify the answers: +

a

1 --5

2 --5

e

4 -----17

i

5 --6

m

17 -----21

+

q

6 -----13

+

+ +

+

b

4 --7

f

3 -----10

j

4 --5

11 -----21

n

19 -----24

+

7 -----13

r

14 -----15

+

9 -----17

5 --6

+ +

c

6 -----13

+

5 -----13

d

1 --9

g

9 -----16

+

3 -----16

h

5 -----14

+

3 -----14

k

3 --8

l

3 -----10

+

9 -----10

11 -----24

o

20 -----31

+

5 -----31

p

7 -----12

+

11 -----12

16 -----15

s

14 -----17

+

20 -----17

t

23 -----25

+

27 -----25

2 --7 5 -----10

3 --5

+

5 --8

+

2 Find the answers to the following additions: a

64

2 --3

+

1 --3

+

2 --3

Maths Dimensions 7

b

2 --9

+

7 --9

+

5 --9

c

1 --8

+

5 --8

+

3 --8

5 --9

3C 4 -----17

d

+

11 -----17

+

16 -----17

9 -----22

e

+

11 -----22

+

19 -----22

f

21 -----40

+

+

37 -----40

39 -----40

3 Add these fractions: a 1 1--- + 3 3---

b 4 3--- + 2 1---

e 1 1--- + 4 5---

f

5 4 - + 3 -----4 -----

5 1 - + 1 -----g 2 -----

5 - + h 2 -----

5 --3- + 6 1---

j

4 5--- + 2 7---

8 7 - + 2 -----k 3 -----

l

7 11 - + 3 -----o 1 -----

8 7 - + 3 -----p 2 -----

5

5

6

i

8

6

4

4

m 8 2--- + 6 2--3

11

9

c 6 3--- + 3 2---

8

7

11

12

9

15

17 ------ + 4 -----n 5 13

3

18

18

13

1 3 - + 2 -----d 5 -----

7

10

12

10

14

15

13

3 -----14

11 ------ + 2 -----1 13 20

20

11

11

4 Add these fractions: a 2 2--- + 3 1--- + 1 4--7

7

b 5 1--- + 2 4--- + 1 3---

7

5

7 11 2 - + 3 ------ + 2 -----d 14 ----15

15

5

c 6 7--- + 5 3--- + 1 5---

5

8

7 11 13 - + 5 ------ + 8 -----e 4 -----

15

20

20

20

f

8

8

9 8 9 - + 11 ------ + 5 -----13 ----11

11

11

5 A pie is cut into eighths. One child eats one piece while another eats three pieces: a What fraction of the pie is eaten by the first child? b What fraction of the pie is eaten by the second child? c What fraction of the pie has been eaten? 6 A shopkeeper has a new roll of fabric. During the day 1 3--- metres and 4 1--- metres of material 4 4 are sold from the roll. How much material in total has been sold from the roll? 7 One week the Lucas family buys 3 2--- kilograms of oranges. The next week they buy 5 2 4--- kilograms of oranges. How many kilograms of oranges has the Lucas family bought? 5

1 7 - metres of shadecloth on Monday, 5 ------ metres on Tuesday and 8 A hardware store sold 3 ----10 10 9 10 ------ metres on Friday. How much shadecloth was sold during the week? 10

9 A lasagna is heated in an oven and takes 3--- of an hour to warm. A pie is heated next and 4 7 - of an hour. What is the total length of time the oven needs to be on? takes ----12

10 Tran earns his pocket money. He spends 2--- of an hour washing the car, 5--- of an hour vacuuming 7 7 the house and 4--- of an hour mowing the lawn. How long does Tran spend doing chores? 7

11 Mark bought a large pizza cut into 12 slices. He ate 1--- , 3 1 - of the pizza. Sue ate 1--- and Dale ate ----6

12

a How many slices did each person eat? b How many slices were left? 12 Sonja bought 2 pizzas and cut each into 8 slices. Sonja ate 1--- , Aleisha ate 3--- and Julia ate 1--- of the slices. 4

8

8

a How many slices did each person eat? b How many slices were left? c What fraction of the pizzas remained? Chapter 3 Fractions

65

3D

Adding with different denominators Fractions with different denominators must be changed so that they have a common denominator before the numerators of the equivalent fractions can be added.

Example

Solution

1 Add these fractions: +

1 --4

a

2 --3

The common denominator is 12. +

1 --4

2 --3

3 2 1 --- × --- + --3 3 4 3 8 ------ + -----12 12 11 -----12

= = =

b 2 4--- + 3 5--5

×

4 --4

The common denominator is 35.

7

2 4--- + 3 5--- = 2 + 3 + ( 4--- × 5

7

5

=5+

-----( 28 35

+

7 --7

+

5 --7

× 5--- ) 5

25 ------ ) 35

-----= 5 53 35

-----= 5 + 1 18 35

=

-----6 18 35

2 Use a calculator to add these fractions: 3 -----17

a

+

8 -----13

b 3 2--- + 2 4--7

3 -----17

+

=

175 --------221

8 -----13

3 2--- + 2 4---

5

7 5 213 = --------35 3 = 6 ----35

Exercise 3D 1 Without using a calculator add the following. Remember to find the common denominators first:

66

a

1 --4

+

2 --5

b

1 --6

+

4 --7

c

2 --3

+

1 --5

d

2 --3

+

7 --8

e

2 --5

+

3 -----10

f

3 --4

+

1 --8

g

2 --9

+

1 --6

h

3 --5

+

2 --7

i

2 --3

+

4 -----15

j

9 -----10

k

8 -----15

l

5 -----18

Maths Dimensions 7

+

19 -----20

+

3 --5

+

7 --9

3D 2 Add the following fractions without using your calculator: a 2 1--- + 4 1---

b 2 2--- + 4 1--7

c 1 2--- + 3 4--9

d 8 3--- + 4 1---

e 2 1--- + 3 1---

f

4 3--- + 2 2---

g 5 1--- + 4 1---

h 1 3--- + 2 1---

j

8 14 + 10 -----

3 2 - + 6 --k 9 -----

2

5

4

i

5

2

16 2--- + 4 8--3

9

7

7

9

6

25

8

2

9

5

10

2 3--- + 4 1---

l

5

8

4

2

3 Find the answers to the following using your calculator. Remember to write your answers in the correct form, not as on your calculator: +

b

2 --5

+

7 -----10

c

6 --7

d

7 --9

+

5 --6

f

5 --8

+

9 -----11

g

12 -----13

h

7 --8

+

4 --5

j

4 5--- + 2 2--3

k 12 7--- + 5

l

1 1--- + 2 3---

7 4 - + 3 --m 4 -----

n 6 5--- + 1 5---

o 4 3--- + 4 --4-

9 3 - + 3 --p 7 -----

q 3 + 4 2--5

8 8 - + 2 --r 5 -----

11 ------ + 3 -----s 2 13 25

9 11 - + 4 -----t 6 -----

11 -----21

v 1 5--- + 2 1--- + 4 2---

7 ------ + 6 -----w 4 11

7 11 - + 2 -----x 5 -----

a

1 --6

11 -----12

e

3 -----11

i

9 1--- + 3

+

7 -----10

2

10

u

5

+

7 -----18

6

8

+

4

9

6

3 --4 1 --2

8

6

15

+

20

2

5

25

2

4

10

7

8

10

20

15

10

25

4 Calculate the following: +

a

3 --4

d

11 -----12

7 --8

+

3 --5

+

3 --5

+

5 --6

g 3 1--- + 1 5--- + 2 3--2

j

6

4

2

2 --5

+

5 --6

+

3 -----10

c

7 -----10

e

1 --7

+

4 --5

+

23 -----35

f

1 --4

7 7 - + 1 --h 2 3--- + 5 -----

i

3 1 ------ + 3 --- + 1 --2 10

9 1 - + 8 --k 4 5--- + 1 -----

l

4 7--- + 2 --4- + 7 2---

5

2 1 3 - + 3 --- + 4 --5 ----13

b

4

8

10

11

8

2

+ +

5 --6

3 --4

+ +

11 8

1 --6

11 -----12

4

5

2

9

5 A caramel chocolate block is split into fifths, while a peppermint chocolate block is split into sevenths. a What is the common denominator for 1--- and 1--- ? 5

7

b If Tony eats 1--- of caramel and 5 2 --- of peppermint, what is the total 7 amount of chocolate eaten by Tony? c If Michelle eats 2--- of caramel and 5 3 --- of peppermint, what is the total 7 amount of chocolate eaten by Michelle? Chapter 3 Fractions

67

3D 6 Michael spends time each night doing homework. a On Monday night, he spent 3--- of an hour on English and 2--- of an hour on French. How 4 5 much time is spent on homework on Monday night? b Michael spends 2--- of an hour on Maths and 8--- of an hour on a Society and Environment 7 9 assignment. What is the total amount of time spent on homework on Tuesday night? c On Wednesday night, Michael does Maths for 1--- of an hour and 3--- of an hour of Science. 4 8 What total time is spent on homework? d Thursday night requires Michael to spend 4--- of an hour on English and 5 French. What amount of time is spent on homework?

2 --3

of an hour on

e On Friday, Michael spends 1--- of an hour on Maths, 2--- of an hour on Society and 3 5 Environment and 1--- an hour on Science. What is the total time spent on homework 2 on Friday? 7 Donald buys 12 1--- metres of 25 mm pipe and 10 2--- metres of 19 mm pipe to use in his 2 3 garden. What is the total length of pipe purchased by Donald? 8 Mr Bracken purchases 2 2--- kilograms of oranges, 4 kilograms of potatoes and 1 3--- kilograms 5 4 of apples. What is the total weight in Mr Bracken’s carry bag? 9 Matthew ate 1--- of a Hawaiian pizza and 2--- of an Aussie pizza. What was the total fraction of 7 4 pizza eaten by Matthew? 10 Shani has three CDs. One plays for 1 3--- hours, the compilation CD plays for 3 2--- hours and 5 4 the single plays for 15 minutes. What is the total length of time for the three CDs to be played back to back? Give your answer in hours. 11 Aleah has a collection of 14 music CDs. Eight play for 1 1--- hours each, four singles play for 4 35 minutes each and two compilation CDs play for 2 hours each. How many hours would it take to play all the CDs back to back?

68

Maths Dimensions 7

Tetral fractions 3E Introduction This game involves adding fractions to give a result of 1. The game is similar to Tetris, except that fractions replace the coloured shapes. Rather than require three or more coloured shapes to occur in a row, the fractions need to add to 1. This can be done with two or more fractions. There are three levels of the game. The first level has all the denominators the same, the second level has very basic changes in the denominator. The third level, Gold, requires a great deal of concentration and an ability to identify common denominators very quickly from a larger range of fractions.

Equipment TI-84Plus with Algebra 1 App™ (free with TI-84Plus) This application is only free to users of the TI-84Plus calculator or TI-83Plus Silver Edition.

Technology activity 3E Press the APPS key and start the Topics in Algebra application (ALG1PRT1). Press any key to continue.

Select option 1: Number Sense.

Select option 2: Rational Numbers.

Select option 3: Activities.

Press

on the welcome screen to continue.

Chapter 3 Fractions

69

3E Press ENTER to select 1. Slide. This game is similar to Tetris. Use the arrow keys to move the fraction into one of three columns as it falls. The object is to put fractions that add to 1 in the same row or column. The column or row can add up to more than 1; however, if two adjacent fractions can combine to equal 1 then a score is registered and the fractions disappear from the screen. Select the level. Bronze is recommended for beginners. Note: Use the up

and down

arrows to navigate between

levels. Press ENTER to select the level.

Fractions start ‘falling’ from the right of screen. Use the arrow keys to move them up or down. Once you are happy with the placement you can press the left arrow to accelerate the rate at which they fall. In the example here, the two fractions do not combine to give an answer of 1. Shift the 4--- either up or down to move it away from 5

the 3--- . 5

The middle row has gradually filled; however, the descending fraction, 4--- , can be combined with 1--- to clear out the two entries, 1 4 --- and --- . 5 5

5

5

Fractions can be matched in either columns or rows.

Once the screen is full and no more factions can be dropped to the screen, the game ends and the score is recorded.

Show your teacher your final score to compete for the highest score in Bronze, Silver or Gold.

70

Maths Dimensions 7

Subtracting with the same denominator

3F

Only fractions with the same denominator can be subtracted. Always simplify the fraction if required. Improper fractions should first be converted to mixed numbers.

Example

Solution

Subtract these fractions:

a

2 --5

−

1 --5

2 --5

−

1 --5

=

1 --5

b

9 --8

−

5 --8

9 --8

−

5 --8

=

4 --8 1 --2

=

c 4 5--- − 1 1--7

When subtracting mixed numbers, subtract the whole number parts and subtract the fraction parts.

7

4 5--- − 1 1--- = (4 − 1) + ( 5--- − 1--- ) 7

7

7

=

d 4 1--- − 1 7--8

7

3 4--7

If the first fraction is too small, take a whole number off and add it to the fraction part.

8

4 1--- − 1 7--- = 3 9--- − 1 7--8

8

= =

8 2 2--8 2 1--4

8

Exercise 3F 1 Find the answers to the following: −

a

5 --8

e

5 -----13

i

8 --7

m

30 -----25

1 --8

− −

2 -----13

1 --7

−

3 -----25

−

c

5 -----11

−

2 -----11

d

11 -----12

15 -----31

g

17 -----19

−

12 -----19

h

97 --------100

−

3 --8

k

20 -----13

−

5 -----13

l

16 -----11

−

10 -----11

−

21 -----9

o

33 -----10

−

13 -----10

p

18 -----5

−

6 --5

b

7 --9

1 --9

f

30 -----31

−

j

12 -----8

n

42 -----9

−

5 -----12

−

37 --------100

2 Find the answers to the following: a 4 7--- − 1 3---

9 3 - − 3 -----b 10 -----

e 3 1--- − 2

f

12 − 2 3---

j

8

8

8

i

5

4 ------ − 3 -----c 6 10

8 4 - − 3 -----d 9 -----

6 - −8 14 -----

g 4−

h 8−

7 16 − 8 -----

4 ------ − 6 -----k 12 10

10

10

15

10

13

13

3 --7

21

21

11

l

11

5 --9

7 ------ − 1 -----8 19 20

20

Chapter 3 Fractions

71

3F 11 ------ − 8 -----m 14 16

n 4 2--- − 1 5---

q 13 2--- − 5 7---

5 8 - − 6 -----r 11 -----

19

19

9

7

9

o 8 1--- − 2 4---

7

11

5

p 12 3--- − 2 7---

5

8

5 11 - − 4 -----s 7 -----

11

12

8

1 7 - − 4 -----t 5 -----

12

10

10

3 Find the answers to the following: a

9 -----11

–

2 -----11

–

5 -----11

b

10 -----13

–

7 -----13

–

2 -----13

c

22 -----25

–

7 -----25

–

2 -----25

d

42 -----15

–

21 -----15

–

8 -----15

e

37 -----24

–

13 -----24

–

7 -----24

f

31 -----36

–

17 -----36

–

11 -----36

i

1 7 ------ – 4 ------ – 3 -----14 11

l

1 17 ------ – 2 ------ – 1 -----5 11

4 10 - – 5 -----g 14 – 2 ----11

j

h 8 – 2 1--- – 3 7---

11

9

5 11 17 - – 8 ------ – 2 -----12 ----18

18

9

7 13 ------ – 3 ------ – 2 -----k 8 11

18

20

20

20

15

15

24

24

15

24

4 A cake is cut into eight pieces. Three pieces are eaten. What fraction of the cake is left? 5 Sonny has 2 --3- bags of lollies. If 1 1--- bags are eaten, how many bags of lollies are left? 4

4

16 7--8

6 A roll of carpet contains metres of carpet. If a family buys 10 3--- metres of carpet for 8 their lounge room, how much carpet will be left on the roll? 3 - full. If the tin is 7 A shop owner orders a new tin of ice-cream when the tin is only ----10 8 currently ------ full, how much more ice-cream needs to be sold before a new tin is ordered? 10

8 A warehouse has 16 7--- boxes of soft drink, 10 3--- boxes of chips and 38 boxes of cups. 8 4 ------ boxes of cups. A milkbar purchases 3 5--- boxes of soft drink, 2 1--- boxes of chips and 10 16 8

4

25

a How many boxes of soft drink are left at the warehouse? b How many boxes of chips are left at the warehouse? c How many boxes of cups are left at the warehouse? 9 A satellite travels through space and uses 7 1--- litres of 5 fuel. If 8 4--- litres of fuel was on board at launch, how 5 much fuel remains? 10 A satellite spent 37 months in space. It spent 5 months taking photographs of Mars and 23 months collecting data of the Moon. The rest of the time was spent travelling. a Convert each of the times to a fraction of a year. b How much time, in years, was spent travelling? c What fraction of the time was spent travelling? d What fraction of the time was spent photographing Mars?

72

Maths Dimensions 7

Subtracting with different denominators

3G

As for addition, fractions with different denominators must be changed to a common denominator before the numerators of the equivalent fractions are subtracted.

Example

Solution

Subtract these fractions:

a

−

7 --8

2 --5

The common denominator is 40. −

7 --8

5 2 7 --- × --- − --5 5 8 35 16 ------ − -----40 40 19 -----40

=

2 --5

= =

b 4 1--- − 1 5--4

×

8 --8

4 1--- − 1 5--- = 4 2--- − 1 5---

8

4

8

8

8

5 ------ − 1 --= 3 10

= -----c 8 − 6 10

8 5 2 --8

8

17 10 ------ = 7 ------ − 6 -----8 − 6 10

17

17

17

17

7 = 1 ----17

Exercise 3G 1 Without using your calculator find common denominators before subtracting: a

2 --5

−

3 -----10

b

7 -----10

−

e

2 --5

−

1 --4

f

3 --4

−

i

2 --7

−

1 --9

j

5 --6

−

3 --5

c

11 -----12

1 --6

g

3 --5

2 --5

k

9 -----11

− −

1 --6

1 --6

−

2 --5

d

2 --3

−

1 --5

h

6 --7

−

2 --3

l

12 -----13

−

1 --2

2 Find the answers to the following without using your calculator: a 3 1--- − 1 1---

b 6 1--- − 2 1---

e 9 2--- − 4 1---

f

9−

j

2

4

5

i

7

14 -----19

c 10 1--- − 4 1---

d 6 7--- − 1 3---

3 3--- − 1 1---

g 6 1--- − 4

7 h 8 − 2 -----

7 1--- −

9 k 7 3--- − 3 -----

2

5

5

4

3 --4

8

2

6

2

4

10

10

5

8

l

7 4 - − 3 --10 -----

d

7 --9

h

12 -----13

l

2 3--- − 1 1---

10

5

3 Use your scientific calculator to find the answers: a

7 --8

−

2 -----11

b

4 --7

e

5 --6

−

3 -----11

f

7 -----10

i

9 2--- − 3 1---

j

3 5--- − 1 3---

5

6

−

7

1 --8

−

5 --8 8

−

c

2 --7

g

9 -----11

2 --9

−

1 --2

k 3 4--- − 1 2--9

7

−

5

3 --4

−

4 --5 8

Chapter 3 Fractions

73

3G m 6 1--- − 4 2--2

n 7 5--- −

5

8

1 --3

7 o 18 − 3 -----

p 8−

10

14 -----17

7 4 - − 3 --q 10 -----

r 10 5--- − 2 1---

s 4 4--- − 2 3---

9 t 7 3--- − 3 -----

u 4 2--- − 1 5---

v 2 1--- − 1 1---

13 ------ − 1 -----w 3 11

9 ------ − 6 -----x 10 11

10

5

5

8

6

9

4

7

3

25

4

8

20

10

15

10

4 Find the answer to the following: a

11 -----18

−

1 --6

–

1 --3

b

14 -----15

−

2 --5

d

35 -----36

−

4 --9

–

1 --4

e

21 -----22

−

5 -----11

g 10 −

2 --5

– 4 3---

j

8

3 --8

–

1 --4

h 19 − 4 1--- – 6 2---

4

2

14 5--- − 4 1--- – 2 2--6

–

3

7

4 3 ------ − 5 --- – 1 --k 8 11 12

7

4

c

19 -----20

−

1 --4

–

2 --5

f

41 -----48

−

3 --4

–

1 -----12

i

12 1--- − 2 2--- – 5 1---

l

9 2--- − 3 5--- – 1 --3-

4

5

5

6

6

8

5 The Harris family purchases 3--- hectare of land on which to build their house. If the house 4 1 - hectare, how much land is left for garden? occupies ----12

6 A chocolate cake takes 2--- hour to bake. Fairy cakes take 1--- hour to bake. How much longer 3 4 does the chocolate cake take to cook? 9 1 - metres left on it. If a fencing contractor uses 842 --- metres in a 7 A roll of wire has 1082 ----10 4 fence, how much wire will be left on the roll?

8 Ramon has 8 2--- metres of material. He uses 5 1--- metres to make a flag and 7 2 banner. How much material is left?

5 -----14

metre for a

9 Jess spent 1--- of her pocket money going roller skating, and used 1--- of her pocket money to 8 5 buy a book. She put the rest of her pocket money in the bank. a What fraction of her pocket money did Jess bank? b If Jess had $160 in pocket money, how much did she spend on each item? 10 Joachim spent 5--- of his pocket money on 8 rollerblades and 1--- of his pocket money 4 on snacks from a supermarket. a What fraction of his pocket money did he spend in total? b What fraction of his pocket money did he have left? c If Joachim had $200 in pocket money, how much did he spend on each item?

74

Maths Dimensions 7

Exploring multiplication of fractions 3H Learning task 3H Kylie had to find the answer to 1--- × 1--- . 2 4 She was having difficulty but managed this diagram.

1 In your own words explain to Kylie what How would you complete the diagram?

1 --2

×

1 --4

means. 1 --2

Kylie was asked to describe a real life situation that would involve finding

2 Demonstrate your understanding of 3 Is

1 --2

×

1 --4

the same as

1 --4

1 --2

×

1 --4

× 1--- . 4

by writing about a real life example.

× 1--- ? Explain your answer using diagrams. 2

4 Use your calculator to determine if your answer is correct. Kim drew the following diagrams to explain how to multiply 3 2 --- × --4 3

3 --4

× 2--- . 3

a Draw a rectangle 4 units tall, because the denominator of a 3 2 --- is 4, and 3 units wide, because the denominator of --- is 3. 4 3

b Colour in 3--- or 9 parts of the rectangle. 4

b

3 9 --- = -----4 12

6 = -----

c Shade in 2--- or 6 parts of the coloured area of the rectangle. c

12 = 1--2

3

2 6 --- = --9 3

d The amount that is coloured and shaded is the result when the two fractions are multiplied.

5 Draw similar diagrams to find the answers to the following: a

3 --4

×

6 --9

b

5 --6

×

2 --5

c

2 --3

×

d

4 --5

×

1 --2

e

1 --4

×

5 --7

f

3 -----10

g

1 --5

×

2 --3

h

3 --4

×

1 --4

i

1 --3

×

3 --4

j

1 --2

×

2 --7

k

1 --3

×

2 --5

l

1 --2

×

2 --5

m

3 --4

×

3 --5

n

2 --3

×

1 --4

o

1 --6

×

2 --3

3 --5

×

3 --7

Chapter 3 Fractions

75

3I

Multiplying fractions To multiply fractions, cancel common factors, multiply the numerators, then multiply the denominators.

Example

Solution

1 Multiply these fractions: a

b

3 --4

5 --6

×

×

c 4×

1

8 --9

1

11 -----25

3 --8

×

3 ----41

×

5 ----6

5 --6

×

4×

3 --8

2

8 ----93

11 -------25 5

×

=

1 --1

=

2 --3

×

=

1 --6

=

11 -----30

5 --6

2 --3

×

11 -----5

1

×

=

4 ----1

=

5 --4

1

3 ----82

×

5 ----62

= 1 1--4

Your calculator can assist you to multiply and simplify fractions.

Example

Solution

2 Simplify the fraction

17 ------ . 68

Enter

17 -----68

into your calculator using the

fraction button. Now press ENTER . 17 -----68

=

1 --4

Exercise 3I 1 a Copy and complete the table below by using a calculator with a fraction button to determine the answers: Question 1 1 --- × --5 2 1 7 --- × -----5 10 3 3 --- × --4 7 5 5 --- × --6 7

76

Maths Dimensions 7

Answer

3I b Write a sentence or two to explain any patterns or strategies you can see between the question and the answer. Discuss your ideas with your group members or others in the class. c Using diagrams to represent multiplication of fractions, describe the method your calculator uses to calculate the answers. 2 Find the answers to the following (remember to simplify your answer fully): a

3 --4

×

8 --9

b

1 --3

×

9 -----10

c

2 --5

×

10 -----11

d

5 --8

×

3 --5

e

2 --7

×

1 --2

f

4 --5

×

10 -----13

g

5 --6

×

7 -----10

h

2 --3

×

4 --5

i

7 -----12

j

2 --9

×

6 --7

k

1 --2

×

4 --9

l

2 --5

×

2 --3

×

4 --5

×

1 --6

×

1 --4

3 Calculate the answers to the following (remember to simplify your answer fully): a 4×

b 6×

3 --8

1 --9

c

e 2 1--- × 4

f

1 1--- × 9

× 4 1---

j

2 1--- × 2 2---

2

2 --3

i

2

3

3

2 --3

×0

d

×1

g 5 × 1 1---

h 3 × 2 2---

k 1 4--- ×

l

4

7

8 --9

5

5

2 --3

7 --9

× 3 2--7

4 Evaluate the following: a 2 1--- × 1 1--- × 6

2

b 4 3--- × 2 2--- × 2 6---

3 --4

4

-----e 2 4--- × 3 4--- × 1 13 5

i

6×

7

3 --4

20

× 2 1--3

3

7

f

-----1 1--- × 2 1--- × 1 11

j

5 --6

6

4

14

× 10 × 1 4--5

c 2 1--- × 1 3--- ×

5 --6

d 1 3--- × 1 7--- × 3

g 4 --1- × 1 3--- ×

1 --3

h

2 --5

l

4 1--- × 5 1--- × 1 1---

4

2

5

5

4

k 2 2--- × 4 1--- × 1 4--3

2

5

9

× 3 1--- × 1 1--2

8

4

3

6

5 Kelvin wants to buy a polar fleece jacket with a price tag of $78. The shop has a sale with 1 --- off the tag price. How much will Kelvin have to pay for the jacket? 3

6 Margaret takes 3--- of an hour to do her Maths project. It takes Carmel 2 1--- times longer than 4 4 Margaret to do the project. How long does it take Carmel? 7 A family in a car travels at 85 kilometres per hour. How far will they go in 2 5--- hours? 6

8 Workers pay

8 -----25

of their wages in tax:

a If a worker’s wage is $520, how much money is paid as tax? b How much money would the worker take home? 9 The ingredients needed to make fairy cakes are listed below: 1 1--- cups of flour

1 teaspoon of vanilla essence

1 --2

1 --- kg of butter 8 3 ------ litre of cream 10

2

cup of jam

3 eggs

2 --3 1 --4

cup of castor sugar cup of milk

a How much of each ingredient would be required to make a double mixture? b How much of each ingredient would be required to make half the mixture? Chapter 3 Fractions

77

3J

Dividing fractions Change all mixed numbers to improper fractions first. Now change the division sign to a multiplication sign and tip the fraction after the sign upside down. Proceed as for multiplication. Cancel down whenever possible and change any answers that are improper fractions back to mixed numbers.

Example

Solution

Divide these fractions:

a

÷

3 --4

1 --2

b 1 2--- ÷ 3

÷

3 --4

1 2--- ÷

5 --6

=

3 --- × 4 = 3--- × 2 = 3--2 = 1 1--2

1 --2

3

=

5 --6

= = =

2 --1 1 --1

÷

5 --3 5 --3 1 --1 2 --1

× ×

5 --6 6 --5 2 --1

=2

c 1 7--- ÷ 2 1--8

1 7--- ÷ 2 1--- =

2

8

2

= =

15 -----8 15 -----8 3 --4

÷ ×

5 --2 2 --5

Exercise 3J 1 Find the answers to the following: a

3 --4

÷

1 --8

b

2 --5

÷

8 -----15

c

1 --2

e

1 --3

÷

4 --9

f

6 --7

÷

7 --9

g

3 -----14

i

4 --9

÷

5 --6

j

3 -----10

÷

11 -----12

k

4 --7

b

8 -----21

÷

2 --7

c

3 --4

f

9 -----14

1 ÷ 1 -----

g

12 -----21

j

2 1--- ÷

k

4 --7

÷

6 -----11

÷

d

1 --8

3 --7

h

2 -----15

÷

10 -----21

÷

9 -----14

l

7 -----12

÷

5 --6

÷

9 -----16

d

1 --8

÷

4 --7

2 Calculate: a

÷

2 --5

4 -----15

e 1 1--- ÷ 4

i

78

5 -----12

5 --6

÷ 1 1--4

Maths Dimensions 7

4

17

3 --4

÷ 1 2--7

÷

9 -----14

÷

4 --7

4 - ÷ h 2 ----11

l

4 1--- ÷ 2

5 --6

1 -----22

3J 3 Evaluate the following: a 2 1--- ÷ 4 1---

b 1 5--- ÷ 1 1--3

c 1 1--- ÷ 1 2--3

d 1 1--- ÷ 3 1---

e 6 1--- ÷ 2 4---

f

1 5--- ÷ 1 1---

g 2 4--- ÷ 2 1---

h 2 2--- ÷ 2 1---

j

4 4 1--- ÷ 2 -----

5 k 6 3--- ÷ 1 -----

l

c 12 ÷

d 8÷

4 --7

h 4÷

1 --3

2

6

8

i

7

5

1 3 3--- ÷ 4 ----4

11

7

2

3

11

9

7

4

28

4

4

2

5

2

-----1 8--- ÷ 1 31 9

54

4 Evaluate the following: a 2÷

b 8÷

1 --8

8 --5

e 16 ÷

4 --9

f

21 ÷

i

14 ÷

7 --6

j

7÷

m

2 --5

÷4

n 8÷

q 1 1--- ÷ 6

7 --9

3 --7

21 -----12

k 27 ÷

2 --7

o

1 r 9 ÷ 1 -----

4

g 9÷

17

3 --4

6 -----11

9 --8

l

÷6

26 ÷

p 8÷

s 20 ÷ 1 2--7

13 -----6

4 --5

4 - ÷2 t 2 ----11

5 Brittany has 2 1--- cakes to divide between 8 people; what fraction does each person receive? 2

6 Nancy’s chooks eat 1 1--- kilograms of wheat each day. How many days will a 40-kilogram 4 bag of wheat last? 7 A party shop sells 25-metre rolls of paper tablecloth. a If the tables are 2 1--- metres long, how many tablecloths can be cut from the roll? 4

b How much paper is left over? 8 An orchard worker packs apples into 1 1--- -kilogram bags. If a bin contains 10 3--- kilograms of 5 4 apples, how many bags can be packed from a bin of apples? 9 A developer buys 16 2--- hectares of land which is to be subdivided into 2--- -hectare house 5 5 blocks. How many blocks will there be? 10 A wheat silo holds 5--- tonne of wheat. The livestock on the 6 1 - of the silo and the remainder is sold. What farm eat ----12 weight of wheat, in tonnes, is used to feed the livestock? 11 The West family has two water tanks with 40 5--- kilolitres 8 of water in each tank. a How many kilolitres of water do they have in total? b If they use 2 2--- kilolitres of water per week, how long will 3 it take to empty the tanks if it does not rain? c If there are four people in the West household, how much water does each person use per week, on average? Chapter 3 Fractions

79

3K

Fractions of quantities The word ‘of’ can be replaced with the multiplication operation.

Example

Solution

1 Find 1--- of 12.

1 --4

4

of 12 = =

1 12 --- × -----4 1 12 -----4

=3

2 Find 3--- of 20.

3 --5

5

of 20 = =

3 --5 3 --1

× ×

= 12

3 Find 1--- of 4 2--- . 2

1 --2

5

of 4 2--- = 5

= = =

1 --- × 2 1 --- × 1 11 -----5 2 1--5

20 -----1 4 --1

22 -----5 11 -----5

Exercise 3K 1 Find: a

1 --4

of 16

b

1 --6

of 18

c

1 --9

of 18

d

2 --- of 5

e

3 --4

of 12

f

2 --5

of 20

g

5 --6

of 30

h

3 -----10

i

2 --5

of 5

j

1 --2

of 8

k

3 --5

of 40

l

3 --8

of 64

m

2 --3

of 45

n

4 --5

of 35

o

3 --7

of 42

p

5 --8

of 64

q

4 --9

of 108

r

7 --8

of 96

s

1 --3

of 60

t

11 -----12

15

of 20

of 132

2 Evaluate the following:

80

a

2 --3

of $60

b

4 --5

of $400

c

2 --9

of $360

d

1 --4

of $1000

e

1 --2

of 500 metres

f

2 --5

of 200 metres

g

5 --6

of 300 kilometres

h

2 --7

of 14 centimetres

i

1 --3

of 60 minutes

j

3 --4

of 15 minutes

k

3 --8

of 40 seconds

l

1 --3

of 24 hours

m

7 --8

of 200 metres

n

5 --6

of $1200

o

7 -----12

p

7 -----10

q

2 --5

of 4800 sheep

r

7 --9

of 4000 lollies

Maths Dimensions 7

of $14 052 of 364 950 ML

3K 3 If Ali slept for 1--- of the day, for how many hours did he sleep? 4

4 Bronwyn took 2--- of her box of chocolates to school to share with friends. If the box 3 contained 21 chocolates when full, how many did Bronwyn take to school? 5 Brendan caught 24 fish but he had to throw 5--- back because they were undersized. How 8 many fish did Brendan take home?

6 Travis earns $87 from his newspaper round. a He decides to save 1--- of his money. How much will he save? 2

b He uses 1--- of his money to buy a DVD. What is the price of the DVD? 3

c The last 1--- of his money he puts in his wallet. How much is that? 6

7 Lime is spread on a farm at 1--- a tonne per hectare. 2

a If the Notts have 16 hectares, how much lime do they need? b The Brackens have 26 hectares, how much lime do they need to spread? c The Townsends have 18 2--- hectares, how much lime do they require for their property? 3

------ metres of polar fleece material to make three tops. Calculate how 8 Mikayla purchased 7 13 20 much material is needed for each top.

9 Amanda wishes to make 3--- of the following butterscotch sauce recipe, which requires 4 440 g brown sugar, 500 mL of cream and 250 g of butter. How much of each ingredient does she need?

Chapter 3 Fractions

81

3L

Squares and square roots of fractions Squaring a fraction is multiplying the fraction by itself, so the rules of multiplication apply. Mixed numbers must be changed into improper fractions. Always cancel down whenever possible and change an answer that is an improper fraction back to a mixed number.

Example

Solution

1 Evaluate: a

b

⎛ 4---⎞ ⎝ 5⎠

2

⎛ 4---⎞ ⎝ 5⎠

2 ⎛ 2 3---⎞ ⎝ 4⎠

2

=

4 --5

=

16 -----25

2 ⎛ 2 3---⎞ ⎝ 4⎠

×

4 --5

⎛ 11 ------⎞ ⎝ 4⎠

=

2

=

11 11 ------ × -----4 4 121 = --------16 9 = 7 ----16

When finding the square root of a mixed number, the mixed number must first be changed into an improper fraction. Finding the square root of a fraction is the same as finding the square root of the numerator and the square root of the denominator. The square root can only be found without the use of a calculator if the numbers are perfect squares. Perfect squares are numbers such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 …

Example

Solution

2 Evaluate: a

4 --9

4 --9

= =

b

9 1 ----16

4 ------9 2 --3

9 - = 1 -----

25 -----16

=

25 ---------16 5 --4 1 1--4

16

= =

3 Use your calculator to find

82

Maths Dimensions 7

3 --- . 8

3 --8

= 0·612 correct to 3 decimal places

3L Exercise 3L 1 Find the answers to the following: a

⎛ 1---⎞ ⎝ 4⎠

2

b

⎛ 2---⎞ ⎝ 5⎠

2

c

⎛ 5---⎞ ⎝ 6⎠

2

e

⎛ 2---⎞ ⎝ 3⎠

2

f

⎛ 1---⎞ ⎝ 2⎠

2

g

9 ⎞2 ⎛ ----⎝ 10⎠

i

5 ⎞2 ⎛ ----⎝ 12⎠

j

7 ⎞2 ⎛ ----⎝ 11⎠

k

⎛ 4---⎞ ⎝ 7⎠

2

d

⎛ 1---⎞ ⎝ 3⎠

2

h

⎛ 3---⎞ ⎝ 8⎠

2

l

⎛ 8---⎞ ⎝ 9⎠

2

2 Evaluate the following: a

2 ⎛ 1 1---⎞ ⎝ 2⎠

b

2 ⎛ 2 1---⎞ ⎝ 4⎠

c

2 ⎛ 1 2---⎞ ⎝ 7⎠

d

2 ⎛ 1 2---⎞ ⎝ 5⎠

e

2 ⎛ 3 1---⎞ ⎝ 3⎠

f

2 ⎛ 1 2---⎞ ⎝ 9⎠

g

2 ⎛ 2 1---⎞ ⎝ 5⎠

h

2 ⎛ 4 1---⎞ ⎝ 6⎠

i

2 ⎛ 1 7---⎞ ⎝ 8⎠

j

2 ⎛ 2 5---⎞ ⎝ 6⎠

k

9 ⎞2 ⎛ 1 ----⎝ 10⎠

l

2 ⎛ 2 6---⎞ ⎝ 7⎠

3 Evaluate the following: a

1 --4

b

1 --9

c

1 -----25

d

9 -----25

e

81 --------100

f

36 -----49

g

144 --------169

h

64 --------100

i

81 --------121

j

4 -----81

k

50 --------338

l

98 --------162

4 Evaluate the following mixed numbers: a

2 7---

b

2 1---

c

1 5 -----

d

-----1 11

e

6 1---

f

11 1---

g

-----5 19

h

12 1---

i

6 3 -----

j

1 7---

k

20 1---

l

2 2 -----

9

25

25

9

4

4

4

4

16

9

25

49

5 Use your calculator to find the answers to the following, correct to 2 decimal places: a

4 --7

b

6 -----17

c

1 --3

d

1 --5

e

8 --9

f

7 --8

g

2 3---

h

5 1---

i

8 1---

j

9 6 -----

k

5 4---

l

1 1---

4

10

3

7

2

8

Chapter 3 Fractions

83

3M

Order of operations The order of operations used for whole numbers in Chapter 1 also applies for fractions.

•••••••••••••

B Work out the calculations inside the brackets first. If there is more than one operation inside the brackets, then they must also follow the rules of BODMAS. O If the question contains a fraction ‘of’ a number, then this is calculated next. D Work out the division and multiplication calculations, working across the page M from left to right. A Work out the addition and subtraction calculations, working across the page S from left to right.

Example

Solution

Evalulate the following:

a

⎛ 2--- + 1---⎞ × 6--- – 1--⎝ 3 4⎠ 7 7

⎛ 2--- + 1---⎞ × 6--- – 1--⎝ 3 4⎠ 7 7

=

⎛ 2--- × 4--- + 1--- × 3---⎞ × 6--- – 1--⎝ 3 4 4 3⎠ 7 7

=

6 1 8 3 ⎛ ----- + ------⎞ × --- – --⎝ 12 12⎠ 7 7 11 6 1 ------ × --- – --12 7 7 11 2 ------ – -----14 14 9 -----14

= = =

b

2 16 3 --- + ------ ÷ --3 25 8

=

2 16 3 --- + ------ ÷ --3 25 8

= = = = =

2 4 8----- + --- × 3 5 3 2 5 32 --- × --- + -----3 5 15 10 32 ------ + -----15 15 42 -----15 -----2 12 15 2 4--5

Exercise 3M 1 Evaluate the following:

84

a

1 --2

of 8 + 2

b

2 --3

of 9 − 2

c 5+

e

2 --7

of 14 − 3

f

2 --5

of 15 − 2

g 3 + 4 3--- × 2

h 2×

8 -----25

i

2 2--- + 1 × 2 2---

j

6+

k 9−

l

2 --5

3

3

Maths Dimensions 7

1 --3

of 18 ÷ 3

2 --5

of 15 4

1 --2

of 6 ÷ 2

d 26 −

4+

5 --6

of 24 ÷ 1 1--5

of 25 ÷ 7

3M 2 Evaluate the following: of 8 −

2 --3

d 6 − 2 1--- ÷

5 --6

a

1 --2

2

g 3 1--- − 8

j

6+

1 --4

8

6 --7

4

2 --3

e

⎛ 2 1--⎝ 2

9 -----10

−

+

1 3 --- ⎞ 3⎠

⎛ 1 1--⎝ 2

2⎞ --5⎠

+ 1 --2

×

13 -----4

c 6 2--- + 1 1--- × 2 2---

1 --4

3

×

2 --7

÷ 1 1---

8 -----25

4

of 12 −

2 --3

×

b

h 2 1--- ×

of 4 3---

of 8 −

1 --3

m 4 5--- +

×

5

2

f

6 2--- ÷ 3

4 --9

− 2 2---

i

2 1--- × 2

2 --3

+ 1 3--- × 2 2---

l

2 1--- −

2 --3

×

k

2 --5

×

2 --5

−

1 -----10

×

n

8 --9

÷

4 --3

+

2 --3

×2

o

3 --4

×

6 --5

b

1 --2

of

2

c

3 --8

+

⎛ 1---⎞ ⎝ 6⎠

e

⎛ 5---⎞ ⎝ 6⎠

f

7 ⎞2 ⎛ ----⎝ 12⎠

i

1 -----25

l

2 1--- ×

c

3 --4

f

2 ⎛ 1 2---⎞ ⎝ 3⎠

i

8 --9

l

2 ⎛ 3 5---⎞ ⎝ 6⎠

o

1 --5

3 --5

5

8

5

4

3

÷ 2 2---

5 --8

3

+

1 -----10

+1

2

×

9 -----11

3 Evaluate the following: a

⎛ 2---⎞ ⎝ 3⎠

2

+

d 1 1--- + 2

36 -----49

g j

5 --8

÷ 1 1---

3 --4

8

⎛ 3---⎞ ⎝ 4⎠

2

−

1 --3

+ 2 1--- × 1 2--4

3

2 1 - × 2 --÷ 2 2--- + 2 ----49

3

k

⎛ 1 1--⎝ 2

b

1 --2

⎛ 2---⎞ ⎝ 5⎠

−

+ 4 2--- ÷ 1 4---

h 3 2--- ÷

3

3

2

3 --4

+

3

5

4 -----81

− 4 3---

2⎞ --5⎠

− 1 7---

4

9

−

5 --8

÷ 2 1--2

÷ 1 2--- + 2 4--3

4

5

÷ 12 1---

4 -----50

4

4 Evaluate: a

⎛ 1---⎞ ⎝ 4⎠

2

d 9 1--- + 2

+

g

5 --6

j

2 ⎛ 2 7---⎞ ⎝ 8⎠

2

⎛ 2---⎞ ⎝ 3⎠

× 2 --3

2

of 5 2--- × 10 1--9

100 --------144

8

− ⎛⎝ 1 1---⎞⎠ 4

+ 1 1--- × 2

m

42 1--- ÷

•••••••••••••••••••••

BODMAS

4

+ 5 1---

26 -----27

+

2

e

of 6 1--- + 49 -----81

×

3 --5

−

× 2 2---

4 --5

4

1 --4

3

×

12 -----13

2

2 7 ------ − ⎛ 2 ---⎞ + 2 --h 8 19 ⎝ ⎠ 20

3

3 --4

k

2 ⎛ 2 1---⎞ ⎝ 4⎠

1 --4

n

⎛ 3---⎞ ⎝ 4⎠

2

÷

⎛ 1--⎝2

9

+

5⎞ --6⎠

2

+ ⎛⎝ 1 1---⎞⎠ − 4

7 --8

6 - − of 3 ----25

−

1 --2

×

2 --5

------ − ÷ 1 19

1 --5

1 --4

81

×

2 --9

2

− ⎛⎝ 1 1---⎞⎠ × 2

+

3 --4

64 -----81

The operations are completed in the order in which you go up these stairs. ■

Brackets first.

■

Fractions or powers of are on the second step, so calculate these next.

■

Multiplication and division are on the third step.

■

Addition and subtraction are on the top step. Calculate these last.

Remember that for division and multiplication, and for addition and subtraction, which are on the same steps, work across the page from left to right. Chapter 3 Fractions

85

Puzzles 1 Express the shaded areas as fractions to answer the riddle:

Did you hear about the man who fell into an upholstery machine?

C Fraction Red

E Fraction Red

H Fraction Red

L Fraction Red

D Fraction Blue

F Fraction Blue

I Fraction Blue

O Fraction Blue

R Fraction Red

U Fraction Red

S Fraction Blue

V Fraction Blue

T Fraction Green

Y Fraction Green

____ ____ 1 --5

____ ____

2 --5

4 --5

____ ____ ____ ____ ____

1 --3

3 --5

1 --8

3 --4

3 --4

1 --2

____ ____ ____ ____ ____ ____ ____ ____ ____ 4 --9

2 --5

1 --6

1 --4

3 --8

2 --5

4 --9

2 --5

5 --6

2 Complete the equivalent fractions to answer the riddle:

Did you hear about the clairvoyant cow?

A 8 1 --- = ----- = ---2 4 C

E 3 2 1 --- = ----- = ------ = ---3 D 12 F

H 2 1 --- = ------ = --5 15 I

P 20 5 --- = ------ = -----6 18 R

T 9 6 3 --- = ---- = ------ = ----7 28 S U

L 2 O 1 --- = ------ = ----- = -----4 20 N 28

____ ____ 10 12 ____ ____ ____ 12 3 4

____ ____ ____ ____ ____ 16 7 21 5 6

____ ____ ____ ____ ____ ____ 9 21 12 21 24 4

____ ____ ____ ____ ____ ____ ____ 15 2 14 12 21 24 4

86

Maths Dimensions 7

____ ____ ____ 14 4 4 ____ ____ ____ 2 8 6

Ch3 3 Calculate the answers to the following problems to find the answer to the riddle:

What do you get when you cross a photocopying machine with a keyboard? a Simplify these fractions using addition or subtraction:

A

1 3 --- + --4 4

E

6 4 --- – --7 7

B

2 1 1 --- + --7 2

G

3 9 8 ------ – 4 --5 10

C

1 4 2 --- + 4 --3 9

H

1 1 5 7 --- – 2 --- – 3 --2 4 8

D

1 4 3 5 ------ + 2 --- + 3 --2 5 10

I

3 4 – --4

b Simplify these fractions using multiplication:

N

3 8 --- × --4 9

S

1 --- of 20 4

O

2 6 × --3

T

3 --- of 500 5

P

4 1 1 --- × --5 6

U

1 --- of 63 3

R

1 1 1 4 --- × 5 --- × --3 6 8

V

3 --- of 128 8

____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 2 3 2 ---------3 2--3 2--4 11 3--21 6 7--300 3 1--48 3

7

10

3

5

9

4

7

____ ____ ____ ____ ____ ____ 3 2 --4 3 2--4 ----1 5 3

10

3

Chapter 3 Fractions

87

Applications Phases of the Moon

It takes the Moon 29·531 days to orbit the Earth, starting as a new moon and continuing through the phases of first quarter, full moon, last quarter and then back again to another new moon. Research the terms ‘first quarter’, ‘new moon’, ‘last quarter’ and ‘full moon’. How does each of these terms relate to the Moon’s orbit of the Earth? Calculate how many days of the orbit have passed at each of the Moon’s phases.

Fraction patterns Calculate each of the sums below: 1 --2 1 --2 1 --2 1 --2

+ + + +

1 --4 1 --4 1 --4 1 --4

= + + +

1 --8 1 --8 1 --8

= + +

1 -----16 1 -----16

= +

1 -----32

=

What do you predict will be the answer for the next sum in the pattern? What happens if the addition sign is replaced with a subtraction sign? 1 --2 1 --2 1 --2

− − −

1 --4 1 --4 1 --4

= − −

1 --8 1 --8

= −

1 -----16

=

Describe the pattern in your own words.

88

Maths Dimensions 7

Ch3 Fraction bingo This is to be played in a group, with one person calling the fractions. Players draw the following 5 × 5 grid in their books. The players then write one fraction from the list below in each square. Write each fraction once only. The caller then reads out fractions equivalent to those in the list below. Players must simplify the fractions by cancelling common factors, then cross out each fraction when it is called.

3 7 9 1 1 2 1 3 1 2 3 4 1 5 1 2 3 4 5 6 1 3 5 7 1 2 4 5 7 8 1 --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , --- , ------ , ------ , ------ , ------ , 2 3 3 4 4 5 5 5 5 6 6 7 7 7 7 7 7 8 8 8 8 9 9 9 9 9 9 10 10 10 10 1 2 3 5 1 5 2 1 2 1 3 ------ , ------ , ------ , ------ , ------ , ------ , ------ , ------ , ------ , ------ , ------ . 11 11 11 11 12 12 13 15 15 20 20

The winner is the first person to call out BINGO when they have crossed out a straight line horizontally, vertically or diagonally.

Chinese tangrams The seven-piece Chinese tangram can be put together to form a square as shown. Calculate what fraction of the whole square each piece of the tangram represents.

1 5 2

6 7 3

4

Squares a Count the number of squares in the diagram. b Calculate the size of each smaller square as a fraction of the large square. c Calculate the size of each triangle as a fraction of the large square.

Chapter 3 Fractions

89

Enrichment 1 Modern fractions can easily be converted to Egyptian or unit fractions using this method. To convert 5--- to an Egyptian fraction, find the largest unit fraction that is smaller than 5--- . 9

9




1 --4

1 --3

5 --9

to start the sum. −

=

1 --3

5 --9

−

3 --9

=

2 --9

is not a unit fraction so this needs to be converted further.

2 --9

+?

To find ?

2 --9

−

=

1 --5

10 -----45

−

=

9 -----45

1 -----45

As a unit fraction is left over, the sum can now be completed: 5 --9

=

+

1 --3

1 --5

+

1 -----45

Convert the following fractions to Egyptian fractions:

a

3 --4

2 --5

b

2 Which is larger,

7 -----10

c

2 --3

d

3 -----10

e

3 --7

f

5 --6

or 5--- ? 7

You could use your calculator or division to convert the fractions to decimals. 7 -----10

= 0·7 and

= 0·71

5 --7

Or use equivalent fractions: 7 -----10

=

49 -----70

and

5 --7

=

50 -----70

However, Egyptian fractions also give the answer effectively: 7 -----10 5 --7

= =

1 --2

+ +

1 --2

1 --5

1 --5

and +

1 -----70

By using the sum of different unit fractions it can be seen that

5 --7

is larger by

1 ------ . 70

Use Egyptian fractions to write each of the following fractions as a sum of different unit fractions. Use this information to state which fraction is larger in each of the pairs:

a

5 --6

or

8 --9

b

8 --9

or

9 -----10

c

3 --4

or

4 --5

d

4 --9

or

5 -----11

e

4 --5

or

3 Copy and complete the following fraction triangles and square: a b c

7 --9

f

or

6 3–4

9 1–4

3 3–4

4 2–5

6

5 1–4

7 5 –– 12

8 4 4–5

90

5 --8

Maths Dimensions 7

1 4 –– 12

7 -----11

Ch3 4 Complete the following magic squares. You will need to use your fraction skills: a

b

1 --4 7 --8 3 --4

1

1 --3

7 -----15

c

d

2 --3

6

3

3 --5

4

5

3 1--2

3

4 1---

1 1---

2 1---

4

2 1---

8

3 1---

1

2

2

2

1

4 1--4

2 3---

2

4

4

4

Can you make up your own 3 × 3 magic square with fractions?

5 Use the diagram of the tennis court to answer the following questions: 6 2– m 5

5 1– m 2

8 1– m

11 m

5

1 4 –– m 10

1 2– m 5

23 4– m 5

a b c d e

Find the area of the service box shaded blue. Find the area of the entire tennis court, including the doubles boxes. What fraction of the total court area is the shaded service box? What fraction of the total court area is the singles court? It is outlined in red. Compare the area of court covered by players in different sports. For example, what area of a netball court is covered by the Centre player compared with that covered by the Goal Shooter? What area of a hockey ground is covered by a Wing player compared with that covered by a Centre Back?

6 The fourteen-piece Greco-Roman tangram called the Loculus of Archimedes or the stomachion is made from a rectangle in which the length is twice as long as the width. What fraction of the whole rectangle is represented by the following pieces. a 8 and 9 1 14 b 1, 2 and 3 4 c 2, 3, 5, 6, 7, 9 and 10 2 3 13 8 d 8 to 14 12 11 6 5 9 e 4 to 11 10 7 f 12 Chapter 3 Fractions

91

Revision Exercise 3A 1 Express the following diagrams as fractions: a b

c

2 Simplify the following fractions: 20 -----60

a

b

32 -----56

35 -----45

c

3 State whether the following are proper fractions, improper fractions or mixed numbers. Label the numerator, denominator and whole number where applicable: 16 -----5

a

b 4 1---

c

2

1 --3

Exercise 3B 4 Fill the gaps: 4 2 18 --- = ------ = ------ = -----25 5

a

b

3 36 --- = ------ = ------ = -----7 21 49

5 For each pair of fractions state which is larger: 2 3 --- , -----5 10

a

b

3 5 --- , -----7 11

c

1 1 --- , --9 6

d

5 5 --- , --8 7

d

37 -----8

6 Change the following improper fractions into mixed numbers: 12 -----5

a

b

11 -----3

c

20 -----7

7 Change the following mixed numbers into improper fractions: a 2 1---

b 4 2---

4

c 3 4---

7

8 d 1 -----

5

11

Exercises 3C and 3D 8 Find the answers to the following: a 1 2--- + 2 3--5

5

b

2 --7

+

4 --7

c 5 1--- +

b

3 --4

+

5 --6

c 2 7--- + 3 1---

8

9 5 - + 1 -----d 3 -----

3 --8

11

11

9 Add the following: 4 --7

a

+

1 --3

8

2

d 1 4--- + 3 4--5

9

10 The Wade family has different types of flour to use in their bread-making machine. They currently have 4 1--- kg of light rye flour, 3 5--- kg of dark rye flour, 6 kg of wholemeal flour and 8 8 2 7--- kg of white flour. How many kilograms of flour do they have altogether? 8

11 Suzie buys 2 3--- kg of apples, 1 --1- kg of bananas and 1--- kg of oranges. What is the total weight 2 4 3 of her fruit?

92

Maths Dimensions 7

Ch3 Exercises 3F and 3G 12 Subtract the following: a

7 --8

−

3 --8

9 ------ − 1 -----b 2 17

e

2 --3

−

4 --7

f

20

c 5−

20

4 1--- − 1 2--2

d 3 1--- − 2

4 -----11

2

g 6 − 2 5---

5

h 7 1--- − 3 7---

8

4

9

3 7 - metres tall. His younger brother Wayne is 1 ------ metres tall. Who is the taller 13 Michael is 1 ----10 10 and by how much?

14 Maria has 2 2--- metres of material but she needs 4 4--- metres of material to make a dress. How 5 5 much more material should Maria buy?

Exercise 3I 15 Evaluate: a

2 --3

×

7 --8

b

3 --4

×

10 -----15

×

c 5 × 2 1--- ×

12 -----25

2

16 James has 6 bottles of soft drink which each hold have altogether?

3 --4

d 1 1--- ×

2 --7

4

2 --5

× 3 1--6

of a litre. How many litres does James

17 A fruit punch mixture needs 2--- litre of orange juice, 1--- litre of pineapple juice and 2 1--- litres 5 3 2 of lemonade. If Ivan wants to make a mixture which is 4 1--- times this volume, how much of 2 each ingredient would he need?

Exercise 3J 18 Evaluate: a

2 --5

÷

b 2 1--- ÷

3 --8

4

c 3 5--- ÷ 2 2---

3 --5

6

4 - ÷4 d 2 -----

9

11

Exercise 3K 19 Calculate: a

1 --4

of 32

b

2 --5

b

⎛ 2---⎞ ⎝ 7⎠

c

3 --8

2

c

2 ⎛ 1 5---⎞ ⎝ 6⎠

d

2 ⎛ 2 1---⎞ ⎝ 5⎠

9 --------100

g

9 7 -----

h

-----1 19

of 40

of 24 hours

Exercise 3L 20 Evaluate: a

⎛ 3---⎞ ⎝ 4⎠

2

16 -----49

e

f

16

81

Exercise 3M 21 Evaluate: a

2 --3

×

4 --5

+ 6 1--3

b 8 1--- − 7 ÷ 1 2--2

5

c

⎛ 3---⎞ ⎝ 4⎠

2

−

2 --3

×

7 --8

÷ 1 2--5

Chapter 3 Fractions

93

CHAPTER

4 Decimals and Percentages

Decimals and Percentages Decimals and percentages are used in everyday life, for example, when buying goods or borrowing money. Interest rates and some taxation systems such as the GST (Goods and Services Tax) are based on percentages. Percentages are used to describe changes in numbers. For example, a recent study showed that 54% of koalas in the Strzelecki region were located in an area managed by the paper manufacturing industry. If this area continues to be logged at the current rates, then approximately 25% of the habitat available to the Strzelecki koala will be lost in the next decade.

This chapter covers the following skills: • Revising place value, notation and estimation of decimals • Comparing decimals • Rounding decimals • Addition, subtraction, multiplication and division skills • Comparing decimals to fractions and percentages • Finding the percentage of a number • Multiplying and dividing by powers of 10 • Calculating percentages

4A

Place value and notation The place value columns used when working with whole numbers can be extended to numbers which are smaller than 1. This is done by using a decimal point to separate the whole number from the fraction part. The place value columns are listed in the grid below. Two numbers, 472·35 and 1·206, are included. 472·35 means 4 hundreds + 7 tens + 2 units + 3 tenths + 5 hundredths 1·20 means 1 unit + 2 tenths + 0 hundredths Hundred

Tens

Units

·

Tenths

Hundredths

4

7

2

·

3

5

1

·

2

0

Exercise 4A 1 Copy the table into your workbook then place the following numbers into their correct place value column. Question Hundreds Tens Units · Tenths Hundredths

a

3·12

b

12·890

c

123·5

d

8·5678

e

2·008

f

56·7071

g

0·1004

h

440·6

i

49·003

j

943·761

k

78·0002

l

0·0643

Thousandths

Tenthousandths

2 Write down the place value of the 3 in each of the following numbers: a 23·678 b 12·4378 c 178·9003 d 0·346 e 349·444 78 f 390·897 51 g 14·003 457 h 2·436 887 i 1278·463 76 j 567·890 32 k 2·467 830 l 0·000 035 3 How many decimal places are in each of these numbers? a 16·890 76 b 346·23 c 1·670 e 0·456 f 56·789 789 789 g 1·234 789 30 i 4·987 j 9643·2300 k 7·909 09

96

Maths Dimensions 7

d 90·006 031 h 2·9 l 2·3

4A 4 Write the following words as decimals: a 2 units + 6 tenths c 8 thousandths e 9 ten-thousandths

b 5 units + zero tenths + 7 hundredths d 9 tens + 2 hundredths + 4 thousandths f 7 tenths + 7 thousandths

5 Which is bigger? For each pair of decimals write down the number which is bigger: a 0·543 or 5·43 b 4·96 or 4·69 c 7·6 or 7·06 d 0·014 or 0·14 e 126·896 or 126·986 f 1·074 or 10·47 g 6 hundredths or 0·006 h 7 tenths or 0·17 6 Write the following sets of numbers in ascending order (that is, from smallest to largest): a 2·345, 3·245, 2·435, 2·543, 2·453 b 27·81, 27·18, 2·718, 27·08 c 19·115, 19·511, 19·151, 19·500 d 0·4, 0·000 04, 0·004, 0·0004 e 1·003, 0·997, 1·909, 0·909, 0·907 f 0·989, 0·9999, 0·9909, 0·9199, 0·999 89 7 One day, Malcolm recorded the temperature every 3 hours between 9 am and 9 pm. The results are listed in the table below: Time Temperature

a b c d

9 am

12 noon

3 pm

6 pm

9 pm

23·4°C

25·8°C

28·5°C

24·3°C

17·8°C

At what time was the temperature highest? Which temperature has a place value of 5 tenths? Write in words the temperature at 6 pm. Arrange the temperatures in ascending order.

8 Monique purchased a hat for $27·92. Martin also purchased a hat, which cost $29·72. Mandy’s hat cost 8 hundredths of a dollar more than Monique’s hat. a How many cents is 8 hundredths of a dollar? b What was the price of Mandy’s hat? c Whose hat was the most expensive? 9 At an athletics carnival, the following measurements are recorded in the Year 7 high jump event: Aaron 1·45 m, Joshua 1·85 m, Karl 1·55 m, Matthew 2·05 m. a Who had the highest jump? b Whose jump was the shortest? c Whose jump can be written as 1 units, 8 tenths and 5 hundredths? d List the first, second and third place-getters. e Find the total height of the combined jumps of all four students. f Find the difference between the highest jump and the shortest jump. Chapter 4 Decimals and Percentages

97

4B

Estimation of decimals In Chapter 1 Whole Number Revision, estimation was used as a quick check of calculations to ensure that a careless mistake had not been made. In some circumstances an estimation is more than adequate. Estimation can also be used to check decimal calculations. To make estimates of an approximate answer we need to round off the numbers.

Example

Solution

Evaluate:

a 123·08 × 2·87

123·08 × 2·87 ≈ 120 × 3 = 360

b 4·789 + 6·5

4·789 + 6·5 ≈ 5 + 7 = 12

c 25·897 − 1·234

25·897 − 1·234 ≈ 26 − 1 = 25

d 0·36 ÷ 2

0·36 ÷ 2 ≈ 0·4 ÷ 2 = 0·2

e 41·63 ÷ 6

41·63 ÷ 6 ≈ 42 ÷ 6 =7

Exercise 4B 1 Copy and complete the following table: Question e.g. 124·78 + 23·23

a

0·89 + 14·78

b

235·8907 + 7·008

c

1·78 + 9·45

d

14·95 − 4·12

e

13·67 − 0·87

f

56·9 − 0·767

g

24·89 × 12·89

h

4·55 × 12·09

i

13·95 × 3·004

j

235·97 ÷ 4

k

637·89 ÷ 8

l

80·586 ÷ 9

Approximate question

Estimate answer

Calculator answer

≈ 120 + 20

= 140

= 148·01

2 Estimate the total cost if Hamish purchased four items at $5·99, $4·49, $3·25 and $1·09. 3 At a factory outlet, Milan buys five items costing $79·95, $16·95, $128·50, $43·70 and $69·95. Estimate the total cost of these items.

98

Maths Dimensions 7

Rounding decimals

4C

Decimals are rounded to a set number of decimal places when the decimal continues forever or when a lesser degree of accuracy is needed.

Steps in rounding a decimal Step 1: Count the number of decimal places required and place a vertical line between these digits and the ones that follow. Step 2: Look at the digit to the right of the line. Step 3: If the digit is 0, 1, 2, 3 or 4, then the digit to the left of the line remains the same and the digits after the line can be dropped off. If the digit is 5, 6, 7, 8 or 9, then the digit immediately before the vertical line is increased by 1.

Example

Solution

Round the following decimals to the number of places indicated in the brackets.

a 23·6342 [2 dp]

23·63 | 42

b 1·234 597 5 [4 dp]

1·2345 | 975 The digit after the line is 9, so the digit before the line has one added to it: 1·2346

The digit after the line is 4, so the digit before the line stays the same: 23·63

Exercise 4C 1 Round each of the following to the nearest whole number (i.e. with no decimal places): a 7·8 b 19·7 c 124·5 d 23·85 e 47·13 f 983·054 g 0·9 h 0·47 2 Write each of the following decimals correct to 1 decimal place: a 2·345 b 0·243 c 4·5721 e 0·09 f 689·0812 g 45·89

d 99·8732 h 0·02

3 Write each of the following decimals correct to 2 decimal places: a 23·693 b 12·809 c 25·006 73 e 56·222 351 f 12·1 g 14·999

d 14·5573 h 88·0984

4 Write each of the following decimals correct to 3 decimal places: a 45·2535 b 97·024 78 c 19·6578 e 2·6097 f 107·9999 g 2·679 679

d 14·234 875 345 h 0·3333

5 Write the following decimals correct to the number of decimal places (dp) indicated in the brackets: a 45·876 124 [5 dp] b 0·087 346 23 [7 dp] c 21·469 023 [4 dp] d 0·004 562 [3 dp] e 34·99 [0 dp] f 17·090 457 [4 dp] g 0·7 [3 dp] h 0·142 857 [4 dp] i 43·467˙ [4 dp] j 1·5426 [2 dp] k 9·154 679 [3 dp] l 0·004 672 [4 dp] Chapter 4 Decimals and Percentages

99

4D

Adding decimals When adding decimals ensure that the digits are placed in their correct place value column and that the decimal points are in line. You may like to fill any empty decimal places with zeros. The decimal point should appear in the answer directly beneath the decimal points in the question.

Example

Solution

Add these decimals:

a 54·13 + 3·25

54·13 + 3·25 57·38 1 1

b 14·37 + 1·66 + 23·8

14·37 1·66 + 23·80 39·83

Exercise 4D 1 Add these decimals: a 23·21 + 13·54

b

127·3 + 45·6

c

1·25 + 6·51

d

45·91 + 12·73

e

237·72 + 56·91

f

12·909 + 93·007

g

34·95 + 6·86

h

1·78 + 8·56

i

1·42 + 92·08

2 Write each of the following sums in columns, making sure the decimal points are in line, then find the answer. Check your answer by using estimation: a 0·65 + 3·98 + 5·12 b 12·763 + 45·601 c 104·69 + 23·28 d 129·87 + 42·90 e 14·89 + 342·09 + 5·01 + 12·77 f 160·876 + 2·801 + 0·083 g 1·05 + 4·99 + 120·12 h 19·3 + 10·6 + 250·9 i 136·8 + 568·01 + 3018·34 j 44·82 + 1·7 + 1000·74 + 0·07 3 What is the sum of 245·98, 34·07 and 1·88? 4 Find the total weight for 45·75 kilograms, 16·25 kilograms and 100·2 kilograms. 5 Angela is training to swim in the school 800-metre freestyle event. During one week she swam 0·67 kilometre on Monday, 2·5 kilometres on Tuesday, 1·45 kilometres on Wednesday, 0·85 kilometre on Thursday and 1·8 kilometres on Friday. How far did Angela swim for the week?

100

Maths Dimensions 7

4D 6 Sian went to the supermarket and purchased a packet of jelly babies $1·86, a block of chocolate $3·28, a bottle of soft drink $1·87, and a packet of chips $2·79. How much did Sian spend? 7 Max took five parcels to the Post Office. The parcels weighed 2·75 kg, 0·58 kg, 0·27 kg, 1·8 kg and 0·95 kg. a What was the total weight of his parcels? b What was the combined weight of the heaviest and lightest parcels? 8 Pia purchases some material to make herself some new clothes. She purchases 2·3 metres of denim to make a skirt, 3·1 metres of cord to make a jacket and 1·8 metres of cotton to make a shirt. What was the total length of material purchased? 9 Rochelle purchases fresh The next week her brother fruit and vegetables from Tyrone purchases: the market. She buys the potatoes $6·49 following items: onions $2·51 potatoes $5·99 capsicum $1·27 onions $1·10 pumpkin $0·86 lettuce $0·97 lettuce $0·99 carrots $1·58 apples $3·95 apples $8·24 bananas $3·23 bananas $2·28 oranges $2·95 kiwi fruit $1·36 a How much did Rochelle spend at the market? b How much did Tyrone spend at the market? c Who spent the most at the market? d What was the total cost of the fruit and vegetables for the two weeks? 10 Morgan decides to make spaghetti bolognese as a surprise for his mother. He places some beef mince $4·26, tomato paste $1·89, an onion $0·31, tomatoes $3·64, and spaghetti $2·75 in his shopping basket. He knows there are herbs and parmesan cheese at home. Morgan has taken $9·60 from his money box and has $5·20 in his wallet. a How much will it cost Morgan to make spaghetti bolognese? b How much money does Morgan have? c Does he have enough money to buy the ingredients for the spaghetti bolognese? Explain your answer. 11 Brent bought an exercise book $3·90, a pencil case $1·65, some pencils $3·95, and a packet of highlighters $3·25. What was the total cost? 12 A family paid $252·15 for their July electricity bill. They had been given a discount of $52·75. What was the original cost of their electricity bill? 13 Emilia is 5 years old and weighs 20·65 kg. At birth she weighed 3·19 kg, how much weight has she gained since birth?

Chapter 4 Decimals and Percentages

101

4E

Subtracting decimals When subtracting decimals check that the digits are placed in their correct place value column and that the decimal points are in line, just as you do for addition. You will need to fill any empty decimal places with zeros. The decimal point should appear in the answer directly beneath the decimal points in the question.

Example

Solution

Subtract these decimals:

a 57·86 − 3·41

57·86 − 3·41 54·45

b 14·371 − 8·826

0 13 13 6 11

14·371 − 8·826 5·545

Exercise 4E 1 Subtract these decimals: a 28·76 − 13·54 d

b

187·9 − 15·6

c

10·87 − 7·54

85·81 − 14·23

e

167·32 − 94·96

f

22·72 − 13·07

64·1 6·9

h

9·56 − 8·78

i

g −

−

92·8 9·7

2 Write the following subtractions in columns, making sure the decimal points are in line, and find the answers. Check your answer by using estimation: a 80·98 − 5·12 b 212·96 − 85·61 c 140·39 − 23·20 d 219·87 − 32·44 e 14·3 − 5·7 f 16·81 − 0·08 g 1·99 − 0·14 h 19·97 − 9·29 i 1018·34 − 568·20 j 1164·9 − 246·86 k 176·8 − 12 l 44 − 3·8 3 What is the difference between 34·7 and 21·8? 4 How much bigger is 65·07 than 30·77? 5 What is the difference between 3·76 metres and 121·05 metres? 6 The Chan family spent $143·35 on groceries at the supermarket. How much change did they get from $200? 7 Ballarat recorded a minimum temperature of 5·8°C at 6:10 am. The temperature rose to a maximum of 23·5°C at 2 pm. How much did the temperature rise?

102

Maths Dimensions 7

4E 8 Ten years ago shares in a particular bank were priced at $9·35. They are now priced at $31·12. By how much has the cost of these shares risen? 9 The Frazer family noted the odometer reading before they started on their holiday. They noted it again at the end of each day of travelling. The odometer readings, in kilometres, are given in the table below: Odometer reading Start

108 372·7

Monday

108 818·8

Tuesday

109 391·3

Wednesday

109 900·5

a Calculate the distance travelled on: i Monday ii Tuesday b On which day did they travel furthest? c What was the total distance travelled on their trip?

iii Wednesday

10 Clint has a savings account with the Citizen Bank. His bank statement comes every month and the top section for April is given below. He had $57·34 in the bank as indicated by his opening balance.

CB Citizen Bank Statement begins 1st April 2003 Closing balance $ Savings Account Number: 124 976 334 554 Name:

Clint Person

Branch: Melbourne, VIC Date

Transaction detail

Debit (−)

Credit (+)

Balance

01 Apr 2003

Opening balance

07 Apr 2003

Interest

$4·57

$61·91 cr

Deposit

$10·00

$71·91 cr

$57·34 cr

The bank pays Clint $4·57 interest and Clint deposits $10 in the bank on 7 April. Using the information given below, calculate Clint’s bank balance after each transaction: a Clint is given $20 for his birthday, which he deposits in the bank on 9 April. b On 14 April he places his leftover pocket money of $5·82 in the bank. c Clint needs $29·95 from the bank to buy a new shirt for himself and a present for his mate’s birthday. He withdraws this money on the 20 April. d Clint deposits $25·85 in the bank on 27 April, which he earned doing odd jobs.

Chapter 4 Decimals and Percentages

103

4F

Multiplying decimals Multiplication of numbers with decimals uses the same skills as you practised in Chapter 1.

Example

Solution

Multiply these numbers:

a 6·9 × 3

6·9 3

×

207 20·7

b 24·59 × 0·5

×

There is 1 decimal place in the question, so the answer needs one decimal place.

24·59 0·5

12 295 12·295

There are 3 decimal places in the question, so the answer must have three decimal places.

Exercise 4F 1 Calculate the answers to the following questions by setting the questions out correctly for multiplication. Check your answers by using your calculator: a 23·6 × 7 b 14·9 × 9 c 0·42 × 7 d 124·75 × 4 e 2·354 × 3 f 56·89 × 6 g 9·98 × 9 h 4·874 × 8 i 45·980 × 7 j 9·807 × 8 k 23·9087 × 5 l 19·087 45 × 8 m 7·6 × 0·4 n 41·5 × 2·5 o 5·8 × 9·3 p 25·3 × 1·2 2 Estimate the answers to these questions, then use your calculator to determine the exact answers: a 12·7 × 14 b 34·9 × 23 c 0·67 × 38 d 6·84 × 19 e 99·5 × 72 f 5·78 × 47 g 4·74 × 53 h 134·87 × 21 i 1·76 × 49 j 84·098 × 124 k 78·934 × 87 l 129·57 × 264 m 2·9 × 21·7 n 44·83 × 5·13 o 149·56 × 32·9 p 3·87 × 1·57 3 Find the answers to the following: a 2·6 × 0·7 b 5·6 × 0·9 e 4·1 × 0·8 f 34·2 × 0·6 i 36·98 × 0·4 j 1·235 × 0·5 4 Calculate the following: a 0·046 × 0·6 d 0·0033 × 0.08 g 0·000 901 × 0·004

104

Maths Dimensions 7

c 9·4 × 0·3 g 96·2 × 0·8 k 3·604 × 0·6

b 0·0065 × 0·7 e 0·005 × 0·008 h 0·067 × 0·009

d 5·9 × 0·4 h 14·8 × 0·9 l 67·908 × 0·7 c 0·000 06 × 0·3 f 0·81 × 0·06 i 0·43 × 0·007

4F 5 Find the answers to the following: a 13·7 × 1·7 b 44·9 × 3·9 e 54·09 × 6·9 f 23·63 × 3·5 i 66·81 × 1·25 j 84·23 × 1·57

c 15·8 × 5·9 g 46·92 × 7·8 k 123·08 × 2·87

d 12·57 × 9·7 h 10·56 × 4·3 l 44·89 × 3·49

6 a Find the product of 45·9076 and 234·5613. b What is 50·8076 times 23·569? c What is the product of 1208·34 and 0·3266? d Multiply 19·0005 and 45·909 090 9. 7 Gordan sold a bull at the market. The bull weighed 856·5 kilograms and the butcher paid him $1·38 per kilogram. How much did Gordan receive for his bull? 8 A plumber charges $38·50 per hour. a The Fletcher family decides to have the bathroom renovated. The plumber spends 4·5 hours connecting pipes. How much will the Fletcher family pay the plumber? b The plumber then unblocks a drain for the Tan family, which takes 0·75 hour. How much will the Tan family pay? c How much did the plumber earn for the day? 9 Marcia filled her car with 42·72 litres of unleaded petrol which costs 116·7 cents per litre. How much did Marcia pay for the petrol? 10 Frank is concreting his garage and orders 13·75 cubic metres of ready-mixed concrete which costs $127·50 per cubic metre. How much will it cost Frank to concrete his garage? 11 Membership to the local tennis club costs $20·50 for the year from January to December. Myra wants to join at the end of May, so the club charges her for 7 months, which is approximately 0·58 of the year. How much will Myra pay for membership? 12 Candice purchases 3·65 metres of material to make a dress. If the material costs $12·99 per metre, how much did Candice pay? 13 A greengrocer sells bananas for $0·97 per kilogram, Granny Smith apples for $3·29 per kilogram, oranges for $2·78 per kilogram, carrots for $0·99 per kilogram, zucchini for $4·29 per kilogram, potatoes for $1·29 per kilogram and tomatoes for $1·97 per kilogram. a Find the cost of the following items: i 2·3 kilograms of oranges ii 4·76 kilograms of apples iii 0·33 kilogram of zucchini iv 1·78 kilograms of tomatoes v 5·2 kilograms of potatoes vi 1·87 kilograms of carrots vii 3·23 kilograms of bananas b Find the total cost of the fruit and vegetables. 14 Les wants to purchase 25 rolls of netting to use when baling hay. If it costs $273·90 per roll, find the total cost of the netting. Chapter 4 Decimals and Percentages

105

4G

Dividing by whole numbers When dividing a decimal number by a whole number: Step 1: Set the division question out as you would normally for division. Step 2: Do the division, ignoring the decimal point. Step 3: The decimal point is placed in the answer directly in line with the decimal point in the question.

Example

Solution

Divide these numbers:

a 2·4 ÷ 6

0·4 6) 2·4

b 2·36 ÷ 4

0·59 4) 2·36

Exercise 4G 1 Calculate the answers to the following by setting the questions out correctly for division. Check your answers by using your calculator: a 15·6 ÷ 4 b 145·5 ÷ 5 c 97·2 ÷ 3 d 21·84 ÷ 7 e 34·24 ÷ 8 f 67·86 ÷ 9 g 0·3474 ÷ 6 h 146·35 ÷ 5 i 0·44 ÷ 11 j 0·045 ÷ 5 k 237·96 ÷ 6 l 14·64 ÷ 8 m 46·8 ÷ 2 n 8·4 ÷ 12 o 10·25 ÷ 25 p 73·71 ÷ 21 2 Estimate the answers to the following, then use your calculator to determine the exact answers. Round your answers to 2 decimal places: a 47·982 ÷ 12 b 17·765 ÷ 9 c 2·064 ÷ 4 d 4·87 ÷ 2 e 71·14 ÷ 5 f 95·092 ÷ 7 g 21·93 ÷ 11 h 7·034 216 ÷ 8 i 24·2424 ÷ 6 j 177·390 ÷ 9 k 143·618 ÷ 12 l 962·3261 ÷ 8 3 Mrs Quick hasn’t enough food in the house to make lunches for her three children so she decides to give them a school lunch order. She has $9·45 to divide equally between them. How much does each child get to spend on their lunch? 4 Elizabeth is given $9·75 to pay for train tickets to school for 5 days. How much does her ticket per day cost? 5 A packet of 100 cups costs $3·56. How much does each cup cost in: a dollars? b cents? 6 A packet of biscuits costs $3·48 and contains a dozen biscuits (1 dozen means 12). How much does each biscuit cost? 7 An aeroplane travels 1555·5 kilometres in 3 hours. a How far does it travel in 1 hour? b How far does it travel in 15 minutes?

106

Maths Dimensions 7

Dividing by decimals

4H

When dividing a decimal by another decimal, convert the divisor to a whole number. For example, 20 ÷ 5 = 4, 200 ÷ 50 = 4, 2000 ÷ 500 = 4 and 2 ÷ 0·5 = 4. Therefore make the divisor a whole number by shifting the decimal point to the right.

Example

Solution

Evaluate 24·68 ÷ 0·4.

Move the decimal point one place to the right in both numbers to give: 246·8 ÷ 4 61·7 4)246·8 ∴ 24·68 ÷ 0·4 = 61·7

Exercise 4H 1 Calculate the answers to the following: a 2·6 ÷ 0·2 b 4·9 ÷ 0·7 e 0·36 ÷ 0·6 f 0·0081 ÷ 0·09 i 1·21 ÷ 0·011 j 0·0008 ÷ 0·4 m 72 ÷ 0·8 n 0·9 ÷ 0·06 q 0·182 ÷ 0·08 r 0·000 64 ÷ 0·4 2 Evaluate the following: a 0·1374 ÷ 0·03 d 0·256 ÷ 0·04 g 1·1304 ÷ 0·9 j 0·008 76 ÷ 0·06 m 0·048 ÷ 0·0008 p 0·186 ÷ 0·12 s 0·076 53 ÷ 0·004

b e h k n q t

c g k o s

1·08 ÷ 0·9 10 ÷ 0·1 0·0024 ÷ 0·06 0·8 ÷ 0·8 0·42 ÷ 0·7

0·28 ÷ 0·08 0·1806 ÷ 0·7 2·7137 ÷ 0·11 54·69 ÷ 0·06 8·4326 ÷ 0·07 4·593 ÷ 0·006 5·492 ÷ 0·11

c f i l o r u

d h l p t

14·4 ÷ 1·2 18·9 ÷ 0·009 0·025 ÷ 0·005 0·6 ÷ 1·2 6·8 ÷ 0·005

1·58 ÷ 0·005 0·3648 ÷ 0·008 0·002 512 ÷ 0·002 0·5499 ÷ 1·2 0·65 ÷ 0·005 1·4562 ÷ 0·09 3·3912 ÷ 0·9

3 Estimate the following, then check your answers using a calculator: a 10·488 ÷ 2·3 b 0·878 64 ÷ 0·56 c 0·007 65 ÷ 0·17 d 0·46 ÷ 0·02 e 0·368 16 ÷ 0·104 f 42·641 13 ÷ 0·57 g 0·002 34 ÷ 0·13 h 0·367 ÷ 0·4 i 25·2978 ÷ 0·22 4 a Find the result when 56·7902 is divided by 0·08. b What is the quotient of 679·847 ÷ 0·7? 5 A clothing factory worker uses 0·4 metre of cotton to sew on a button. How many buttons can be sewn from a 98-metre reel of cotton? 6 How many pieces of wood 0·3 metre long can be cut from a piece of timber 5·4 metres in length? 7 How many 1·2-litre bottles of soft drink are needed if 22·2 litres is to be added to the punch? Chapter 4 Decimals and Percentages

107

4I Exploring powers of 10 Learning task 4I 1 Copy the table and use your calculator to evaluate the following: Question 1·8 × 10

Answer

Comments

18

The answer is larger than the decimal number in the question and the decimal point has moved one place to the right

0·6 × 10 7·3 × 10 3·68 × 100 5·008 × 100 0·005 × 1000 1·612 × 1000 1·75 × 1000

2 Use the information from the table to answer the following questions: a Describe the effect when a decimal number is multiplied by 10. b Describe the effect when a decimal number is multiplied by a power of 10, say 100 or 1000. 3 Copy the table below and use your calculator to solve the following divisions: Question 40·3 ÷ 10

Answer

Comments

4·03

The answer is smaller than the decimal number in the question and the decimal point has moved one place to the left

50·4 ÷ 10 0·24 ÷ 10 650·48 ÷ 100 346·24 ÷ 100 19·36 ÷ 100 1087·4 ÷ 1000 471·28 ÷ 1000 0·24 ÷ 100

4 Use the information from the table to answer the following questions: a Describe the effect when a decimal number is divided by 10. b Describe the effect when a decimal number is divided by a power of 10, say 100 or 1000.

108

Maths Dimensions 7

4I 5 Find the answers to the following without doing a formal calculation. (Hint: Move the decimal point.) Check your answers by using a calculator: a 12·74 × 10 b 76·54 × 10 c 1·456 × 10 d 5·983 × 10 e 0·65 × 100 f 15·8739 × 100 g 0·3589 × 100 h 0·0067 × 100 i 123·678 × 1000 j 0·008 76 × 1000 k 2·789 04 × 1000 l 42·78 × 1000 6 Find the answers to these questions, then check your answers by using estimation: a 34·987 × 1000 b 65·0098 × 10 000 c 0·000 340 08 × 1 000 000 d 87·764 02 × 10 e 2·857 162 × 100 f 5·679 011 × 100 000 g 235·890 × 100 h 34·125 67 × 1000 i 456·999 87 × 10 000 j 345 × 1000 k 2·348 712 45 × 10 000 000 l 34·78 × 100 000 7 Evaluate: a 3·475 × 500

b 0·341 195 × 30 000

8 Evaluate the following by simply moving the decimal point: a 3·983 ÷ 10 b 123·876 ÷ 100 d 345·98 ÷ 10 e 56·99 ÷ 100 g 980·54 ÷ 100 h 234·123 ÷ 1000 j 6060·98 ÷ 100 k 235·9087 ÷ 1000

c 15·347 × 50 c f i l

145 832·45 ÷ 1000 0·87 ÷ 10 56 732·9 ÷ 1000 34·90 ÷ 10

9 Find the answers to these problems, then check your answers by using estimation: a 23·87 ÷ 1000 b 0·098 ÷ 10 c 234·98 ÷ 100 000 d 1·23 ÷ 10 000 e 67·908 ÷ 10 f 0·876 ÷ 100 g 23 4567·9 ÷ 100 000 h 12 765·4 ÷ 1 000 000 i 12·45 ÷ 10 000 j 345·76 ÷ 10 000 k 3·456 ÷ 100 l 1257·45 ÷ 10 000 m 4·789 ÷ 100 n 3·7652 ÷ 1000 o 19·567 ÷ 10 p 0·098 ÷ 100 q 34·567 ÷ 1000 r 0·0876 ÷ 1000 s 23 456·23 ÷ 1000 t 0·0987 ÷ 100 u 0·105 ÷ 100 000 10 Calculate the following: a 24·86 × 20 d 444·88 ÷ 200 g 3434·55 × 5000 j 8560 ÷ 400 m 1·68 × 2000

b e h k n

124·65 × 500 435·84 ÷ 30 346·84 × 40 9160 ÷ 200 2·95 × 500

c f i l o

19·25 × 50 99·54 ÷ 600 9478·35 × 50 21 570 ÷ 300 10·04 × 400

Chapter 4 Decimals and Percentages

109

4J Decimal Defender Introduction This game provides lots of practise in multiplying or dividing by a power of 10.

Equipment TI-83Plus/TI-84Plus with Decimal Defender App™. TIConnectV1.6™ and Cabri® Jr can be downloaded from the Student CD in the back of this book. For information on other applications such as Decimal Defender see ‘Software Downloads’ on the Companion Website.

Technology activity 4J Press the APPS key and start the Decimal Defender application.

arrow keys to select either multiplication, Use the division or both.

Twelve digits appear at the top of the screen. An equation appears at the bottom of the screen. In the example shown: 943.6 × 10000 = 9436000 This can be written as 9436000.000 The decimal needs to be placed three zeros from the right. Use the arrow key to move the spaceship and the key to fire the decimal place.

arrow

Put the decimal in the wrong place and the number will come down and blow up your ship.

The amount of ‘life’ remaining is shown in the bottom right-hand corner of the screen. Time erodes away your life; put the decimal in the wrong place and your life erodes away a lot faster!

The more questions you get correct, the higher the score. Good luck!

110

Maths Dimensions 7

Fractions and decimals

4K

Every fraction can be changed to a decimal by dividing the numerator by the denominator. The result will be a decimal that has no remainders, or one which has a pattern of remainders.

Example

Solution

1 Change these fractions to decimals: a

3 --8

3 --8

=3÷8 = 0·375

0·375 64 8 3·000

b

1 --9

1 --9

=1÷9 = 0.1˙

0.111˙ 11 9 1·000

) )

The decimal 0·111 . . . is a recurring or repeating decimal. Place a dot above the digit to indicate that it repeats. If two or more digits repeat, use a line above the digits which repeat.

Example 2 Change

2 -----11

Solution 2 -----11

to a decimal.

= 2 ÷ 11 = 0·181 818… = 0.18

0·18181 92 92 11 2·00000

)

Exercise 4K 1 Convert the following fractions to decimals: a

1 --3

b

2 --5

c

1 --4

d

1 --8

e

7 --8

f

5 -----12

g

5 --6

h

3 --4

i

4 --5

j

3 -----10

k

5 --9

l

3 -----11

m

5 --8

n

8 --9

o

5 --6

p

2 --3

q

2 --9

r

1 --5

s

1 --6

t

5 -----11

2 Convert these fractions to decimals: a

16 -----9

b

10 -----3

c

9 --5

d

16 -----11

e

13 -----6

f

17 -----10

g

12 -----11

h

13 -----12

3 Convert these fractions to decimals and round to 2 decimal places: a

4 --7

b

1 --7

c

14 -----3

d

17 -----7

e

4 --9

f

3 --8

g

2 --7

h

15 -----9

Chapter 4 Decimals and Percentages

111

4L

Converting decimals to fractions To convert a decimal into a fraction you need to remember your place value table and your knowledge of fractions, and to follow the steps listed below. Step 1: The digits to the right of the decimal point become the numerator of your fraction. Step 2: The place value of the last digit becomes the denominator. Step 3: Simplify the fraction if possible. Hundreds

Tens

Units

·

Tenths

Hundredths

0

·

3

5

1

·

2

0

Example

Thousandths

6

Solution

Convert these decimals to fractions:

a 0·35

The decimal part becomes the numerator and the last digit (in this case 5) is in the hundredths column so the denominator is 100. 0·35 = =

b 1·206

35 --------100 7 -----20

1·206 = 1 + =

206 -----------1000 --------1 103 500

Exercise 4L

112

1 Write the following decimals as fractions in their simplest form: a 0·2 b 0·04 c 0·9 e 0·13 f 0·45 g 0·066 i 0·25 j 0·375 k 0·75

d 0·003 h 0·088 l 0·125

2 Write the following decimals as fractions in their simplest form: a 8·5 b 3·6 c 2·07 e 1·06 f 2·875 g 3·625

d 8·008 h 4·07

3 Write these decimals as fractions in their simplest form: a 0·104 b 0·909 c 0·705 e 0·135 f 0·996 g 0·548 i 1·25 j 2·75 k 1·02 m 5·315 n 7·343 o 9·384 q 10·105 r 12·018 s 15·360

d h l p t

Maths Dimensions 7

0·340 0·625 3·65 12·648 20·045

Percentages

4M

Percentage means per one hundred: 20% means 20 out of 100. The symbol for percentage is %. To convert from a percentage to a fraction, write as a fraction out of 100 then simplify.

Example

Solution

1 Convert these percentages to fractions: a 9%

9 --------100

b 45%

45 --------100

=

c 125%

125 --------100

25 1 - = 1 --= 1 --------

= 17%

This means 9 out of every 100. 9 -----20 100

4

2 Write the following as percentages: a

17 --------100

17 --------100

b

4 -----20

4 -----20

×

5 --5

0·73 =

c 0·73

=

20 --------100

= 20% 73 --------100

= 73%

Exercise 4M 1 Write each percentage as a fraction: a 21% b 97% e 65% f 173%

c 50% g 159%

d 25% h 333%

2 Change each of the following fractions into percentages: a

3 --------100

b

81 --------100

c

7 -----10

d

3 -----10

e

23 -----50

f

1 -----50

g

1 -----25

h

12 -----25

i

1 --2

j

1 --5

k

1 --4

l

1

3 Rewrite each of the following percentages as decimals: a 16% b 47% c 7% d 88% g 35% h 5% i 114% j 425%

e 50% k 250%

f l

90% 315%

4 Convert each of the following decimals to percentages: a 0·33 b 0·45 c 0·13 d 0·93 g 0·01 h 0·02 i 1·63 j 1·59

e 0·19 k 1·0

f l

0·05 1·06

5 Arrange each of the following sets of numbers from smallest to largest by converting the fractions and percentages to decimals: a 0·8, 82%, 7--- , 8--- , 0.87˙ b 67%, 2--- , 0·6, 65%, 0.67˙ 8

c 0·96,

9 ------ , 10

9

98%, 0·923, 92·5%

3

d 29%,

7 ------ , 25

24%, 0·26,

1 --4

Chapter 4 Decimals and Percentages

113

4N

Finding percentages of quantities To find a percentage of a quantity, follow these steps. Step 1: Write the percentage as a fraction with a denominator of 100. Step 2: Change the ‘of’ to multiplication. Step 3: Write the amount as a fraction. Step 4: Simplify.

Example

Solution

Find:

a 10% of $340

10% of $340 10 340 = --------- × --------100 1 = $34

b 120% of $90

120% of $90 120 90 = --------- × -----100 1 = 12 × 9 = 108

Exercise 4N 1 Find: a 20% of 90 e 10% of 40 i 70% of 490

b 25% of 88 f 15% of 250 j 36% of 175

c 10% of 400 g 5% of 60 k 19% of 30

d 30% of 660 h 12% of 350 l 2% of 400

2 Find: a 110% of 80 e 106% of 750

b 200% of 150 f 105% of 1200

c 130% of 70 g 155% of 75

d 225% of 100 h 350% of 12

c 3·3% of 11 000 g 9·8% of 3500

d 40·5% of 400 h 20·4% of 1250

3 Use a calculator to find these percentages: a 6·5% of 200 b 10·2% of 500 e 0·25% of 800 f 0·01% of 80 000

4 Tan has savings of $10 000 in the bank. He decides to use 60% of his savings to buy a motorbike. How much does the motorbike cost? 5 a Lake Eppalock holds 312 000 megalitres of water when full. If it is 65% full, how much water is in it? b Eildon Weir holds 3 390 000 megalitres when full. If it is only 4% full, how much water is that? c Pine Lake near Horsham holds 64 200 megalitres when full. If it is only 4% full, how many megalitres of water is in Pine Lake? 6 Sonja obtained 67% on her Maths test. If her test was out of 85 marks, how many marks did Sonja obtain?

114

Maths Dimensions 7

Calculating percentages Best Jeans is having a sale and all their jeans are discounted by 10%. How much would you pay for the jeans if they are $109·50? Two methods can be used to determine the discount price: subtracting the discount, and finding the percentage paid.

Example

4O

SA

L

dis 10% E co un t

Solution

Tickets to the circus cost $52 each. Abdi is entitled to a 20% discount. How much is Abdi’s ticket?

Method 1 Subtracting the discount: 20% of $52 0·20 × 52 = 10·40 52·00 − 10·40 = 41·60 So, Abdi pays $41·60 for his ticket. Method 2 Finding the percentage paid: 100% − 20% = 80% 0·8 × 52 = 41·60 So, Abdi pays $41·60 for his ticket.

Exercise 4O 1 Use either method to determine the sale price of the following items: a b c

20% Off

$199 15%

Off

$100

10% Off

$12 5% Off

$30

e

f

75 % $6 of 0 f

d

2 Calculate the value of each of the following: a 15% of 50 kilometres b 17% of 1000 metres d 95% of 20 litres e 62·5% of $400

12 21 % Off $999

c 125% of $70 f 35% of 8 centimetres

3 The Australian Government has introduced a Goods and Services Tax (GST) and this tax is added to the cost of most goods and services. The GST rate is 10%. Calculate the total cost of each of the following items after GST has been added: a $5 bag of cement b $20 pair of sunglasses c $120 pair of jeans d $200 pair of runners e $25 compact disc f $36·50 tie Chapter 4 Decimals and Percentages

115

Puzzles 1 Determine the place value of the 5 in each of the numbers below and then match the letter to the answer to solve the riddle:

Why did the egg cross the street?

L E

50 10·15

O 105 H 537·2

____ ____ 1000 1

S T

5346·2

____ ____ ____ 1 -----------1000

1 --------100

I 0·52 M 51 246·2

3·695

____ ____ ____ ____ 1 1 1 ----------------------10 000 -----

1 --------100

10

1000

1000

____ ____ ____ ____ ____ 1 1 -------------------100 10 10 1000

100

2 Complete the decimal multiplication problems to solve the riddle:

Why did the student find decimals so difficult?

A

1·3 ×3

B

2·4 ×5

E

16·5 ×4

H

3·4 × 5·5

N

5·12 × 2·5

O

15·3 × 0·5

T

4·2 × 1·4

U

5·6 × 1·2

C

2·13 ×6

D

10·5 ×2

I

6·8 × 0·5

L

2·4 × 1·6

P

10·48 × 0·5

S

4·1 × 0·9

V

9·1 × 0·3

Y

7·9 × 1·2

____ ____ ____ ____ ____ ____ 12 66 12·78 3·9 6·72 66 ____ ____ ____ ____ ____ ____ ____ 12·78 7·65 6·72 3·84 21 12·8 5·88 ____ ____ ____ 5·88 18·7 66

116

Maths Dimensions 7

____ ____ 18·7 66 ____ ____ ____ 3·69 66 66

____ ____ ____ ____ ____ 5·24 7·65 3·4 12·8 5·88

Ch4 3 Complete the following decimal division problems to solve the riddle:

What is the longest piece of furniture in the school?

A 30 ÷ 0·2 H 10 ÷ 2·5 N 45·6 ÷ 0·12 U 10·35 ÷ 0·45

B 45 ÷ 1·5 I 8·4 ÷ 0·12 O 124 ÷ 0·8 V 51·2 ÷ 6·4

C 26 ÷ 0·4 L 50·4 ÷ 1·2 P 31·2 ÷ 5·2 W 1·48 ÷ 3·7

E 48 ÷ 1·2 M 65·4 ÷ 0·6 T 69·7 ÷ 1·7 Y 4·55 ÷ 0·65

____ ____ ____ 41 4 40 ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 109 23 42 41 70 6 42 70 65 150 41 70 155 380 ____ ____ ____ ____ ____ 41 150 30 42 40

4 Convert the following decimals to percentages to solve the riddle:

What do you get when you cross a calculator with a snake?

E 0·5 I 0·89 A 3·25 E 1·75

R 0·17 D 0·1 D 1·0 C 0·55 ____ ____ 63% 25%

A 0·3 H 1·2 M 2·13

N 0·25 A 0·63 N 1·05

____ ____ ____ ____ ____ 30% 10% 100% 50% 17%

____ ____ ____ ____ ____ ____ ____ 213% 325% 55% 120% 89% 105% 175%

Chapter 4 Decimals and Percentages

117

Applications The price is right? In each case use the clues given to find the correct price for each grocery item.

1 10 mL bottle of tea-tree oil $5·26, $5·03, $4·47, $4·81, $4·67, $4·96, $4·77, $4·36, $4·49, $4·57 Clue 1: The number is not divisible by 2. Clue 2: The number is smaller than the result of 2·6 × 1·77. Clue 3: The sum of the digits is odd. Clue 4: The number has four tenths. Clue 5: The last two digits form a prime number. 2 750 g packet of cereal $3·28, $3·15, $2·97, $3·49, $2·99, $3·38, $3·60, $3·39, $3·47, $3·29 Clue 1: The digit in the hundredth column is greater than 4. Clue 2: The sum of the tenths and hundredths columns is not 10. Clue 3: The last two digits form a composite number. Clue 4: The number has 9 hundredths. Clue 5: The number in the tenths column is even. 3 A 4-pack of yoghurt $3·18, $3·12, $2·95, $2·87, $3·03, $3·14, $3·21, $3·08, $3·29, $3·19 Clue 1: The number has 3 units. Clue 2: The number is greater than $3·05. Clue 3: The number is smaller than 3·84 ÷ 1·2. Clue 4: The sum of the digits is odd. Clue 5: The number has the digit 1 in either the tenths or hundredths column.

Magic squares Magic squares have every row, column and diagonal adding up to the same magic number. For each magic square below, discover the magic number and then use it to complete each of the missing boxes. You will be practising your addition and subtraction skills using decimals. 12·78 4·26 2·84

5·68

37·8

25·2

7·1

100·8 18·9

31·5

11·36 15·62 17·04 1·42 19·88

118

Maths Dimensions 7

12·6

81·9 75·6

6·3

Ch4 Calculator games 1 You will need a partner and one calculator between you. The first eleven prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31. Take turns in dividing 213 by a number less than 12. The first person to reach the prime number 2 receives one point. The process is repeated for each prime number result. The winner is the person with the most points. 2 You will need a partner and one calculator between you. Starting with the prime number 2, you and your partner are to take it in turns multiplying by a number less than 12. The winner is the first person to reach 151 and receives one point. The process is repeated using each of 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 as starting points. The overall winner is the person with the greatest number of points.

Currency 1 Australia has 5, 10, 20 and 50 cent coins in circulation. In how many ways can you make 50 cents using these coins? List all the possibilities. 2 In the United States they have 1, 5, 10, 25 and 50 cent coins in circulation. In how many ways can you make 50 cents using these coins? List all the possibilities. 3 In Canada they have only 1, 5 and 25 cent coins in circulation. How many of each of these coins can make 50 cents? 4 Brazil has 1, 5, 10 and 50 centavos coins in circulation. How many of each of these coins are required to make 50 centavos? 5 Israel has only 5, 10 and 50 agorot coins in circulation. In how many ways can you make 50 agorots using these coins? 6 How many $2 coins would you need to stack to your height? What would be the value of the stack? You will need to know the thickness of a $2 coin and your height. You may be able to research the thickness of a coin from a website such as that for the Royal Australian Mint or otherwise approximate the thickness by measuring a stack of $2 coins and then dividing by the number of coins in the stack. 7 What value of 5 cent, 10 cent, 20 cent or $1 coins would be needed in each case to reach your height?

Chapter 4 Decimals and Percentages

119

Enrichment In Exercise 4L we looked at changing decimals to fractions; however, you did not change any decimals which continue on forever. These are called recurring decimals. Changing recurring decimals to fractions is more challenging and requires some basic algebra.

Example

Solution 0·45 = 0·45454545…

Convert 0·45 to a fraction.

Let N represent the fraction we are trying to find. Equation 1 N = 0·454545454545… 100N = 45·45454545… Equation 2 100N − N = 45·454545… − 0·4545454545… Equation 2 − Equation 1 99N = 45 The recurring decimal 45 part is subtracted. N = -----99

1 Convert the following recurring decimals to fractions by using the example above: b 0·8˙

c 0·72˙

g 0·18˙

h 0·456

i

d 0·5˙

0·06˙

j

2·47˙

e 0·345

f

0·245

k 0·308˙

l

3·673

450 mm

2 Find the cost of building a dog kennel using the design below. You have the following materials available for use: Dressed hardwood: 42 mm × 19 mm $2·64 per metre 65 mm × 19 mm $3·40 per metre 65 mm × 32 mm $7·95 per metre m 250 mm Dressed pine 0m 45 70 mm × 35 mm $2·60 per metre 35 mm × 35 mm $2·23 per metre 42 mm × 42 mm $3·48 per metre Pine lining 89 mm × 19 mm $4·11 per metre 108 mm × 19 mm $5·28 per metre 250 mm 800 m 100 mm m 140 mm × 12 mm $1·60 per metre 600 mm Wheat-coloured Laser-Lite waterproof roofing 86 cm × 1·8 m $26·91 per sheet Calculate how much of each material is used and the total cost of the kennel.

120

Maths Dimensions 7

575 mm

a 0·23˙

Ch4 3 The 1994 Academy Awards will be remembered for the ‘American Express dress’ worn by Lizzy Gardiner who won an Oscar for costumes for Priscilla Queen of the Desert. The dress was made entirely out of American Express cards. a Estimate how many Australian $100 or $50 dollar notes would be required to make an evening dress. b Design an evening dress or a pair of boxer shorts, then measure a $100 or $50 note or find its dimensions. c Decide how you will join the notes together and how much overlap they will have. d Calculate the number of notes required for each pattern piece. e What is the total monetary value of the dress or shorts?

4 A fencing contractor builds fences for people. The owners of a block of flats want to replace the fence along the front of the flats. They decide to use a series of pine structures shown below.

a Calculate the cost of each structure if the posts are 1·8 metres long and cost $8·25 each. The 3·6-metre rails cost $17·60 each. b If the front boundary is 126·6 metres long and the ends of the rails are 0·5 metre apart, calculate how many structures are needed to cover the distance if the first and last structures are exactly in line with the side boundaries. c How many posts are needed? d How much will the posts cost? e A bolt is placed through each post where the rail crosses. If each bolt and nut costs $3·50, what is the total cost of the nuts and bolts? f What is the total cost of materials for the fence? g The contractor also charges for his labour. He estimates he will spend two 8-hour days using his machinery to put in the posts. He will charge the owner $50 per hour. It will take another four 8-hour days to assemble the rails and he will charge $25 per hour for those days. What is the total cost of labour? h What is the total cost of materials and labour for the fence? i The contractor also has to charge GST, an additional 10%. Calculate the total cost of materials and labour, including GST. Chapter 4 Decimals and Percentages

121

Revision Exercise 4A 1 Write down the place value of the 5 in each of the following questions: a 45·907 b 38·0056 c 12·508 d 50·893 e 102·456 870 f 12·268 405 g 88·587 h 4·087 354 879

Exercise 4C 2 Round each decimal to the number of decimal places indicated in the brackets: a 34·876 234 [4 dp] b 1·345 89 [2 dp] c 19·606 060 [3 dp] d 0·008 84 [2 dp] e 0·04 [1 dp] f 0·5 [4 dp]

Exercise 4D 3 Find the answers to the following questions: a 23·89 + 45·12 b 508·909 + 689·234 c 0·98 + 1·45 d 0·876 + 12·687 e 23·78 + 974·67 + 3·984 f 17·09 + 34·67 + 0·08 4 Sonja has a bank balance of $204·78, Michael’s bank balance is $389·56 and Nello’s bank balance is 5 tens and 6 hundredths more than Sonja’s. a How many dollars and cents is 5 tens and 6 hundredths? b How much is Nello’s bank balance? c Who has the most money in the bank? d How much more money than Sonja does Michael have?

Exercise 4E 5 Find the answers to the following questions: a 134·78 − 34·54 b 0·873 − 0·063 d 809·6 − 6·9 e 67·000 − 2·568

c 12·569 − 1·999 f 687·2 − 2·85

Exercise 4F 6 Find the answers to the following questions: a 13·5 × 8 b 0·34 × 6 d 19·6 × 0·5 e 34·76 × 4·2

c 0·45 × 5 f 234·6 × 3·5

7 Max needs some help to decide which is the best buy: ■ 20 kg of dog food for $35·80 ■ 10 kg of dog food on special for $17·50 normally $18·90 ■ 2·5 kg of dog food for $6·25 a Calculate the cost per kilogram for the 20 kilogram bag of dog food. b Calculate the cost per kilogram for the 10 kilogram bag of dog food when on special. c Calculate the cost per kilogram for the 2·5 kilogram bag of dog food. d Which product should Max buy? Explain your answer.

122

Maths Dimensions 7

Ch4 Exercises 4G and 4H 8 Calculate the answers to the following questions: a 100·16 ÷ 8 b 12·6 ÷ 6 d 62·5 ÷ 2·5 e 23·04 ÷ 0·4

c 308·61 ÷ 3 f 56·4 ÷ 0·08

Learning task 4I 9 Calculate: a 3·789 × 100 d 7·98 × 30

b 12·345 × 100 000 e 234·875 ÷ 10

c 100·56 × 4000 f 124 567·432 ÷ 100 000

Exercise 4K 10 Convert the following fractions to decimals: a

7 -----12

b

3 --8

c

1 -----11

d

7 --9

9 --5

f

11 -----8

e 5·809

f

12·058

e 225%

f

1%

e

Exercise 4L 11 Convert the following decimals to fractions: a 0·5 b 0·007 c 0·67 d 0·346

Exercise 4M 12 Write the following percentages as fractions and decimals: a 23% b 90% c 35% d 300%

Exercise 4N 13 Evaluate (mentally if possible): a 10% of $80 b 20% of 30 cm d 5% of 20 cents e 120% of $10

c 25% of $100 f 250% of $50

Exercise 4O 14 Kelly purchased 126·75 metres of pipe which cost $2·89 per metre. She also bought some pipe fittings and joiners which cost $6·98, $7·02, $2·33, $15·87 and $10·33. a How much did the pipe cost Kelly? b How much did the pipe fittings and joiners cost? c What was the total cost of the pipe, pipe fittings and joiners? d What would the cost be if Kelly was given a 10% discount? e How much change would Kelly get from $500?

Chapter 4 Decimals and Percentages

123

CHAPTER

5 Length and Perimeter

Length and Perimeter Australia, the largest island in the world, measures 4000 kilometres from east to west and 3700 kilometres from north to south. The coastline stretches for about 36 735 kilometres. As Australia is such a large country, measurements of this kind are very important when building new transport systems and taking care of our environment. Photographs taken from satellites can be used to find an accurate estimate of Australia’s coastline.

This chapter covers the following skills: • Selecting appropriate units to specify a quantity • Using common prefixes and notation, and converting between units 1 cm = 10 mm 1 m = 100 cm = 1000 mm 1 km = 1000 m = 100 000 cm • Using measuring instruments accurately • Devising ways to accurately measure objects too big or too small to measure individually • Reading a variety of scales accurately • Calculating perimeters of shapes Perimeter: total distance around the outside of shape

5A

Units used to measure lengths The standard unit that we use when stating the length of a small object in the metric system is the metre. Small objects are measured using parts of a metre: ■

centimetres (cm; there are 100 in a metre)

■

millimetres (mm; there are 1000 in a metre)

Large objects are measured in metres and long distances are measured in kilometres (km; or thousands of kilometres).

Exercise 5A 1 What unit would be best to measure the following? (Choose from mm, cm, m, km.) a the height of a golf tee b the height of the Sydney Harbour Bridge c the length of a cricket pitch d the length of a bull ant e the height of a desk above the floor f the depth of a coal mine g the distance from a golf tee to the hole h the length of a fast bowler’s run up at the cricket i the length of a stitch in a row of knitting j the distance from the Earth to the Sun 2 List five things that would be best measured by using the following units: a kilometres b metres c centimetres d millimetres 3 List five things that you would find in your bedroom that are longer than 5 centimetres but shorter than 10 centimetres. 4 List five things you would find in the kitchen that are longer than 30 centimetres.

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Estimating by using known values

5B

When we put our mind to it it’s amazing what things we are able to estimate. Often we can use things of known measure to compare to other unknown objects.

Example Estimate the height of the girl in the photo given that the boy is 1·8 metres tall.

Solution The girl appears to be approximately 80% or 0·8 times the height of the boy. 0·8 of 1·8 = 1·4 The girl is approximately 1·4 metres tall.

Exercise 5B 1 Use a suitable strategy to estimate the following, explaining your method for each question. You might like to work in groups to complete this exercise—other people’s opinions are important. a Estimate the length of b Estimate how high the boy c Estimate the height of the bicycle if the girl has jumped off the ground, the blackboard if the is 1·4 metres tall. if his opponent is known child is known to be to be 1·8 metres tall. 1·2 metres tall.

d Estimate the height of the e Estimate the height of the Great Southern Stand above top of the clock tower if the the ground if the light tower person is 1.6 metres tall. is 85 metres high.

f

Estimate the length of the watermelon if the girl’s little finger is 5 centimetres long.

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5B 2 Estimate the following lengths, explaining your method for each question: a Estimate the height of the diving b Estimate the height of the silos if the board above the water if William cockatoos are about 25 cm long. is 1·5 metres tall.

128

c Estimate the height of the sculpture if the boys are about 1·4 m tall.

d Estimate the height of this sculpture if the boys are about 1·4 m tall.

e Estimate the length of the paddlesteamer, if the person on the deck is about 1·5 m tall.

f

Maths Dimensions 7

Estimate the height of the sculpture if the girl is 1·4 m tall.

Estimating lengths

5C

Shapes and lines can be made to look larger or smaller by drawing other distracting lines or designs around them. For each of the following decide which shape you think is the larger and then measure both to check your guess.

Exercise 5C 1 Compare the two shapes. Are they circular? Are their diameters the same length?

2 Compare the two lines by measuring their lengths.

B

A A B

3 Compare the two shapes by measuring the length of each edge.

4 Compare the two rectangles by measuring the length and width of each.

A

B

B

A

5 Compare the cylinders by measuring their heights and diameters.

A

6 Compare the sizes of the triangles by measuring the height and the base length of each one.

B

C A

D B

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5C 7 Compare the sizes of the kookaburras by measuring each one’s height.

8 Compare the sizes of the shells by measuring the width and length of each one.

9 Estimate the lengths of the following lines by using the scale given below, then measure them to check your answer. List the lines in order of easiest to most difficult to estimate. Explain the order. E A F

B

C G

D

0 CM 1

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Maths Dimensions 7

2

3

4

5

6

7

8

Reading scales when measuring

5D

Instruments that we use to measure have scales marked on them which are read to determine the measurement. When reading a scale, count the spaces between the whole numbers to work out how much each mark represents.

Example Find the length of each pencil to the nearest millimetre using the ruler. pencil A

0 CM 1

2

3

4

5

6

7

8

9

10

pencil B

Solution Use the information on the ruler to work out the scale: ■ The whole numbers are 1 centimetre apart. 1 ■ There are 10 spaces between each centimetre, therefore each small space represents ------ of 10 a centimetre, which is 1 millimetre. Pencil A The zero of the ruler matches the start of the pencil, so read the scale where the pencil ends. Pencil A is 75 mm long. pencil A

0 CM 1

2

3

4

5

6

7

8

9

10

pencil B

Pencil B The start of the pencil is on the 2 cm or 20 mm mark. The end of the pencil is on the 80 mm mark, so the length is 80 mm less the 20 mm on the scale. Pencil B = 80 − 20 = 60 mm long.

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5D Exercise 5D 1 Measure the lengths of the lines below to the nearest millimetre using the rulers: a 0 CM 1

2

3

4

5

6

7

8

9

10

0 CM 1

2

3

4

5

6

7

8

9

10

0 CM 1

2

3

4

5

6

7

8

9

10

0 CM 1

2

3

4

5

6

7

8

9

10

0 CM 1

2

3

4

5

6

7

8

9

10

0 CM 1

2

3

4

5

6

7

8

9

10

b

c 11

12

13

11

12

13

d

e

f

2 Use the scale below to find the length of each line: units 1

2

3

4

5

a b c d e 3 In a shed, Horatio found an old ruler that looks different from the one he uses at school. He counted 16 spaces between the whole numbers, so each small space stands for a sixteenth of an inch. On this ruler the position of point F is 4 inches and 9 small spaces; 9 - inches. that is, length = 4 ----16 Using fractions, identify the positions of the letters shown below:

0

132

A

B

Inches

1

Maths Dimensions 7

C

D

2

E

F

3

4

G

5

6

Measuring lengths accurately

5E

Lengths need to be measured accurately when materials are to be cut to a precise length. Dressmakers and plumbers, for example, need to measure carefully and accurately before cutting the materials with which they are working. If the pieces are cut too short, then that piece may be not be able to be used to complete the job. When you measure an object make sure that the zero of the ruler is placed at the beginning of the object. Trace an imaginary line across from the end of the object to the scale of the ruler. Work out the scale and so determine the length of the object. Curved lines can be measured by placing a piece of string over the line and then stretching the string tightly against a ruler.

Example

Solution

Measure the height of this light globe by using the centimetre ruler provided.

5 4·2 4

Place the ruler in the same direction as the height of the light globe. Make sure that the bottom of the globe is on the same level as the zero of the ruler. The height of globe is 4·2 cm.

3 2 1 0

Exercise 5E 1 Use a ruler to measure the indicated lengths to the nearest millimetre. a b c

d

e

f

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5E 2 Measure the outside of these silhouettes as accurately as possible. A piece of string may help to measure the curved parts. a

b

c

d

3 Measure these lengths of wire as accurately as possible. a

b

4 Measure each mobile carefully to the nearest millimetre to find the length of wire that is needed to make it. If the special wire costs $12 per centimetre, how much will the wire cost for each mobile? a b

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Converting length units

5F

The standard length is that of the metre, and the smaller and larger length units are related to it via a series of conversion factors. Milli means ‘a thousandth’, so a millimetre (mm) is a thousandth of a metre (m) or there are 1000 millimetres in 1 metre.

••

1000 mm = 1 metre

Centi means ‘a hundredth’ and so a centimetre (cm) is a hundredth of a metre (m) or there are 100 centimetres in 1 metre.

••

100 cm = 1 metre

Kilo means ‘thousands of’ and so a kilometre (km) is a thousand metres (m) and so there are 1000 metres in 1 kilometre.

•• The units can be thought of as being connected by either multiplication or division using multiples of 10. You can use the metric ladder as a visual guide.

1000 m = 1 km

÷ 1000

■

When climbing down the ladder, multiply by the factor.

÷ 100

■

When climbing up the ladder, divide by the factor.

÷ 10

kilometre (km) metre (m) centimetre (cm)

× 1000 × 100 × 10

millimetre (mm)

Example

Solution

1 Find the factor that can be applied to convert: a metres to centimetres

Multiply by 100 to convert metres to centimetres.

b millimetres to centimetres

Divide by 10 to convert millimetres to centimetres.

2 Use a conversion factor to express: a 200 m in kilometres b 30 cm in millimetres

200 m ÷ 1000 = 0·2 km 30 cm × 10 = 300 mm

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5F Exercise 5F 1 Show the factor that would have to be applied to convert the following quantities: a metres to centimetres b millimetres to centimetres c metres to kilometres d millimetres to metres e centimetres to metres f kilometres to metres g kilometres to centimetres h centimetres to millimetres 2 Fill in the spaces: a 3·9 km = _____ m = _____ cm c _____ m = 14 cm = _____ mm

b _____ km = 0·3 m = _____ mm d _____ m = _____ cm = 570 mm

3 Convert the following to the units indicated: a 120 cm = _____ m c 0·2 m = _____ cm e 470 cm = _____ mm g 120 mm = _____ cm

b d f h

1500 m = _____ km 120 m = _____ km 0·02 m = _____ mm 800 m = _____ km

j

3 1--- m = _____ cm

i

20 mm = _____ cm

k

4 1--2

km = _____ m

l

2 2 2--- cm 5

= _____ mm

4 Express the total length of the path in metres. Finish

6·98 m 4·2 m

Start

368 cm 245 cm

3800 mm

5 Fred lives 1·08 km from the local swimming pool. If he walks to the swimming pool and back to his house twice every day during his 2-week holiday break, find the distance he will have walked to and from the pool over the holiday period. Express your answer in both metres and kilometres. 6 The diameter of a 5-cent coin is about 20 millimetres. If forty-two 5-cent coins were placed in a straight line, how long would the line be? Express your answer in both centimetres and metres. 7 The thickness of a 20-cent coin is 2 mm. Find the height in metres of vertical stack of 20-cent coins worth: a $100 b $212 c $583·60

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Adding and subtracting lengths

5G

When we measure lengths in different units, we need to convert them to the same unit before we can combine or compare them. Usually the smaller unit is chosen to avoid the use of decimals.

Example

Solution

Find the answer to each of the following:

a 5 km + 2300 m

Convert units to metres. 5 km + 2300 m = 5000 + 2300 5 km × 1000 = 5000 m = 7300 m The answer is 7300 m or 7·3 km.

b 5·8 cm − 26 mm

Convert units to millimetres. 5·8 cm − 26 mm = 58 − 26 5·8 cm = 58 mm = 32 mm The answer is 32 mm or 3·2 cm.

Exercise 5G 1 Find the sums or differences of the following lengths: a 56 mm + 25 mm b 26 m + 5 m d 598 km + 12 km e 365 mm − 25 mm g 125 m − 12 m h 12 mm − 9 mm j 2 mm + 3 mm + 23 mm k 14 km + 2 km + 56 km

c f i l

32 mm + 48 mm 29 cm − 14 cm 32 m + 25 m + 236 m 69 cm + 23 cm + 95 cm

2 Find the sums of the lengths, expressing your answers in the smaller unit: a 7·9 m + 68 cm b 7·9 km + 390 m c 58 cm + 3·4 m d 59 mm + 4·2 cm e 54 cm + 130 mm f 73 mm + 2·9 cm g 436 m + 0·3 km h 93 mm + 5·3 cm i 56 km + 789 m j 23 m + 980 cm k 45 cm + 251 mm l 569 cm + 5·63 m + 231 cm 3 Find the sums or differences of the lengths, expressing your answers in the larger unit: a 7·09 km + 34 m b 57 cm + 4·3 m c 87 mm + 8·3 cm d 86 m + 5·9 km e 78 m − 270 cm f 7·3 cm − 59 mm g 736 m − 0·234 km h 93 m − 783 cm i 600 m + 2·6 km j 256 mm + 56·1 cm k 78·9 km + 9800 m l 5·6 m − 450 cm 4 A team of shot-put athletes in the local little athletics club had their throws measured: Group 1: 3·2 m, 210 cm, 4·5 m, 380 cm Group 2: 3850 mm, 3·9 m, 450 cm, 630 mm Group 3: 520 cm, 4·75 m, 6980 mm, 0·005 km For each group state the longest and shortest throw, in metres. Find the difference between the length of the shortest and the longest throw for each group.

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5G 5 Sam kicked a football 23·2 m with his first kick and then 1989 cm with the next. Which kick travelled further, and by how much did it beat the other attempt? 6 Bill found a worm which was 123 mm long. Harrison’s worm was 9 cm long. Who found the longer worm and how much longer was it than the other one? 7 Shan lives 2.5 km from school. Jin lives 560 m from Shan’s house. If the girls meet at Shan’s house before walking to school, what distance does each girl walk? 8 Wally was playing golf. His first shot travelled 45 m. His subsequent shots travelled 3220 cm, 56·2 m and finally 30 cm. The distance from the tee to the hole was 97 metres. a Find the total distance of his shots. b How much further did he play than the golfer who scored a hole-in-one? 9 Arrange the following lengths from shortest to longest: a 3 cm, 27·3 mm, 0·045 m b 43 mm, 153 mm, 8 cm, 0·0045 km c 6·8 km, 7932 m, 1450 mm d 3 m, 180 cm, 0·0045 km e 0·0058 km, 3·5 mm, 0·12 cm, 0·009 m f 15 mm, 0·2 cm, 0·006 km, 0·04 m 10 Bricks which measure 20 cm long by 15 cm deep by 10 cm high are stacked so that there are no gaps between them. a Find the length, depth and height, in metres, if the blocks are stacked in the following ways: i ii

b Ali, the landscape gardener, has a design that he recommends to people who want to have a border along their garden beds. It is a simple line of the above bricks, two bricks tall as shown here. Ali has estimated the number of bricks he needs for a number of jobs. Find the length of the borders for each of the jobs detailed below: ■ job at Flemington: 5000 bricks ■ job at Richmond: 2100 bricks ■ job at South Yarra: 41 500 bricks ■ job at South Melbourne: 65 250 bricks

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Maths Dimensions 7

Perimeter of shapes with straight sides

5H

The perimeter of a shape is the distance around the outside of the shape. If the lengths are measured in a mixture of units such as centimetres and metres, convert all units to the same unit before adding the lengths. The perimeter of Australia is our coastline, which has been estimated to be about 36 735 kilometres.

Example

Solution

Find the perimeters of the following shapes:

a

The length of the rectangle is 3 cm and the width is 2 cm. Perimeter = 2 + 3 + 2 + 3 = 10 cm The perimeter of the rectangle is 10 cm.

1 cm

1 cm

b

390 cm

2·7 m

Convert the units to metres. 390 cm ÷ 100 = 3·9 m 3100 mm ÷ 1000 = 3·1 m Perimeter = 2·7 + 3·9 + 3·1 = 9·7 m

3100 mm

c

25 mm

1.9 cm

1.9 cm

Change the units to millimetres. 1·9 cm × 10 = 19 mm Perimeter = 19 mm + 14 mm + 19 mm + 25 mm = 77 mm or 7·7 cm

14 mm

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5H Exercise 5H 1 Find the perimeter of the following shapes shown on the centimetre-dot paper: A

B

C

D

2 Find the perimeter of these shapes made up of squares of different sizes: a

b

square size = 10 cm

c

square size = 2·5 cm

d

square size = 4·2 cm

square size = 0·5 cm

e

square size = 0·25 cm

3 Find the perimeter of each of the following shapes. Describe a shortcut method that could be used to find their perimeter. Regular means that all the sides are the same length. a equilateral triangle b regular hexagon c regular pentagon d rectangle 11 cm

16 cm 14 cm

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Maths Dimensions 7

8 cm

17 cm

5H 4 Find the perimeter of each of these shapes, expressing your answer in the units used: a

b

9m 14 m 11 m

21 m

35 cm 12 cm

26 cm

22 cm 15 cm

28 cm 21 m 31 cm

5 Find the perimeter of these shapes. All the angles are right angles. Express your answer in each of the units used: a b 30 mm

10 cm 35 mm

15 cm 14 cm 9 cm

6 Lucinda has to replace the border tiles around each of her six rectangular garden beds. She measured the garden beds and drew them on a piece of paper. Find the total length of tiles that she needs to buy, in centimetres and metres: Bed 1 Bed 2 Bed 3 840 cm

0·6 m

950 cm

1950 cm 1·2 m

950 cm

Bed 4

Bed 5

Bed 6 750 cm

11 m

380 cm

32 m 950 cm

4·2 m

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5H 7 Sketch the following, then find the perimeter expressed in centimetres: a equilateral triangle with side length 10·03 cm b regular pentagon with side length 2·06 m c regular hexagon with side length 0·9 km d regular nonagon (9 equal sides) with side length 154 mm e rectangle of length 124·06 m and width 0·06 km 8 Find the perimeters of these shapes. Express your answer in each of the units used: a

b

2·2 m

32 mm 2·7 cm

135 cm

1·8 m

6·3 cm

150 cm 1·9 cm

c

d

6900 mm

1500 cm

3·8 m 450 cm

64 m

780 cm

6·3 m

4800 cm 32 m

e

520 cm 4·9 m

280 cm 3·9 m 3·7 m

2·5 m

3800 mm

490 cm 980 cm

f

g

0·0045 km

5 ·03

00 29

495 cm

2800

242 cm

m

2700 m

20

00

m

m

m 3·8

2·9

m

2·9

cm

25 m

m

150 cm

200 cm 2m

12 m

3.5 m

i

8m

4.1

4200 mm 283 cm 7.8 m

Maths Dimensions 7

mm

32 00 cm

cm

1·8 m 250 cm

142

000

0

2·3 m

h

25

km

0.003

km

Exploring measurement in the past 5I The first measurement systems were based on body measurements. This proved difficult at times due to the differences between individuals. The Egyptian royal cubit was the first widely used standard unit in the ancient world. It was developed in about 3000 BC and was based on the length of the forearm from the elbow to the outstretched fingertips. At regular intervals, all cubit sticks were checked against the royal master cubit made of black granite, to ensure uniformity. The cubit was divided into 28 digits; each digit is equal to the width of a middle finger. Each digit was divided into parts to allow for very accurate fractional measurements. A royal cubit was approximately 524 mm long and subdivided into smaller units. Royal cubit

524 mm

Digit

28 digits in a cubit

Approx. 19 mm

Palm

4 digits in a palm

Approx. 76 mm

Hand

5 digits in a hand

Approx. 95 mm

Small cubit

24 digits or 6 palms in a small cubit

Approx. 456 mm

The Great Pyramid of Giza was built by thousands of people who measured with cubit sticks. The sticks were very accurate because the sides of the pyramids vary by less than 10 cm over a height of approximately 230 metres.

Learning task 5I 1 a Measure the distance from your elbow to outstretched fingers. b Compare the length of your cubit with that of the royal cubit. c A digit is the width of your middle finger. Compare your digit with that used in ancient Egypt. Explain any possible differences. 2 Choose an appropriate ancient Egyptian unit to measure each of the following lengths: a the length of the classroom b the height of a friend c the length of your pen d the width of your desk e the thickness of your mathematics text book f the length of your calculator screen

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5J Rhino Introduction Use basic coordinates to try and find a hidden rhino. The distance to the rhino is then provided by the calculator. The set of points ‘equidistant’ from the coordinates can be marked. The distance, in this case, is not in a straight line. (If it were, a circle would be transcribed). The purpose of this activity is to measure distances and to determine the minimum number of guesses required to solve the problem.

Equipment TI-83Plus/TI-84Plus with Smile Maths application. TIConnectV1.6™ and Cabri® Jr can be downloaded from the Student CD in the back of this book. For information on other applications such as Smile Maths see ‘Software Downloads’ on the Companion Website.

Technology activity 5J 1 Try and find the hidden location of the rhino. Press the APPS key and start the Smile Mathematics (SmileMth) application.

Press ENTER repeatedly to bypass all the welcome information until you arrive at the main menu (shown opposite). Select Rhino from the main menu.

Help is available should you require it, otherwise, select 2 to start the game.

A rhino is hidden behind one of the points in this 9 × 9 grid. You must try to find the rhino in as few moves as possible. Enter the coordinates of one of the points on the grid. Use the number key pad to enter the coordinates and the key to move from the x-coordinate to the y-coordinate. Press ENTER once you have completed your guess.

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Maths Dimensions 7

5J The number of blocks to the rhino will be displayed. In the example shown, the rhino is three blocks away from the coordinate (4, 5). A block is the distance from one point on the grid to another. For example, the point (1, 5) is 3 blocks away, so too is (3, 3).

2 Use a grid like the one shown below to help work out where the rhino is hidden. 9 8 7 6 5 4 3 2 1 0

1

2

3

4 5

6

7

8

9

a Plot your first point on the grid and use the information returned by the calculator to help determine each of the possible locations for the rhino. b Write down each of the coordinates for the possible location of the rhino. c Use one of the coordinates you have written down as your next guess. 3 Repeat the above process with the information you receive about the rhino’s location. 4 How does this help you find the rhino? 5 What is the maximum number of guesses you need to find the rhino?

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Puzzles 1 Convert the units of length to find the answer to the riddle:

What is the definition of an autobiography?

A 30 cm = ____ mm E 100 mm = ____ cm I 240 mm = ____ m O 0·1 mm = ____ cm T 0·1 km = ____ m ____ ____ ____ 100 3·6 10

C F L R U

D 24 cm = ____ mm H 3600 m = ____ km M 0·15 cm = ____ m S 3·6 cm = ____ m Y 1500 cm = ____ km

150 cm = ____ m 1·5 km = ____ m 3·6 mm = ____ cm 240 cm = ____ m 0·36 km = ____ m

____ ____ ____ ____ 0·36 0·24 1500 10

____ ____ 0·01 1500

____ 300

____ ____ ____ ____ ____ 0·036 100 0·01 2·4 0·015

____ ____ ____ 1·5 300 2·4

2 Find the perimeter of the shapes below to solve the riddle:

What happened to Raymond when he met the man-eating monster?

A

B

5

4

C

8

E

5

18

3 10 4

F

5

6

H

M

8

3

4

N

3

5

8

9

R

X

4

5

Y

5

Z

15

3

5

5

7

7 10

10

146

10

____ ____ 26 38

____ ____ ____ ____ ____ ____ 24 38 12 13 20 38

____ ____ 13 16

____ ____ ____ ____ 50 34 13 45

Maths Dimensions 7

10

Ch5 3 Calculate the perimeter of the shapes below. Match the letters to the correct value below to find the answer to the riddle:

How much do pirates pay for their earrings? A

C B N E

U

R

____ 20 cm

____ ____ ____ ____ ____ ____ ____ ____ 32 cm 12 cm 22 cm 22 cm 20 cm 32 cm 18 cm 18 cm

____ 8 cm

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Applications Coins Measure the diameter of the following coins correct to the nearest millimetre:

a one-dollar coin

b 20-cent coin

c 10-cent coin

d 5-cent coin

Use your measurements to find the distances to the nearest millimetre, centimetre, metre and kilometre that one million of each of the coins would reach if placed in straight lines.

Pace lengths Go out onto the oval and design an experiment to find: ■

the length of your walking pace over 100 metres

■

the length of your running pace over 100 metres.

Determine the pace lengths for each member of your class and find the average (or mean) pace length for the class for walking and running. Obtain a video of a recent 100 metres sprint both for men and women, from either the Olympic Games or the International Grand Prix, and count the number of paces the winner and loser from each race take to complete the race. Compare the results within the race between the winner and loser, and with your results.

Presents You have been given the job of wrapping a birthday present which is a book that is 20 cm long, 15 cm wide and 2 cm thick. Unfortunately you don’t have much wrapping paper. Experiment with paper rectangles of different sizes to find the rectangle with smallest perimeter that will allow you to wrap the present. Submit the most successful attempt as well as your other attempts to your teacher.

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Maths Dimensions 7

Ch5 Brick walls Find a brick wall and count the number of bricks contained in a square metre of wall, including mortar joints. Use this information to find the number of bricks required to build a brick wall such as a wall of your house, your school or some other building. Remember to allow for any windows in that wall. Submit a photograph of the wall along with your calculations.

Clinometer In pairs, determine the height of tall objects around your school by using a tape measure and a clinometer. Possible objects are the flagpole, basketball or netball post, buildings, trees etc.

1 Choose one object and mark out a suitable distance (e.g. 5 or 10 metres) from the base of the object on level ground. 2 From that point, measure the angle from your eye to the top of the object. 3 Record the size of the angle and the distance. 4 Measure your height. 5 Using a suitable scale, draw a scale diagram to help you determine the height of each object.

Are you a rectangle or a square? Work in a group and, with a tape measure, measure the armspan and the height of each person as accurately as possible.

Armspan

A person is a square if their armspan is the same as their height. Your group can decide what difference in these measurements is allowed (5 millimetres is reasonable). A person is a rectangle if their armspan and height are not the same. They are a tall rectangle if their height is greater than their armspan or a short rectangle if their armspan is greater than their height.

Height

Record the results for your group, then combine them for your class. Draw a chart of the results. Are the rectangular people tall or short rectangles?

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Enrichment 1 A piece of plastic water pipe needs to be cut and then glued between two flexible end pieces already in place. The pipe needs to overlap 15 mm inside the larger end pieces and then be glued into position.

Pipe A Pipe B Pipe C

a Measure the diameter of pipes A, B and C and determine which pipe should be used for the job. b What length of pipe needs to be cut to complete the job if the diagram above has a scale of 1:10? 2 Find the perimeter of these shapes in which all angles are right angles: a

b

65 mm

70 cm 10 cm

100 mm

2 cm

70 cm

3 Regular hexagons and octagons with edge lengths of 15 centimetres are to be made out of wire and hung as mobiles. Twice as many hexagons as octagons are to be made for an order for 330 shapes. What weight of wire is required if the wire weighs 200 grams per metre? 4 These figures are made of lengths of wire soldered together. If they are made up of regular shapes with all sides equal, find: i the total length of wire needed ii the perimeter of each figure for each figure. a

b

12 mm 14 mm

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Maths Dimensions 7

Ch5 5 The local under 10s hockey club plans to decorate its hockey field by constructing a series of coloured rectangles on the pitch as shown in the diagram. 3m 2·5 m

3m

2·5 m

3m 3m

3m 3m 3m

6m

2·5 m 2·5 m 3m 40 m

a Find the total length of each colour required for the job. b If the tape costs 25 cents per metre how much will the tape cost to complete the job? 6 Linda wants to replace the border tiles which go around each of the five rectangular beds in her garden. She measured the garden beds and drew them on a piece of paper. Bed 1

Bed 3

Bed 2 9·9 m

14·4 m

1350 cm

2250 cm 900 cm Bed 4

9·45 m Bed 5 630 cm

5·4 m 12·15 m 585 cm

Find the number of tiles she needs to buy if each tile is 45 cm long.

7 A pen manufacturer claims that each of its pens can write for a total length of 10 kilometres. As part of a trial, straight vertical lines are drawn between darker horizontal lines that have been especially printed on pieces of paper. The dark lines are 1·5 centimetres apart and the pen lines need to be 4 millimetres apart. If the paper is 30 centimetres wide and 40 centimetres long how many sheets of paper will be needed for each pen?

4 mm 1·5 cm

Chapter 5 Length and Perimeter

151

Revision Exercise 5A 1 What length unit would be best to measure: a the height of an ant? c the distance from Euroa to Benalla?

b the length of a football field? d the height of a computer screen?

Exercise 5B 2 Estimate the length of the plane, if the doorway at the top of the stairs is approximately 2 metres tall.

Exercise 5C 3 Without using a ruler, draw straight lines with the following lengths: a 3 cm b 4 cm c 10 cm d 30 cm Check the accuracy of the lengths and note how far out your estimates were. You may have estimated exactly!

Exercise 5D 4 Mark the following lengths on the ruler: a 2·5 cm

b 1 1--- cm

c 4·2 cm

d 4 cm

e 0·2 cm

f

4

0

10

20

2 cm

Exercise 5E 5 Use a ruler to measure the following lines to the nearest millimetre: a b c d e

f

152

Maths Dimensions 7

g

30

40

50

Ch5 Exercise 5F 6 Complete these length conversions: a 48·9 cm = ____ mm b 32·88 km = ____ m d 0·021 km = ____ m e 8·56 cm = ____ m g 4 1--- cm = ____ mm 2

h 1 1--- km = ____ m

c 214 mm = ____ cm f 514 m = ____ km 1 --5

i

4

m = ____ cm

7 The cards shown below are placed in a line. 12 cm

15 cm

Find the length (in metres) of the cards when the following number of cards are used: a 3 b 5 c 20 d 24 e 38

Exercise 5G 8 Find the sum or difference in lengths and express the answer in the smaller unit: a 23 mm + 1·8 cm b 4500 m − 2·4 km c 1·2 m + 56 cm − 450 mm 9 The heights of some students are shown below: Student

Height

John

1591 mm

Manuel

1·67 m

Lulu

1365 mm

Valda

142 cm

a How much shorter than 2 metres is each person? b What is the sum of their heights expressed in each of the units used in the table? 10 Len’s hobby is collecting silver chains. For Christmas he was given chains 12 cm, 230 mm, 30 3--- cm and 512 mm long. The chains in his collection prior to this totalled 2 m. 4 What is the length of his chain collection now? Express the answer in centimetres.

Exercise 5H 11 Find the perimeter of the following shapes: a b c 3 cm 2·5 cm

d

46 cm

2·5 cm 1·6 cm

Chapter 5 Length and Perimeter

153

CHAPTER

6 Area and Volume

Area and Volume Pieter Cornelis Mondrian (1872–1944) was an abstract painter who used lines and shapes to represent reality. He moved the lines around until they had energy. The areas were then painted with strong colour to add tension between the lines. These lines had the quality of movement which could show the force of a thunderclap or the delicacy of a cat. He wanted to depict the infinite. A straight line is infinitely extendable and so the open-ended space between two parallel lines is infinitely extendable. Each painting seems to be a small part of the larger, infinite whole work, which is the universe.

This chapter covers the following skills: • Finding and comparing areas • Using grids to find area • Calculating the area of triangles and quadrilaterals Area rules Rectangle: length × width Parallelogram: base × height Triangle: 1--- × base × height 2

• Calculating the volumes of prisms Volume rules Rectangular prism: length × width × height General prism: area base × height

6A

Finding and comparing areas The area of a shape is the amount of two-dimensional space inside that shape. We can measure the area of a shape by dividing it into square units and counting the number of squares that we can fit inside it.

Example Using the square grid below, find which shape has the larger area:

B

A

Solution Step 1: Divide the shapes into squares using the grid, combining half squares to make whole squares. Step 2: Count up the number of squares within each shape. The larger the number of squares inside the shape the larger its area. Step 3: Shape A has 10 squares. Shape B has 10 squares. So they have the same area.

B

A

156

Maths Dimensions 7

6A Exercise 6A 1 Order the following shapes from largest area to smallest area:

F A C

B

E

D

2 Order the following shapes from largest area to smallest area:

F

A C

B

D

E

3 Use the dot paper to estimate the shaded area of the following shapes:

C A

B

Chapter 6 Area and Volume

157

6B

Using grids to find areas Usually the shapes we find in nature are irregular and don’t have straight sides that can be easily placed on a grid. We can only find the area of these shapes by counting the squares that fit inside it in an approximate way.

Example Find the area of this animal footprint which was found in the Australian desert.

Solution Step 1: Place a centimetre grid over the shape.

Step 2: Count the number of squares in the shaded area in the following way: Those squares which are at least half filled count as a whole square. Those squares which are not at least half filled do not count at all. Count this square Don’t count this square

Area is approximately 4 square centimetres or 4 cm2.

Exercise 6B 1 Use the centimetre grids to estimate the area of the following shapes: a b

c

158

Maths Dimensions 7

6B d

e

f

g

2 Use the centimetre grids to estimate the area of the following shapes: a

b

c

d

Chapter 6 Area and Volume

159

6C Exploring areas of faces Areas of shapes can be estimated using centimetre grids.

Learning task 6C Use the centimetre grids to estimate the area of the faces below:

160

a

b

c

d

e

f

Maths Dimensions 7

Area of rectangles

6D

A rule can be used to find the area of a rectangle. Multiply the length by the width. Make sure that the measurements for length and width have the same units. The units for area are squared, for example square centimetres or cm2, because area is the measure of two-dimensional space.

••

Area = length × width

Example

Solution

Find the areas of these rectangles: Area = length × width = 30 × 14 = 420 mm2

a 14 mm 30 mm

b

First convert the width into centimetres: 0·26 × 100 = 26 cm Use the formula Area = length × width = 32 × 26 = 832 cm2

0·26 m 32 cm

Exercise 6D 1 Find the area of each of the following rectangles by: i counting the squares ii using the rule a b 2 cm 4 cm 4 cm 3 cm

c

d 5 cm

6 cm

6 cm 4 cm

2 Find the area of the following rectangles: a b c

d 22 cm

4 cm 4 cm

12 cm

21 cm

32 cm

3 cm

15 cm

Chapter 6 Area and Volume

161

6D 3 Find the area of the following rectangles: a b

c 8 mm 14 mm

3 km

18 m

13 m

3 km

d

e 32 km

f 39 cm 51 cm

18 km

5 km 2 km

g

h

i

160 m

47 km 83 m

25 km 87 mm 34 mm

4 Divide the following shapes into rectangles and use the rule to find the total area: a b c 6m 18 cm 21 km 2m 12 km

3m

18 cm

3m

21 cm

8m 2m

2 cm

10 km 2 km

25 km

3 cm 20 cm

5 km

3m

7 km 6 km

5m 11 km

5 Find the area of the following shapes expressed in square centimetres: a b c 0·2 m 0·9 m

0·12 m 352 cm 61 cm 2 cm

162

Maths Dimensions 7

6D 6 Find the area of the following shapes: a a rectangle with length 32 mm and width 56 mm b a rectangle with length 1·2 km and width 960 m (in square kilometres) c a rectangle 30 cm long by 260 mm wide (in square millimetres) d a square with side length 45 mm 7 Find the shaded area in the following rectangles: a b 20 m 17 m

c

48 m 45 m

16 m

14 m 12 m

23 m

37 m

33 m

20 m

13 m

8 Find the shaded area in the following shapes: a b 205 cm 57 m

c

19 m 201 cm

101 cm 122 cm

31 m 17 m

33 m

27 m

40 m

23 m 7m

20 m

14 m 35 m

9 A backyard in the shape of a rectangle measures 14 metres by 20 metres. In it there are three garden beds which measure 1·2 m by 5 m, 7 m by 4·5 m and 3·8 m by 14 m. The rest of the backyard is grass. a Find the area of the garden beds. b Find the area of grass in the backyard. 10 A rectangular envelope which is 15 cm by 25 cm has a rectangle drawn inside it 3 cm from the outside edge. The margin is coloured red. Find the area of the red margin. 11 A chess board contains 64 squares. Each square is 4 cm by 4 cm. a What is the area of one square? b What is the area of all the squares? c What is the area of the white squares? d The border around the chess board is 2 cm wide, what is the area af the border?

Chapter 6 Area and Volume

163

6E

Area of parallelograms A parallelogram can be made from a rectangle by shifting a triangular piece as shown below. The rule for finding the area of a parallelogram is similar to that for a rectangle.

width

Shift the triangular end to the left

height base

length

•••

Rectangle Area = length × width

Parallelogram Area = base × height

Example

Solution

Find the area of these parallelograms:

a

The units are the same, so you can use the rule straight away. Area = base × height =7×4 = 28 m2

4m 7m

b 90 mm

Change the units of the base to centimetres to make the units the same. 90 mm ÷ 10 = 9 cm Use the rule: Area = base × height = 14 × 9 = 126 cm2

14 cm

Exercise 6E 1 Find the area of the following parallelograms: a b 6m

c

9 cm

18 cm 7m

10 cm

14 cm

d

e

f

43 km

19 m 50 km

72 mm

17 m 89 mm

164

Maths Dimensions 7

6E 2 Find the area of the following parallelograms in square centimetres: a b c 30 cm

60 mm

4 cm

300 mm

4 cm

d

28 mm

e

f 2·3 m

0·8 m

130 cm 400 cm

120 cm 1·2 m

3 Find the area of the shaded regions in the following shapes: a b 19 mm

16 mm

24 m

12 mm

20 m

15 mm 16 m

c

d

13 cm

38 mm

11 cm 8 cm 132 mm 5 cm

e

12 cm

38 mm 10 cm 79 mm

f

124 cm 27 cm 21 cm

18 cm

98 cm 159 cm

Chapter 6 Area and Volume

165

6F Cabri triangles Introduction The aim of this activity is to investigate the factors that affect the area of a triangle. Measurements can be obtained from dynamic geometry software such as Cabri or Sketchpad.

Equipment TI-83Plus/TI-84Plus with Cabri® Jr App™. TIConnectV1.6™ and Cabri® Jr can be downloaded from the Student CD in the back of this book. For further details see ‘Software downloads’ on the Companion Website.

Technology activity 6F Press the APPS key and start the Cabri Junior application.

Press F2 and use the arrow keys to select Triangle and then press ENTER .

The cursor changes to a pencil and is ready to draw your triangle. Move the pencil to a point near the bottom left-hand corner of the calculator screen and press ENTER . This locates the first vertex (corner) of the triangle. Use the right arrow key only and drag the pencil towards the right of the screen. This ensures the base of the triangle is horizontal. Press ENTER to place the bottom right vertex of the triangle. Use the left and up arrows to place the top vertex of the triangle. Press ENTER to locate the top vertex.

Press F5, select Measure then Area. Press ENTER on one of the sides of the triangle.

166

Maths Dimensions 7

6F A small hand will appear, holding a measurement; this measurement is the area of the triangle. Use the arrow keys to place the measurement at the top of the screen.

Press CLEAR and move the cursor back to the top of the triangle. Once the arrow is over the top vertex press the ALPHA key to lock onto the top vertex.

Use the up and down arrows to change the height of the triangle.

1 What happens to the area of the triangle as the height is increased? 2 What happens to the area of the triangle as the height is decreased? Move the top vertex so that it is directly over the bottom vertex as shown here.

3 Use the left and right arrows only to move the top of the triangle. The height of the triangle does not change as the vertex is moved parallel to the base. What happens to the area of the triangle as the top vertex is moved left and right parallel to the base? 4 Which side length stays constant while the top vertex is moved left and right? 5 The area of a triangle is half that of a rectangle with the same base and height. The rectangle shown on the right has an area of 10 cm2. a What is the area of the triangle? b Write down a possible height and width for the triangle. 6 Four triangles have been drawn inside rectangles of the same dimensions. Determine the area of each of the triangles and discuss your results in relation to the activity you have just completed. a b 3 cm

3 cm

7 cm

7 cm

c

d 3 cm

3 cm 7 cm

7 cm

Chapter 6 Area and Volume

167

6G

Area of triangles Two identical triangles can be joined together to make a parallelogram as shown below. The area of the triangle is half that of the resulting parallelogram.

Cut the parallelogram exactly in half. height

height base

base

Area of each triangle is half that of the parallelogram.

•••••

Parallelogram

Triangle

Area = base × height

Area = 1--- × base × height 2

Example

Solution height = 14 m base = 26 m The units are the same so use the rule.

Find the area of the triangle. 14 m

Area = =

26 m

1 --2 1 --2

× base × height × 14 × 26

= 182 m2

Exercise 6G 1 Find the area of these triangles: a b

c 12 mm

6 cm 6 cm

8m 11 m

14 mm

d

e

40 mm

f

12 km

86 m

61 mm 15 km

65 m

g

h 62 mm

68 mm

i

9 km

15 m 7 km 13 m

168

Maths Dimensions 7

6G 2 Find the area of these triangles: a b

c

49 mm

29 m 52 mm

12 m

28 m

d

5m

18 cm

e

f 58 mm

15 cm

46 mm 5 cm

6 cm

3 Find the area of the shaded regions in the following shapes: a b 40 mm 14 mm

10 mm 26 mm

28 m 15 m 20 m 38 m

c

d

4m

10 m

45 km

18 m

20 km

3m

18 km 32 km

e

12 mm

f

28 m 14 mm

14 m

12 m

12 m

9 mm

Chapter 6 Area and Volume

169

6G 4 Find the shaded areas in each of the following: a b 28 m 15 m 18 m

45 km

42 m 18 km 32 km

c

d

3m

24 mm 8m

18 mm

10 m

e

18 mm 28 mm

2m

f

12 mm

20 m 14 mm

14 m 12 m

9 mm

18 m

5 Find the areas of each of the different colours used in these designs: a b

12 cm 27·7 cm

6 cm 6 cm

c

d

15 cm 15 cm 15 cm 15 cm 15 cm

100 cm

170

Maths Dimensions 7

90 cm

Volume as a measure of space

6H

The volume of a solid is the amount of space contained in it. Cubic units are used because volume is the measure of three-dimensional space. The volume of a beetle is measured using cubic millimetres or mm3.

The volume of a fish tank is measured in cubic centimetres or cm3. 1 cm

1 mm 1 mm 1 mm 1 cm

The volume of soil carried in a trailer is measured in cubic metres or m3.

1 cm

The volume of a mountain is measured in cubic kilometres or km3.

1m 1 km 1m 1m 1 km 1 km

Example

Solution

Find the volume of the following solids if each cube is a cubic centimetre:

The cubes are all the same size so you can add them up.

a

Two rows of 5 cubes = 10 cubes Volume is 10 cm3.

b

Five rows of 4 cubes = 20 cubes Volume is 20 cm3.

c

Back section has four rows of 5 cubes = 20 cubes Front section has two rows of 5 cubes = 10 cubes Volume = 20 + 10 = 30 cubes Volume is 30 cm3.

Chapter 6 Area and Volume

171

6H Exercise 6H 1 Count the number of cubes to find the volume of each shape. Each cube is a cubic centimetre: a

e

b

c

f

g

2 Find the volume of these structures by counting the centimetre cubes:

172

a

b

c

d

Maths Dimensions 7

d

Exploring volume 6I A short-cut method can be used to find the volume of a rectangular prism. The method can be found by building a number of rectangular prisms to find a pattern using blocks.

Learning task 6I 1 a Build the following rectangular prism with your blocks. Each block is 1 cubic unit. Copy and complete the following statements: Length is ________ units. Width is ________ units. Height is ________ units. The volume of the rectangular prism is ________ cubic units. b Now add another layer onto the rectangular prism. Length is ________ units. Width is ________ units. Height is ________ units. The volume of the rectangular prism is ________ cubic units. c Now add another complete layer onto the rectangular prism. Length is ________ units. Width is ________ units. Height is ________ units. The volume of the rectangular prism is ________ cubic units.

d Copy and complete the following table about the rectangular prism you have constructed:

e

Prism

Units in length (l)

Units in width (w)

Units in base (area) (A)

Units in height (h)

Volume of prism (V)

1

3

2

6

1

6

2

2

3

3

i Write a general statement in words about the length, width and area of the prism’s base. ii Write a rule in symbols for the area, A, of the base in terms of length, l, and width, w. iii Write a general statement in words connecting the area of the base and the height and volume of this prism. iv Write two rules for the volume, V, of the prism, the first in terms of A and the height, h, and the second in terms of l, w and h. v Check your findings using prisms of different sizes.

2 Try this for a prism with a length of 4 cubes and a width of 3 cubes. First, build the rectangular prisms. Then complete a table and check that your rule works. 3 a In your own words, describe a rule that would assist you to determine the volume of a rectangular prism of any size. b Does this rule hold for a cube? Construct a variety of cubes to apply and verify your rule. Chapter 6 Area and Volume

173

6J

Volume of rectangular prisms A rectangular prism has the base of a rectangle with straight sides. Finding the volume by counting the cubes takes a lot of time. The short cut is to find the number of cubes in the base layer (which is the same as the area of the base, length × width) and multiplying that by the number of layers (the height).

•••

Volume = area of base × height = length × width × height

Example

Solution

Find the volume of the following solids:

a

Base: 4 cubes across the front by 3 cubes deep 4 × 3 = 12 cubes in the base There are 4 layers. Number of cubes: 4 × 12 = 48 Volume of solid is 48 cubes or 48 cubic units.

b

Base: 8 cubes across the front by 4 cubes deep 8 × 4 = 32 cubes in the base There are 4 layers. Number of cubes: 4 × 32 = 128 Volume of solid is 128 cubes or 128 cubic units.

c

Base: Area = 12 × 4 = 48 cm2 Height is 5 cm Volume = area of base × height = 5 × 48 = 240 cm3 Volume of solid is 240 cm3.

5 cm

4 cm 12 cm

Exercise 6J 1 Use the rule Volume = number of cubes in base × number of layers to find the volume of the following prisms: a b c

174

Maths Dimensions 7

6J 2 Use the rule Volume = area of base × height to find the volume of these solids: a b c 6 cm 4 cm

10 cm

14 cm

20 cm

6 cm 12 cm

7 cm

8 cm

d

e 15 cm

f 10 cm 6 cm

9 cm

10 cm

10 cm

21 cm

10 cm

g

10 cm

h 7 cm 38 cm

4 cm 19 cm

21 cm 23 cm

3 Find the total volume of each of these groups of blocks: a b 16 cm

15 cm

15 cm

15 cm 30 cm 20 cm

15 cm

15 cm

30 cm 20 cm

20 cm

4 State the dimensions of three rectangular prisms with volumes of: a 54 cm3 b 36 cm3 c 48 cm3

d 144 cm3

5 State the dimensions of groups of blocks with the following volumes: a 2 blocks with volume 24 cm3 b 3 blocks with volume 42 cm3 Chapter 6 Area and Volume

175

6K

Volume of prisms A prism is a three-dimensional shape with a regular cross-section. The volume of any prism can be found by using the rule:

••

Volume of a prism = area of base × height

Example

Solution

Find the volume of the following prisms:

a 12 cm

Volume = area of base × height = 24 × 12 = 288 cm3

Area of base: 24 cm2

Volume = area of base × height

b 7 cm

4 cm

Base area =

1 --2

×4×5

= 10 cm2 Volume = 10 × 7 = 70 cm3

5 cm

Exercise 6K 1 Find the volume of these solids: a

b 32 cm

11 cm

Area of base: 49 cm2

Area of base: 54 cm2

c

d 40 cm

73 cm

Area of base: 98 cm2

176

Maths Dimensions 7

Area of base: 123 cm2

6K 2 Find the volume of the following prisms—they have been tipped over on their sides so that the base is facing you: a b c d 21 cm

21 cm 12 cm

Area of base: 35 cm2

Area of base: 12 cm2

Area of base: 54 cm2

e

f

Area of base: 42 cm2

Area of base: 30 cm2

g

12 cm

21 cm

h

9 cm

18 cm

Area of base: 40 cm2

19 cm Area of base: 102 cm2

Area of base: 51 cm2

3 Find the volume of the following solids by finding the area of the triangular base and multiplying by the height: a b 10 cm 7 cm 12 cm 17 cm 8 cm 9 cm

c

d

18 cm

8 cm

16 cm 6 cm 7 cm 7 cm

e

42 m

f 58 m 22 cm

46 m

20

cm

26 cm

Chapter 6 Area and Volume

177

Puzzles 1 Find the area of each coloured shape then match the letter to the answer below to solve the riddle:

What has two wheels and flies? A

B

R U

H

I

S

N

____ 40

____ ____ ____ ____ ____ ____ ____ 44 10 13 13 7 9 2

____ ____ ____ 13 7 20

2 Calculate the area of each letter then match the letter to the correct area to solve the riddle:

What did the woman in the clothing shop say when she answered the phone? 24

18

20

10

6 12 30

10 30

30

30

8

20

10

30

10 30

5

10 6

8

4

6 16

5

12

8 24

8

25

24

5 12

30 15

178

Maths Dimensions 7

24

6

Ch6 ____ ____ ____ ____ 475 576 432 300

____ ____ ____ ____ ____ ____ 475 576 432 512 464 400

____ ____ ____ ____ ____ ____ 464 480 560 456 420 432

3 Calculate the volume of each prism and match the letter to the answer below to solve the riddle:

What happened to the optometrists who fell over? (All measurements are in centimetres.)

C

8

2

4

1

A

11 18

8 13

2

5

5 3

12

H

20 8

L

P

2

M 3

1

5

4

5

2

3

4 10

T

S

3 1

1 6

9

____ ____ ____ ____ 27 6 24 30

3

8

6

Y 1

E

D

1

____ ____ ____ ____ 75 32 66 24

____ 32

____ ____ ____ ____ ____ ____ ____ ____ ____ 18 160 24 36 27 32 36 80 24

Chapter 6 Area and Volume

179

Applications How big is your hand? Trace around your hand onto a sheet of grid paper to cover the maximum number of whole squares as possible. Estimate the area of your hand by counting the squares. Trace around your foot onto a piece of grid paper and estimate the area of your foot. How much bigger is your foot than your hand? Compare your results with those of other students in the class.

The open box activity You will need several sheets of grid paper that measures 12 cm by 8 cm. Cut square corners out of the paper and fold up the sides to make an open box.

Make three boxes by cutting out squares with edge lengths of 1 cm, 2 cm and 3 cm and calculate the volume of each box. List the boxes in order from largest volume to smallest volume. Cut different-sized squares out of the piece of paper to find the box with the largest volume.

Estimating quantities Cut out squares of paper with edge lengths of 10 cm, 30 cm and 50 cm and put them on the wall next to each other. Write the area of each square on it in bright letters. Choose a rectangular object such as a book or a folder and judge which square has the closest area to it—you can place the objects against the square to help you estimate. Measure the lengths and widths of your chosen object and calculate its area. Record the difference between its area and the area of the square of paper. Repeat this with other rectangular objects. What is the volume of your classroom? Find a classroom which is a large rectangular prism. Your task is to estimate the number of boxes with a volume of 1 cubic metre that will fit into the room, and so estimate the volume in cubic metres. Improve your estimate by estimating the volume of the room in cubic centimetres. Volume = 1 cm3

180

Maths Dimensions 7

Ch6 Mondrian models This Mondrian-style dress shows bold areas of colour. Estimate the area of each of the colours red, blue, yellow and white used to make this dress. Design your own dress or boxer shorts using a Mondrian design.

Triangles The triangles shown below all have a base length of 2 grid units and are 6 grid units high. On dot paper, draw another three different-shaped triangles with the same base length and height. Draw blue lines inside the triangles between the dots. Cut the triangles out and then cut the triangles into smaller shapes along the blue lines. Combine these small shapes to make a rectangle and so find the area of each triangle. Glue your shapes into your book and write a report showing what you found. Repeat this investigation using the parallelograms shown on the grid below.

Rectangles This rectangle has a perimeter of 20 cm. Draw as many other rectangles as possible which have perimeters of 20 cm. Use whole numbers of centimetres only. Record your results in a table and identify a pattern. Repeat the table for rectangles with perimeters of 30 cm, 40 cm and 50 cm. A farmer has 2000 metres of fencing and needs to construct a temporary rectangular enclosure for his sheep. What sized enclosure should he make so that it has the maximum area?

2 cm 8 cm

Chapter 6 Area and Volume

181

Enrichment 1 A path 1 metre wide is to go around each of the following garden beds. Find the area of path for each: a

b

4m

c

22 m

10 m 12 m

14 m 12 m

10 m

16 m 20 m

5m 4m

40 m

2m

4m

5m 4m

8m

38 m

18 m 20 m

14 m

42 m

2 Find the areas of the following shapes. Give your answer in square centimetres: a

b

13 cm

240 mm

260 mm

48 cm

26 cm

320 mm 13 cm

3 The area of a hectare is defined as being that contained in a square 100 metres by 100 metres. Express the area of the following paddocks in hectares: a

b 4000 m

c

100 m 100 m

12 000 m

24 000 m 9000 m 4000 m 4000 m

8000 m

d

7000 m

e

12.6 km

3.2 km 6.2 km 5.8 km

4 Find the area of the shaded regions: a

40 mm

b

c 4m

24 mm

28 m 18 mm

20 m 10 m

15 m 38 m

18 m

182

Maths Dimensions 7

3m

Ch6 5 Bill is about to tile an external wall with tiles that are 20 cm long and 10 cm wide. a How many tiles will be needed if the wall measures 2·4 m by 6·2 m. b If the tiles are sold in boxes of 20, how many boxes should he buy? After checking the tiles he finds that 10% of the 1800 mm tiles have cracks and can’t be used. c How many boxes of tiles should he order now? d How many good tiles will he have left over? 2·4 m Bill now tiles a neighbour’s wall, which includes a door as shown. He uses 20 cm by 40 cm tiles. 10 cm 900 mm 3·8 m e Find the number of tiles needed to complete the job. 6 Find the area of all the surfaces of the following solids: a

b

c

9m

25 m 34 cm 24 m

9m 9m

15 m 7m

12 cm 12 cm

7 A flat rectangular life form from another galaxy measures 6 mm by 8 mm at noon. It doubles its dimensions (length and width) every 2 hours. a Find its area at 2 pm, 4 pm, 6 pm and so on to midnight. b Express the area at each time interval as a percentage of its area at noon. Another organism under investigation is in the form of a cuboid which measures 2 cm tall by 3 cm wide by 4 cm long at noon. It also doubles its length, width and height every 2 hours. c Find the organism’s volume at 2-hourly intervals from noon to midnight. d Express the volume at each time interval as a percentage of its volume at noon. 8 A liquid is poured into this container so that its level rises 1 centimetre every 10 seconds. a How long will it take for the container to be completely full? b What volume of liquid will there be in the container after 10, 20, 30, 40 seconds … until it is full? c How many millilitres of liquid does the container hold? (1 cm3 = 1 millilitre)

8 cm 10 cm 6 cm

12 cm

8 cm

Chapter 6 Area and Volume

183

Revision Exercise 6A 1 Order the following shapes from largest area to smallest area: A

B

C

D

F E

Exercise 6B 2 Estimate the area of the following shapes by using the grids shown: a

b

3 Find the area of each of the rectangles by: i counting the squares ii using the rule a

b 4 cm 5 cm 8 cm 12 cm

c

d

10 cm

4 cm 8 cm

6 cm 4 cm 2 cm 6 cm 8 cm

184

Maths Dimensions 7

Ch6 Exercise 6D 4 Find the area of each of the following rectangles: a

b

c

33 cm

1·8 m

42 cm

7 cm

220 cm 4 cm

5 Find the shaded area: a

b

48 m

32 cm 48 cm

32 m 27 m

37 m

90 cm

70 cm

Exercise 6E 6 Find the area of the following parallelograms: a

b

c 23 cm

21 m

93 km 32 m

12 cm 81 km

7 Find the shaded area: a

b 36 cm 42 cm

56 cm

5·3 cm 11·6 cm

60 cm

c

d

80 cm

24 cm

58 cm 18 cm

42 cm

Chapter 6 Area and Volume

185

Ch6 Exercise 6G 8 Find the area of the following triangles: a

b

32 m

29 cm

21 m

38 cm

c

d

12 cm

6 cm 32 km

16 km

Exercise 6H 9 Count the number of cubes in each solid and so find its volume. The volume of each cube is 1 cubic centimetre: a

b

Learning task 6I 10 Use the rule Volume = number of cubes in base × number of layers to find the volume of this prism:

186

Maths Dimensions 7

Ch6 Exercise 6J 11 Use the rule Volume = area of base × height to find the volume of these solids: a

b

c

6 cm 20 mm 5 cm 12 cm

21 cm 18 mm 19 mm

9 cm 12 cm

Exercise 6K 12 Find the volume of these solids: a

b

c

15 cm 49 mm

42 m

Area of base: 68 cm2

Area of base: 248 mm2

Area of base: 120 m2

13 Find the volume of the following solids by finding the area of the triangular base and multiplying by the height: a

b 25 m

11 mm

21 m 8 mm 18 m

9 mm

Chapter 6 Area and Volume

187

CHAPTER

7 Time and Mass

Time and Mass Galileo’s work in physics and astronomy laid the basis for our space program today. The Mir space station took 4 hours to re-enter the Earth’s atmosphere and splash into the ocean. Compare this with the time it took for a rock to fall from the top of the Leaning Tower of Pisa. The rock took a little more than 3 seconds to fall from the tower which is 55 metres tall.

This chapter covers the following skills: • Using clocks, calendars, timetables and schedules, including use of seconds and the 24-hour day • Producing and using timelines • Calculating time intervals when working with daylight saving, Australian and world time zones • Expressing units of time and mass using different units Time 1 minute = 60 seconds 1 hour = 60 minutes = 3600 seconds 1 day = 24 hours 1 fortnight = 14 days 1 year = 12 months 1 year = 365 days 1 leap year = 366 days

• Using and constructing timetables and calendars • Working with questions involving mass Mass 1 tonne = 1000 kg 1 kg = 1000 g 1 g = 1000 mg

7A

Timelines A timeline is a line with a time scale which shows the position of events in order.

Example 1 Sally found the following timeline for tennis players who have won the Grand Slam since the International Tennis Federation was founded in 1912. From this timeline list the players who have won and the year in which they did so. Maureen Connolly (USA)

1900

1910

1920

1930

1940

1950

Don Budge (USA)

Rod Laver (Aus)

1960

Steffi Graff (Ger)

1970

Margaret Court (Aus)

1980

1990

2000

Martina Navratilova (USA)

2 Place the following TV programs on the timeline below: 10 pm The Fat, 7 pm News, 3:30 pm Play School, 6 am Open Learning, 7 am Mixy, 12:30 pm Catalyst, 4:15 pm Rob, 5 pm Rugrats, 8:30 pm Four Corners, 6:30 pm As Time Goes By, 11 am Pathways to Australian Science, 4 pm Bananas in Pyjamas

Solution 1 Read the timeline from left to right, listing the names and the dates in order.

2

Open Learning

7 am

8 am

Mixy

9 am

11 am

Pathways

1 pm

Year

Don Budge

1938

Maureen Connolly

1953

Rod Laver

1962, 1969

Margaret Court

1970

Martina Navratilova

1984

Steffi Graff

1988, 1994

Rugrats Rob

Catalyst

6 am

Name

3 pm

5 pm

Bananas in Pyjamas Play School

190

Maths Dimensions 7

News

7 pm

Four Corners

9 pm

The Fat As Time Goes By

7A Exercise 7A 1 Place the dates for these food inventions on the timeline below: 1484: Hot dogs 1700: French fries 1890: Peanut butter 1554: Camembert cheese 1923: Vegemite 1870: Margarine 1762: Sandwiches 1861: Jelly beans 1853: Potato chips

1400

2000

2 Lilly enjoys playing netball and researched its history. She found the following information. Show the main events on a timeline. a Netball was adapted from basketball in about 1891 in the United States. b The Ling Association drew up a code of rules in 1901. c First national association was formed in New Zealand in 1924. d England formed a national association in 1926. e Australia formed a national association in 1927. f The International Federation of Women’s Netball Associations was founded in 1960. g The World Netball Championships were first held in 1963. 3 The television programs on Channel 2 from 6 am to 5 pm have been jumbled up. Those in green are shown in the morning, while the ones in red are shown in the afternoon. Place them in order and so find the length of each program. 4:55 Zoo Olympics 5:00 Rugrats 3:30 Play School 4:15 Bob the Builder 6:00 Lifelong Learning 10:45 Africa’s Child 2:00 The Geisha 4:30 Arthur 6:30 Tennis 10:30 Le Club 1:00 Two Years in Galapagos 4:25 Little Monsters 3:00 Tweenies 12:30 Catalyst 9:30 Play School 11:30 Discovering Science 4:00 Bananas in Pyjamas 12:00 World at Noon 10:00 For the Juniors

Chapter 7 Time and Mass

191

7A 4 Draw a timeline and mark on it the year in which the following lighthouses were built in Victoria: 1880: Cape Nelson (Portland) 1884: Cape Otway 1859: Cape Schank 1862: Gabo Island 1890: Point Hicks 1859: Wilsons Promontory 5 Draw a timeline and on it mark the years that correspond to the following events: 1974: Cyclone Tracy hits Darwin. 1908: Canberra selected to be the capital city of Australia. 1930: Phar Lap wins the Melbourne Cup. 1911: The Northern Territory is transferred from South Australia to the Commonwealth. 1961: Sydney’s tram system closes. 1932: The Sydney Harbour Bridge opens. 1956: The Olympic Games are held in Melbourne. 1948: The first Holden car is produced in Victoria. 1985: Floodlights allow night play at the MCG. 1973: The Sydney Opera House opens. 1982: Brisbane hosts the Commonwealth Games. 6 Place the following television programs in telecast order. The programs in green are shown in the afternoon while the programs in black are shown in the morning. 11:40 News 5:00 News 6:00 Digswell Dog Show 4:00 The Making of Rugrats in Paris 9:40 MOVIE: Powder Burn 6:30 Dennis the Menace 5:30 Sports Tonight 8:30 Where on Earth is Carmen Sandiego? 6:00 Blockbuster Entertainment 7:00 The Magic School Bus 2:00 Motorcycle Racing 8:00 Totally Wild 6:30 7th Heaven 9:00 Video Hits 7:30 MOVIE: Robin Hood: Men in Tights 11:30 Geisha 12:00 Bright Ideas

192

Maths Dimensions 7

Time conversions

7B

Time is measured in a number of units: seconds for short times, and days and years for longer events. The ladder shows how to convert between different units of time.

Year The units are connected by either multiplication or division using the numbers on the ladder. When climbing down the ladder, multiply. When climbing up the ladder, divide.

÷ 12

× 12 Month

Week ÷7

×7 Day

÷ 24

× 24

••••••••••

Hour

For years: 1 decade = 10 years 1 century = 100 years 1 year ∼ − 52 weeks 1 year ∼ − 365 days 1 leap year ∼ − 366 days

Example

÷ 60

× 60 Minute

÷ 60

× 60 Second

Solution

Complete the following conversions:

a 2 minutes = ______ seconds b 3 hours = ______ minutes

2 × 60 = 120 seconds

c 5 days = ______ hours d 3 years = ______ days e 240 seconds = ______ minutes

5 × 24 = 120 hours

3 × 60 = 180 minutes 3 × 365 = 1095 days 240 ÷ 60 = 4 minutes

Exercise 7B 1 State the units from the unit ladder above that could be used to measure the time: a to toast a piece of bread b for an oak tree to fully grow from an acorn c between sunrise and sunset d to travel from Melbourne to Geelong by car e to travel from Perth to Melbourne by plane f to play a netball game g between seeing lightning and hearing the thunder h you spend at school each day Chapter 7 Time and Mass

193

7B 2 Convert the following by filling in the spaces: a 9 centuries = ______ years

b 5 1--- centuries = ______ years

c 50 years = ______ decades

d 7 1--- decades = ______ years

e 28 decades = ______ years g 350 years = ______ centuries i 575 years = ______ centuries

f 45 years = ______ decades h 75 years = ______ decades j 2·5 centuries = ______ decades

2 2

3 Convert the following by filling in the spaces: a 2 years = ______ days b 5 years = ______ months c 35 days = ______ weeks d 9 weeks = ______ days e 108 months = ______ years f 5 weeks = ______ days g 2 1--- years = ______ months 4

h 7 1--- years = ______ weeks 2

4 Convert the following by filling in the spaces: a 4 days = ______ hours b 5 hours = ______ minutes c 36 minutes = ______ seconds d 420 seconds = ______ minutes e 120 hours = ______ days f 180 minutes = ______ hours g 3 1--- days = ______ hours 2

i

2 3--- minutes = ______ seconds 4

h 5 1--- hours = ______ minutes 4

j

200minutes = ______ hours

5 Bert boasts that he can balance a ball on his nose for 2 1--- minutes, while Guido says he can 2 do the same for 140 seconds. Who can balance the ball the longest? 6 Una and Colin spend 19 days with their daughter Helen at Warrnambool and 3 weeks with their son Ian at Flemington. With which child did they spend the most time? 7 The Best Bake recipe book states that a particular fruit cake needs to be cooked for 70 minutes. The Aussie Cookbook states that the baking time should be 1 1--- hours. Which 2 book’s recipe uses the shorter cooking time? 8 Samantha has been collecting stamps for 2 1--- years while Sean has been collecting for 2 29 months. Who has been collecting stamps for the longer time? 9 Chris is able to run around an obstacle course in 48 minutes, while Matthew takes threequarters of an hour to complete the course. Who is quicker at completing the course? 10 Students grew crystals for a science experiment at home for the following times: Sam: 6 1--- days Suzie: 150 hours Quoc: 9240 minutes 2 Convert the times into minutes and so state which person’s experiment lasts: a the longest b the shortest

194

Maths Dimensions 7

Time differences and the calendar

7C

•••••

The years before the birth of Christ are represented by the initials BC starting from 1 BC. The years after the birth of Christ, anno Domini, are represented by the initials AD starting from 1 AD.

In a year the months have different numbers of days: Days

Months

31

January, March, May, July, August, October, December

30

April, June, September, November

29

February in a leap year

28

February in a non-leap year

These facts can be remembered using the rhyme: 30 days have September, April, June and November. All the rest have 31, except February Which has 28 days, and 29 days in a leap year. The calendar for the year 2011 is shown here.

January S

M

T

W

February T

F

30 31 2

3

S

S

M

1 4

5

6

7

8

6

7

March

T

W

T

F

S

1

2

3

4

5

8

9 10 11 12

S 6

M 7

April

T

W

T

F

S

1

2

3

4

5

8

9 10 11 12

S 3

M 4

T

W

5

6

T 7

F

S

1

2

8

9

9 10 11 12 13 14 15

13 14 15 16 17 18 19

13 14 15 16 17 18 19

10 11 12 13 14 15 16

16 17 18 19 20 21 22

20 21 22 23 24 25 26

20 21 22 23 24 25 26

17 18 19 20 21 22 23

23 24 25 26 27 28 29

27 28

27 28 29 30 31

24 25 26 27 28 29 30

May

June

S

M

T

W

T

F

S

1

2

3

4

5

6

7

8

9 10 11 12 13 14

S 5

M 6

T 7

July

W

T

F

S

S

1

2

3

4

31

8

9 10 11

3

M 4

T 5

W 6

August T 7

F

S

1

2

8

9

S 7

M

T

W

T

F

S

1

2

3

4

5

6

8

9 10 11 12 13

15 16 17 18 19 20 21

12 13 14 15 16 17 18

10 11 12 13 14 15 16

14 15 16 17 18 19 20

22 23 24 25 26 27 28

19 20 21 22 23 24 25

17 18 19 20 21 22 23

21 22 23 24 25 26 27

29 30 31

26 27 28 29 30

24 25 26 27 28 29 30

28 29 30 31

September S 4

M 5

T 6

W 7

October

T

F

S

S

1

2

3

30 31

8

9 10

2

M 3

T

W

November

T

F

S

S

M

1 4

5

6

7

8

6

7

December

T

W

T

F

S

1

2

3

4

5

8

9 10 11 12

S 4

M 5

T 6

W 7

T

F

S

1

2

3

8

9 10

11 12 13 14 15 16 17

9 10 11 12 13 14 15

13 14 15 16 17 18 19

11 12 13 14 15 16 17

18 19 20 21 22 23 24

16 17 18 19 20 21 22

20 21 22 23 24 25 26

18 19 20 21 22 23 24

25 26 27 28 29 30

23 24 25 26 27 28 29

27 28 29 30

25 26 27 28 29 30 31

Chapter 7 Time and Mass

195

7C Example 1 Using the year 2011 calendar, state the number of days from 1 January to 1 February.

Solution Count the number of days (including the starting date) to the finishing date. A shortcut is to group the number of days in whole months and add them up. 31 + 1 = 32 days (including 1 January and 1 February)

2 How many years are there between: a 75 BC and 15 BC?

75 to 15 BC: 75 − 15 = 60 years

b 23 BC and 56 AD?

23 BC to 56 AD: 23 + 56 = 79 years

3 Which months have 30 days?

From the rhyme above, September, April, June, November have 30 days.

Exercise 7C 1 Use the year 2011 calendar to find the number of days from midnight on 19 March to and including the following dates: a 1 April b 25 April c 30 June d 1 August e 16 August f 24 December 2 Find the number of days between and including the following dates: a 15 December and 10 January b 24 March and 1 August c 16 August and 24 December d 7 February and 24 August in a leap year e 7 February and 24 August in a non-leap year f 20 October and 19 February g 30 June and 1 September h 1 January and 22 May in a leap year i 3 July and 8 December j 20 March and 11 October 3 For each of the following people calculate the number of years they lived: a Aristarchus of Samos (c. 310–c. 250 BC), a Greek astronomer, was reported as being the first to assert that the Earth revolves around the Sun. b Aristotle (384–322 BC) was a Greek philosopher and scientist. He shares with Plato (428–347 BC) and Socrates (470–399 BC) the distinction of being the most famous of the Ancient Greek philosophers. c Eudoxus (408–355 BC), Decartes (1596–1650), Archimedes (287–212 BC), Fermat (1601–1665), Pascal (1623–1662), Newton (1642–1727), Leibnitz (1646–1716), Euler (1707–1783), Lagrange (1736–1813) and Gauss (1777–1855) are all mathematicians.

196

Maths Dimensions 7

7C 4 In the year 2000 the following dates were noted: New Year’s Day: Saturday 1 January Labour Day: Monday 13 March Queen’s Birthday: Monday 12 June Melbourne Cup Day: Tuesday 7 November Christmas Day: Monday 25 December ANZAC Day: Tuesday 25 April Boxing Day: Tuesday 26 December Find the number of days between (i.e. not including) the following days: a Labour Day and the Queen’s Birthday b Labour Day and Boxing Day c ANZAC Day and Queen’s Birthday d Labour Day and ANZAC Day e New Year’s Day and Queen’s Birthday f Melbourne Cup Day and Christmas Day 5 The following dates show the length of school terms for secondary schools in New Zealand for the year 2000: Term 1: 1 February to 7 April Term 2: 26 April to 30 June Term 3: 17 July to 22 September Term 4: 9 October to 8 December Find the number of days that students were at school. 6 Find the number of years the following Kings of Assyria ruled. All the years are BC: a 1132–1115: Ashurreshishi I b 1114–1076: Tiglath-pileser I c 1074–1076: Asharidapil-Ekur d 1073–1056: Ashurbelkala e 1055–1054: Eriba-Adad II f 1053–1050: Shamshi-Adad IV g 1049–1031: Ashurnasirpal I h 1030–1019: Shalmaneser II i 1081–1013: Ashurirari IV j 1012–972: Ashurrabi II k 971–967: Ashurreshishi II l 966–935: Tiglath-pileser II 7 Find the number of years the following Roman Emperors ruled: a Augustus: 27 BC to 14 AD b Tiberius: 14–37 AD c Caligula: 37–41 AD d Nero: 54–68 AD Chapter 7 Time and Mass

197

7D

Time differences in hours and minutes The time of day is usually given using either the older clockface system (such as half past 5 pm) or the digital system (such as 5:30 pm), working in groups of two lots of 12 hours, am for morning and pm for afternoon. Sometimes the time of day is given using a single block of 24 hours known as the 24-hour clock time system. This system starts at midnight and the hours are numbered from 1 to 24 to the end of the day.

Example

Solution

1 Give the digital time for the following: a quarter past 3 pm b 23 minutes to 4 in the morning c noon d 5 minutes past midnight 2 Give the clockface time for these digital times:

3:15 pm 3:37 am 12:00 12:05 am

a 3:07 am b 7:43 pm 3 Give the 24-hour clock time for these times:

7 minutes past 3 am

a 4:23 am b 2:17 pm c a quarter to 5 pm d 13 minutes past 9 am 4 How long is it from 7:35 am to 3:20 pm on the same day?

0423 hours

17 minutes to 8 pm

2 + 12 = 14. Time is 1417 hours. digital: 4:45 pm = 1645 hours 0913 hours 7:35 to 8 am = 25 minutes 8 am to 3:20 pm = 7 hours 20 minutes Total = 7 hours 45 minutes

Exercise 7D 1 Give the digital time for the following: a 0420 hours b 0931 hours c 0832 hours d 1204 hours e 1600 hours f 1912 hours g 2121 hours h 2302 hours i 23 minutes past 4 pm j quarter past 6 am k 14 minutes past 11 am l 24 minutes to 11 pm m 12 minutes to 4 am n 4 minutes to 8 pm 2 Give the 24-hour clock time for the following: a 5:10 am b 7:51 am c 5:11 am e 5:20 pm f 7:38 pm g 8:21 pm i 16 minutes past 5 am j quarter past 7 am l 13 minutes to 10 pm m 8 minutes to 7 pm

198

Maths Dimensions 7

d 1:13 pm h 11:52 pm k 17 minutes past 4 am n 17 minutes to 11 pm

7D 3 Give the clockface time for the following: a 4:15 am b 7:30 pm c 5:10 am f 0900 hours g 0118 hours h 1310 hours

d 9:35 am i 1525 hours

e 11:21 am j 1819 hours

4 Complete the following table: Clockface time

Digital time

24-hour time

Half past 7 in the morning 2:15 am 1425 hours 8:10 pm 2111 hours 16 minutes past 5 pm 23 minutes to 11 in the morning 0011 hours 12:29 pm

5 Give the digital and 24-hour clock time for the time shown on these clockfaces. Each one shows the time in the morning: a

11

12

b

1

10 9 8

11

12

10 9 8

2 3 4

c

1 2 3 4

12

d

1

10 9 8

7 6 5

7 6 5

11

2 3 4 7 6 5

11 12 1 10 2 9 3 8 4 7 6 5

e

11 12 1 10 2 9 3 8 4 7 6 5

6 Give the digital and 24-hour clock time for the time shown on these clockfaces. Each one shows the time in the afternoon: a

11 12 1 10 2 9 3 8 4 7 6 5

b

11 12 1 10 2 9 3 8 4 7 6 5

c

11 12 1 10 2 9 3 8 4 7 6 5

d

11 12 1 10 2 9 3 8 4 7 6 5

e

11 12 1 10 2 9 3 8 4 7 6 5

7 Below is a part of the schedule for a new television channel. Find the length of each television program if there are no commercials: 6:15 am Return of the Nerds 7:42 am The Bouncing Bees 8:15 am Cooking with Gas 9:11 am Home Shopping 10:32 am The Nerds Fight Back 11:01 am Battling Bullants 12 noon The Bullsons 12:49 pm Baffling Bullherds 2:15 pm Carpet Madness 3:10 pm Inky Darkness 4:05 pm Tickling the Tonsils 5:00 pm Yell and Scream 6:15 pm The Nerds are Everywhere

Chapter 7 Time and Mass

199

7E

Using timetables Timetables are used to let people know when services, varying from school lessons to the movements of trains, are available.

Example This is the morning train timetable going from the City to Broadmeadows. Spencer St station

8:39 am

8:56 am

9:16 am

9:35 am

North Melbourne

8:42

8:59

9:19

9:38

Kensington

8:45

9:02

9:22

9:41

Newmarket

8:47

9:04

9:24

9:43

Ascot Vale

8:49

9:06

9:26

9:45

Moonee Ponds

8:51

9:08

9:28

9:47

Essendon

8:53

9:10

9:30

9:49

Glenbervie

8:55

9:12

9:32

9:51

Strathmore

8:56

9:13

9:33

9:52

Pascoe Vale

8:58

9:15

9:35

9:54

Oak Park

9:00

9:17

9:37

9:56

Glenroy

9:03

9:20

9:40

9:59

Jacana

9:05

9:22

9:42

10:01

Broadmeadows

9:08

9:25

9:45

10:04

1 Use the timetable to find the time taken to travel from: a Spencer St station to Essendon b Ascot Vale to Broadmeadows 2 Which stations are closest, assuming that the train travels a similar speed between stations? 3 If you needed to be at Strathmore station by 9:30 am, which trains could you take from the city?

Solution 1 a 8:39 to 8:53, or 8:56 to 9:10, or 9:16 to 9:30, or 9:35 to 9:49 Time taken = 14 minutes b 8:49 to 9:08 Time taken = 19 minutes 2 Glenbervie and Strathmore, the train only takes 1 minute to travel between them. 3 The 8:39 and the 8:56 arrive at Strathmore station before 9:30, the others arrive too late.

200

Maths Dimensions 7

7E Exercise 7E 1 There is a ferry service which goes across the Phillip Heads from Queenscliff to Sorrento seven days a week according to the following schedule: Queenscliff

7 am

9 am

11 am

1 pm

3 pm

5 pm

7 pm

Sorrento

7:35

9:35

11:35

1:35

3:35

5:35

7:35

Sorrento

8 am

10 am

12 noon

2 pm

4 pm

6 pm

8 pm

Queenscliff

8:35

10:35

12:35

2:35

4:35

6:35

8:35

a b c d

How long does the ferry crossing take? How long is it between the arrival and the departure of the ferry? How long is it between the first departure and the last arrival for the day? Joylene lives in Sorrento. She is meeting her friend Mai for lunch at 1:15 in Queenscliff but needs to be home by 5 pm. Plan Joylene’s use of the ferries for the day.

2 This is the train timetable for the service between Canberra and Melbourne: Mon–Fri

Sat

Sun

12 noon

8:30 am

9:20 am

Wodonga (arr.)

3:29

12:08 pm

12:55 pm

Wodonga (dep.)

3:45

12:20

1:10

Albury (dep.)

3:55

12:30

1:20

Woomargama

4:35

1:10

2:00

Holbrook (dep.)

6:05

1:20

2:10

Gundagai (arr.)

6:15

2:50

3:40

Gundagai (dep.)

6:45

3:20

4:10

Yass

7:50

4:25

5:15

Canberra City

8:35

5:10

6:00

Melbourne (Spencer St)

a When does the train leave Woomargama for Canberra on: i Wednesday? ii Saturday? iii Sunday? iv Friday? b At what time do the trains travelling from Melbourne to Canberra arrive at Wodonga during the week? c What is the time taken to travel from Melbourne to Canberra for each service? d How long does the train spend at Gundagai between arrival and departure?

Chapter 7 Time and Mass

201

7F

Time zones Australia is a large continent and the different states have different times. The Earth spins on its axis, so the east coast is the first part of Australia to see the sunrise. The west coast moves into the sunlight later. The time zones of Australia and New Zealand are shown on this diagram, which also shows the time differences between the zones. Western Standard Time (WST)

Central Standard Time (CST)

Eastern Standard Time (EST)

New Zealand Standard Time (NZST)

NT QLD WA SA NSW NEW ZEALAND VIC TAS

– 1 12 h

– 12 h

+2h

Example

Solution

If it is 2 pm in Melbourne, what time is it in:

a Sydney?

Sydney (EST) is in the same time zone as Melbourne (EST): 2 pm

b Adelaide?

Adelaide (CST) is 1--- hour behind 2 Melbourne (EST): 1:30 pm

c Perth?

Perth (WST) is 2 hours behind Melbourne (EST): 12 pm

d Auckland?

Auckland (NZST) is 2 hours ahead of Melbourne (EST): 4 pm

Exercise 7F

202

1 If it is 3 pm in Melbourne find what the time is in: a Alice Springs (NT) b Perth (WA) d Christchurch (NZ) e Launceston (TAS)

c Canberra (ACT) f Brisbane (QLD)

2 If it is 6 pm in Auckland find what the time is in: a Sydney (NSW) b Hobart (TAS) d Perth (WA) e Canberra (ACT)

c Adelaide (SA) f Darwin (NT)

Maths Dimensions 7

7F 3 Complete the following table: Time in Perth

Time in Adelaide

Time in Melbourne

Time in Auckland

4:00 am 5:30 pm 7:15 am 12 noon 12 midnight

4 A plane takes 45 minutes to travel from Melbourne to Sydney. If it leaves at 2:17 pm what time will it touch down at Sydney airport? 5 A family travels by car from Adelaide to Canberra non-stop. If they left at 0430 hours (Adelaide time) and arrived at 1749 hours (Canberra time), how long did the journey take? 6 A plane leaves Auckland at 3 pm and flies directly to Perth. If the flight takes 5 hours and 19 minutes, at what local time does the plane touch down in Perth? 7 A plane leaves Melbourne at 1715 hours and arrives in Auckland at 2230 hours local time. Quoc Tran wants to travel on this flight. It is recommended that people check in at the airport 30 minutes before the flight leaves. It is also known that it will take about 45 minutes for people to pick up their luggage and leave the airport at Auckland. a At what time should Quoc Tran arrive at Melbourne airport? b At what time will he leave Auckland airport? c How long will the journey take in total? 8 During summer in Australia some states Queensland Northern and territories adopt daylight saving Territory time, during which the clocks are turned –1 hour forward 1 hour so that 9 am becomes New South Wales Western 10 am. Victoria, New South Wales, the Australia Victoria ACT Australian Capital Territory, Tasmania and South Australia adopt daylight South saving time. Queensland, Western Australia Tasmania Australia and the Northern Territory do not change their time. a Using the above information, complete the time connections shown on this diagram with Victoria and New South Wales as the reference points. b If it is 5 pm in Melbourne what is the time in the following? i Perth ii Canberra iii Alice Springs iv Adelaide v Hobart c If it is 11 am in Adelaide what is the time in the following cities? i Perth ii Sydney iii Darwin iv Brisbane v Canberra

Chapter 7 Time and Mass

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7G

Ordering events and flow charts In order to complete a task efficiently, the order in which events need to occur must be planned carefully. This is true whether we are solving a maths problem, building a dog kennel or making a cup of coffee. Careful planning leads to successful completion of tasks. In simple flow charts the following boxes are used: Used for Start or Stop

Used for decisions: Yes or No

Used for instructions

Example

Solution

Explain the steps in this chart for tuning in to a favourite radio program that you think starts some time after 7 pm Start at 7 pm

1 NO

Can I find the radio?

Look for the radio.

2

Turn off the radio and wait for 10 minutes.

5

YES

Turn on the radio and tune into the station.

3

4 NO

Is the show that I want to hear playing? YES Listen and enjoy the show.

Stop

204

6

7

Maths Dimensions 7

Step 1: Can you find the radio? If not keep looking for it (Step 2). Step 3: Turn on the radio and tune in to the station. Step 4: Is the show playing? If not wait 10 minutes and try again (Step 5). If it is, listen to the program and enjoy it (Step 6). Step 7: Turn off the radio.

7G Exercise 7G 1 Explain the steps in the following flow charts: a

b Start

Start

2

1 NO

Is the dog hungry? YES

Wait for 2 hours.

NO

Go to bed now?

Watch television for another 15 minutes.

YES

Give the dog some dry food.

3

3

Sleep well.

4

4 NO

Is the dog still hungry?

YES

NO

Did the alarm clock ring?

YES

Feed the dog until it has had enough to eat.

Stop

2

1

6

5

6

Get up, have a shower, have breakfast and go to school.

Stop

5

Get up late, get dressed and rush to school.

7

2 Place these steps in the correct order in a flow chart: a Getting up in the morning and b Boiling and eating an egg going to school ■ Take egg out of boiling water after ■ Get out of bed 3 minutes ■ Go to the kitchen ■ Break shell and eat the egg ■ Woken by the alarm clock ■ Boil water in saucepan ■ Get dressed ■ Put egg into boiling water ■ Go to school ■ Get egg from fridge ■ Eat breakfast ■ Put hot egg under cold water ■ Get ready for school

Chapter 7 Time and Mass

205

7G 3 Complete this flow chart, which tests whether a number is odd or even:

4 Complete this flow chart, which tests whether a number is divisible by 6:

Start

Start

Divide number by 2.

Is the number even?

YES Is there a remainder?

YES

Is the sum of the digits divisible by 3?

Number is not divisible by 6.

NO

5 The photographs below were taken from exactly the same position at various stages during a race. What is the correct order of the photographs? a b

c

206

Maths Dimensions 7

d

7G 6 The flow chart below shows one strategy for staying up past your bedtime. Create your own original flow chart to describe an interesting feature of your life.

Permission has not been granted to reproduce the illustration for this page on this CD. Please see Maths Dimensions 7 Coursebook for the flow chart ‘How to stay up past your bedtime’.

Chapter 7 Time and Mass

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7G 7 The photographs below show a double-storey renovation of a house at Phillip Island. The photographs show four stages: O original house 1 frame 1st storey 2 frame 2nd storey F finished 2nd storey Each stage is photographed from three directions: R rear FL front left FR front right For each of the 12 photographs identify the building stage and the direction from which it was photographed.

208

a

b

c

d

e

f

Maths Dimensions 7

7G The plan shows the original ground floor plan and the new ground floor plan.

Rear

Rear

rear view

rear view

original building

renovated building

front left view

front right view

Front

g

h

i

j

k

l

front left view

Front

front right view

Chapter 7 Time and Mass

209

7H

Mass and conversion of units of mass The mass of an object is a measure of how heavy it is. The metric system uses the units of grams, kilograms and tonnes to measure mass. The mass of the Earth is about 5 940 000 000 000 000 000 000 tonnes, while the mass of the largest animal to have lived, the blue whale, is 190 tonnes. Here are some examples of objects that are 1 unit of mass: A grain of sand

Two buttons

1 milligram

1 gram

This bag of sugar

A small car

1 kilogram

1 tonne

The relationship between these units is: 1 tonne = 1000 kilograms

1 kilogram = 1000 grams

1 gram = 1000 milligrams

tonne (t) The units are connected by either multiplication or division using the numbers on the ladder. When climbing down the ladder, multiply. When climbing up the ladder, divide.

Example

× 1000

÷ 1000 kilogram (kg)

× 1000

÷ 1000 gram (g)

× 1000

÷ 1000 milligram (mg)

Solution

Complete the following conversions:

a b c d e f

210

2 tonnes = ______ kilograms

2 × 1000 = 2000 kilograms

5 kilograms = ______ grams

5 × 1000 = 5000 grams

9 grams = ______ milligrams

9 × 1000 = 9000 milligrams

6400 kilograms = ______ tonnes

6400 ÷ 1000 = 6·4 tonnes

3900 grams = ______ kilograms

3900 ÷ 1000 = 3·9 kilograms

12 000 milligrams = ______ grams

12 000 ÷ 1000 = 12 grams

Maths Dimensions 7

7H Exercise 7H 1 List the unit that is most appropriate to express the mass of the following: a an ant b a brick c a car d a person e a pencil f a trailer load of sand g a feather h twenty big books i an elephant j 3000 litres of petrol k a truck l a piece of paper 2 a 2 tonnes = _______ kilograms c 1·02 t = ________ kg e 540 kg = _______ t

b 3·6 t = _______ kg d 3090 kg = ________ t f 21 500 kg = ______ t

3 a 5 kilograms = ______ grams c 10·02 kg = _______ g e 3254 g = _______ mg

b 6·08 kg = _______ g d 7 grams = _______ milligrams f 990 g = _________ mg

4 a 4000 milligrams = ______ grams c 1200 mg = _____ g e 5·4 g = ________ mg

b 8·3 g = ________ mg d 620 mg = ______ g f 310 mg = ______ g

5 a 4300 grams = _______ kilograms c 450 g = _______ kg e 340 kg = ______ t

b 8720 g = ______ kg d 2600 kilograms = _______ tonnes f 9900 kg = _______ t

6 Helen went shopping for varieties of mushrooms and bought bags containing 450 g, 1·0 kg and 672 g from three different shops. What was the total weight, in: a grams? b kilograms? 7 Julia bought a 3-kilogram container of ice cream. Julia ate 435 grams, Juan ate 50 grams and 1·2 kilograms was used in milkshakes. Find the amount of ice cream that was left, in: a grams b kilograms 8 Enormous Egbert the elephant was put on a short diet. Before beginning the diet he weighed 1323·4 kg. Over the next five weeks he lost 1·2 kg, 0·45 kg, 1·02 kg, 0·7 kg and 2·04 kg. Find the weight in grams and kilograms that: a Egbert has lost altogether b Egbert weighs at the end of the diet 9 Chips are sold in packets weighing 50 g, 100 g, 200 g and 250 g. Each size of packet is packed in boxes of 200 packets. Each empty box weighs 4·5 kg. Find the weight of a box of 200 packets for each packet size. Give your answer in grams and kilograms. 10 William, Robby and Harrison plan to start a worm farm and are shown here choosing worms from a compost heap. They find that on average the worms weigh 12 g. Find the total weight of worms if William wants 1200, Robbie 1500 and Harrison 580 worms. Chapter 7 Time and Mass

211

Puzzles 1 Complete the cross-number puzzle. 1.

2.

3.

4.

5. 6. 9.

7.

8. 10. 12.

11. 13.

14.

Across

Down

Find the number of seconds in: Find the number of minutes: 1. in 3 days 1. 67 weeks 1 3. in 6 --- hours 2. 40 minutes – 36 minutes 3 seconds 4 5. between 4:30 am and 5:15 pm 3. 0·55 minutes 6. in 1·3 hours – 50 minutes 4. 95 weeks 8. in 3 hours 10 minutes – 1 hour 45 minutes 7. 1 minute 23 seconds 9. between midnight and 2:15 am 8. 15 minutes 16 seconds – 1 minute 12 seconds 11. between 11:38 am and 12:24 pm 10. 1 h 25 minutes 21 seconds 12. in 19 weeks and 3 days 13. 36 minutes 5 seconds – 22 minutes 14. in 10 weeks 37 seconds

212

Maths Dimensions 7

Ch7 2 Calculate the elapsed time in each question below. Use the answer and the matching letter to solve the riddle:

Why didn’t the alarm clock work?

D 1 pm to 7 pm U 3 am to 7 am S 5:30 am to noon B 11:30 am to 7:30 pm E 3:10 pm to 7:40 pm A midnight to 1:20 pm ____ 8h

____ 4 1--- h

____ 10 h

____ 14 h

____ 4 1--- h

____ 4 1--- h

2

2

2

C noon to 10 pm I 12:30 am to 8 am N 4 am to 6 pm H 5:25 am to 10:15 am T 6:15 am to 11:55 pm

____ ____ 13 h 4h 20 min ____ 6h

____ 4 1--- h 2

____ 6 1--- h 2

____ 6h

____ 4 1--- h

____ 7 1--- h

2

2

____ 13 h 20 min

____ 17 h 40 min ____ ____ ____ 4h 13 h 14 h 50 min 20 min

____ 6h

3 Complete the mass conversions below and match the letters to the correct answers to solve the riddle:

What do you get when you cross a frog with a calendar?

A 1250 mg = ______ g C 250 kg = ______ t G 0·64 t = ______ kg P 1200 mg = ______ g T 0·001 25 kg = ______ mg U 950 g = ______ kg _____ _____ _____ 12 250 0·95 _____ _____ _____ _____ 120 6·4 1·25 1·2

B 0·95 kg = ______ t E 6400 kg = ______ t L 0·12 t = ______ kg R 0·002 g = ______ mg O 0·25 t = ______ kg Y 0·012 kg = ______ g _____ _____ _____ 640 6·4 1250

_____ 1·25

_____ _____ _____ _____ 12 6·4 1·25 2

Chapter 7 Time and Mass

213

Applications 1 Make a poster by drawing a timeline that represents the following sequence of the Lunar spacecraft expeditions: 1959: March, Pioneer 4 (flyby) 1961: August, Ranger 1 (attempted test flight) November, Ranger 2 (attempted test flight) 1962: January, Ranger 3 (attempted impact) April, Ranger 4 (impact) October, Ranger 5 (attempted impact) 1964: January, Ranger 6 (impact) July, Ranger 7 (impact) 1965: February, Ranger 8 (impact) March, Ranger 9 (impact) 1966: May, Surveyor 1 (landed) August, Lunar Orbiter November, Lunar Orbiter 2 1967: February, Lunar Orbiter 3 April, Surveyor 3 (landed) May, Lunar Orbiter 4 August, Lunar Orbiter 5 September, Surveyor 5 (landed) November, Surveyor 6 (landed) 1968: January, Surveyor 7 (landed) December, Apollo 8 (manned orbiter) 1969: May, Apollo 10 (orbiter) July, Apollo 11 (manned landing) November, Apollo 12 (manned landing) 1970: April, Apollo 13 (manned landing—aborted) 1971: January, Apollo 14 (manned landing) July, Apollo 15 (manned landing) 1972: April, Apollo 16 (manned landing) 1972: December, Apollo 17 (manned landing) Find out any interesting events that occurred on these missions and show them on your display. 2 Make a poster which shows Sir Donald Bradman’s century innings. Innings with * were ‘not out’ innings. Tests against England 1929: December, 112 runs March, 123 runs 1930: June, 131 runs June–July, 254 runs July, 334 runs August, 232 runs 1933: December ’32–January ’33, 103 runs 1934: July, 304 runs August, 244 runs 1937: January, 270 runs January–February, 212 runs February–March, 169 runs 1938: June, 144* runs June, 102* runs July, 103* runs 1946: November–December 187 runs December, 234 runs 1948: June, 138 runs July, 173* runs Tests against West Indies 1931: January, 223 runs February, 152 runs Tests against India 1947: November–December, 185 runs 1948: January, 132 runs January, 127* runs January, 201 runs Tests against South Africa 1931: November–December, 226 runs December, 112 runs December ’31–January ’32, 167 runs 1932: January–February, 299* runs Research the events that occurred during any of the above innings and include them on your display.

214

Maths Dimensions 7

Ch7 3 The Chinese calendar operates on a 12-year cycle using 12 animals. The cycle begins with a rat and ends with a pig. List the twelve animals in the correct order beginning with the rat. Ask a number of people who live in your area the year they were born. The year 1900 was the year of the rat, so work out the animal that governs the birthday of the people in your survey. People born in the years with different animals are said to have different characteristics. List the characteristics of people born using the Chinese calendar and write a report saying whether the characteristics are true for the people in your survey. 4 The world is divided into 24 different time zones. The prime meridian running through Greenwich, near London, is where the time zones start. The International Date Line is on the opposite side of the Earth. It is the line which divides one day from another. East of the line it is 1 day earlier than to the west of the date line.

–3

+10 –9

+5

–7

+12 +9

+7

–8

+4

–6

+8

+3 +6

ATLANTIC

International Date Line

+1

+8

OCEAN

Prime meridian

–3

PACIFIC OCEAN –3

1 AM –11

2 AM –10

3 AM –9

4 AM –8

5 AM –7

6 AM –6

7 AM –5

Standard time zones (even)

8 AM –4

9 AM –3

10 AM –2

11 AM –1

12 PM 0

INDIAN OCEAN

1 PM +1

2 PM +2

3 PM +3

Standard time zones (odd)

4 PM +4

5 PM +5

6 PM +6

7 PM +7

8 PM +8

9 PM +9

10 PM +10

11 PM +11

12 PM +12

Non-standard time zones

a Make a poster showing the international time zones above and explain what the map shows. Research the non-standard time zones and add them to your poster. b State the countries that are in the same time zone as your town or city. c Find a city in another country that is 1, 2, 3, 4 and 5 hours ahead of the time where you live. d Find a city in another country that is 1, 2, 3, 4 and 5 hours behind the time where you live. e Select a variety of cities from different zones and give examples of how the times in these zones relate to each other. f You have friends living in London, Tokyo, Beijing, Los Angeles, New York, Budapest, Toronto, Paris, Athens, Singapore and Cape Town. At what time should you phone them at home if they are only available from 10 to 10:30 am on Monday morning? Chapter 7 Time and Mass

215

Enrichment 1 Draw a timeline and mark on it the times for the rise and set of the following planets: Planet

Rise

Set

Mercury

5:08 am

6:35 pm

Venus

8:59 am

7:38 pm

Mars

11:09 pm

1:48 pm

Jupiter

12:52 pm

10:44 pm

Saturn

12:05 pm

10:19 pm

Find the length of time that each planet is above the Earth’s horizon.

2 Our calendar does not exactly fit a regular mathematical system. Every 4 years, 1 day needs to be added. The length of 1 year is approximately 365 1--- days. 4 a Find the number of days in: i 5 years in a row ii 6 years in a row iii 7 years in a row We have 3 years in a row of 365 days and a leap year of 366 days, which occurs every 4 years. A leap year is usually a year that is divisible by 4. The rule is different for turn of the century years like 1900, 2000. These are leap years only if divisible by 400. b The first Olympic Games in the modern era was held in the year 1896 in Athens. Show that this was a leap year. c State the years in which the following Olympic Games were held. The cities in which they were held are given in order. No games were held in 1916, 1940 or 1944. Second Games, Paris, St Louis, London, Stockholm, Antwerp, Paris, Amsterdam, Los Angeles, Berlin, London, Helsinki, Melbourne, Rome, Tokyo, Mexico City, Munich, Montreal, Moscow, Los Angeles, Seoul, Barcelona, Atlanta, Sydney 3 The following are the term dates for the states and territories in Australia for one year (not a leap year). Use the information in the table to find the number of school days in each state or territory.

216

State or territory

VIC

NSW

QLD

SA

WA

Term 1

25 Jan to 7 Apr

28 Jan to 14 Apr

31 Jan to 20 Apr

31 Jan to 14Apr

1 Feb to 7 Apr

Term 2

26 Apr to 1 May to 23 Jun 30 Jun

2 May to 30 Jun

1 May to 26 Apr to 19 Jun to 1 May to 17 Apr to 7 Jul 30 Jun 14 Sep 30 Jun 23 Jun

Term 3

10 Jul to 15 Sept

17 Jul to 8 Sept

18 Jul to 15 Sept

24 Jul to 15 Sept

17 Jul to 22 Sept

Term 4

2 Oct to 19 Dec

3 Oct to 19 Dec

2 Oct to 15 Dec

3 Oct to 15 Dec

9 Oct to 15 Dec

Maths Dimensions 7

TAS

ACT

15 Feb to 28 Jan to 2 Jun 14 Apr

2 Oct to 20 Dec

NT 31 Jan to 7Apr

17 Jul to 8 Sept

24 Jul to 29Sept

3 Oct to 19 Dec

9 Oct to 15 Dec

Ch7 4 This is the tram timetable for the Moonee Ponds–Footscray line. Most times are missing in the schedule for some tram stops. Copy and complete the table, assuming that the routes travelled are the same for each run. Stop Moonee Ponds Maribyrnong Rd

5:33 am

6:45 am

5:34

6:14

Hotham St

6:17

Maribyrnong River

6:20

Highpoint

5:46

6:44

Monash St

5:51

6:49

Western Automatics

6:32

Geelong Rd

6:35

Footscray Station

6:59 am

5:57

6:50

7:04

6:55

7:09

5 An equilateral triangle has three equal sides, an isosceles triangle has two equal sides and a scalene triangle has no equal sides. Complete this flow chart, which can be used to find the type of triangle.

Start

Are the three sides different?

NO

YES

6 A delivery truck has 20 dozen bottles of lemonade. Each bottle weighs 800 grams and the crate, which holds a dozen bottles, weighs 1·2 kilograms. Four dozen bottles in crates are delivered to the local milk bar. Nine dozen bottles are delivered to the school canteen and nine loose bottles are delivered to the takeaway-food shop. What is the total weight of bottles and crates that were delivered to each place? What is the weight of the load that is left on the truck. Give your answer in both grams and kilograms. 7 The density of a material is a measure of the amount of material per unit volume. The unit of density is usually grams per cubic centimetre. The density of water at 4 degrees Celsius is 1 gram per cubic centimetre, that is a volume of 1 cubic centimetre of water weighs 1 gram at this temperature. Find the weight of water, in grams and kilograms, needed to fill these fish tanks to the top: a

b

28 cm

20 cm 80 cm 30 cm

10 cm

c

1m 2·8 m

28 cm 3·5 m

Chapter 7 Time and Mass

217

Revision Exercise 7A 1 Draw a timeline and mark on it the events that correspond to the following dates: 1981: Australia’s population reaches 15 million 1956: Olympic Games held in Melbourne 1946: The Hills Hoist (clothes line) is invented 1926: Speedo bathers are launched 1933: Australia plays England in the ‘bodyline series’ 2000: Sydney hosts the Olympic Games 1911: VFL players receive payment for the first time 1940: Howard Florey carries out first successful experiments with antibiotics 1930: Sir Donald Bradman scores 334 runs against England 1974: Cyclone Tracy hits Darwin

Exercise 7B 2 Complete the following conversions: a 5 centuries = ______ years c 400 years = ______ centuries e 3 weeks = ______ days g 32 minutes = ______ seconds

b d f h

54 years = ______ decades 3 years = ______ months 7 days = ______ hours 420 seconds = ______ minutes

Exercise 7C 3 Find the number of days between (not including) these dates: a 13 May and 4 June b 1 August and 1 November c 13 December and 1 January d 19 March and 19 May 4 Find the number of years between and including: a 12 BC and 2 BC b 1300 AD and 2001 AD c 230 BC and 540 AD d 120 BC and 1 AD

Exercise 7D 5 Give the digital time for these times: a 0930 hours b 1230 hours d 2142 hours e a quarter past 3 am 6 Draw clockfaces to show the following times: a 5:15 am b 11:31 pm c 0900 hours

c 1600 hours f 20 minutes past 4 am d 1730 hours

e 1720 hours

7 Find the number of hours and minutes between the following times on the same day: a 9:00 am and 2:17 pm b 9:32 am and 5:35 pm c 1500 hours and 1724 hours d 1423 hours and 1900 hours e 0920 hours and 1328 hours f 1431 hours and 1835 hours

218

Maths Dimensions 7

Ch7 Exercise 7E 8 This is the morning bus timetable for Saturday between Northland and St Helena East: Northland

Rosanna

Yallambie

Greensborough

St Helena

St Helena West

St Helena East

7:48

7:58

8:13

8:21 E

8:25

–

8:37

8:18

8:28

8:43

8:51 W

8:55

9:05

–

8:48

8:58

9:13

9:21 E

9:25

–

9:37

9:18

9:28

9:43

9:51 W

9:55

10:05

–

9:48

9:58

10:13

10:21 E

10:25

–

10:37

10:18

10:28

10:43

10:51 W

10:55

11:05

–

10:48

10:58

11:13

11:21 E

11:25

–

11:37

11:18

11:28

11:43

11:51 W

11:55

12:05

–

11:48

11:58

12:13

12:21 E

12:25

–

12:37

The letters E (east) and W (west) represent two different routes through Greensborough. a Jill needs to be in St Helena West by 11 o’clock for a hairdresser’s appointment. Which buses could she catch from Rosanna to make the appointment on time? b How long does the bus take to travel from: i Rosanna to St Helena West? ii Northland to Greensborough? iii Rosanna to Yallambie? iv Northland to St Helena?

Exercise 7F Eastern Standard Time (EST) is 2 hours ahead of Western Standard Time (WST) and half an hour ahead of Central Standard Time (CST). New Zealand Standard Time (NZST) is 2 hours ahead of EST.

9 If it is 4 pm in Sydney, what is the time in: a Brisbane? b Launceston? d Alice Springs? e Christchurch?

c Perth? f Melbourne?

Exercise 7H 10 Complete the following conversions: a 5 tonnes = ______ kilograms c 7·06 t = ______ kg e 3298 g = ______ mg g 4380 milligrams = ______ grams i 9400 mg = ______ g k 780 kg = ______ t

b d f h j l

6·9 t = ______ kg 9 grams = ______ mg 770 g = ______ mg 1760 mg = ______ grams 7620 kilograms = ______ tonnes 5800 kg = ______ t

Chapter 7 Time and Mass

219

CHAPTER

8 Angles

Angles Thousands of years ago, the Ancient Babylonians believed that the Sun moved around the Earth and recorded this journey as 360 days. From this they decreed that 360 units or 360 degrees would be used to describe anything that turned through a full circle. An angle is formed when two lines, or rays, meet at a point called a vertex. Angles are measured in degrees. Angles are a useful mathematical tool in many areas of employment, such as architecture, design, surveying, building and construction, navigation and astronomy.

This chapter covers the following skills: • Naming and classifying different types of angles • Identifying the relationships of complementary and supplementary angles Complementary angles add to 90° Supplementary angles add to 180° Angles in a circle add to 360° • Solving simple problems involving complementary and supplementary angles and angles in a circle • Measuring an angle of any size using a protractor • Using a protractor to draw an angle of any size

8A

Naming angles An angle is formed when two lines, or rays, meet at a point called a vertex. L line or ray a

vertex M

N

When naming angles we can use the three letters positioned at the endpoints of the lines which form the angle. The letter at the vertex is placed in the middle. The symbol for angle is ∠, so the ˆ . angle LMN shown above could be named ∠LMN or ∠NML. It can also be named a°, ∠M or LMN

Exercise 8A 1 Give two names for the angles shown below: a C b

M S K

B

O

c

d

G

P

K

L

M F

2 Name as many angles as possible using the diagram shown below: E

A C B

F

G

D

3 Draw clearly labelled diagrams to represent the angles known as: a ∠ABC b ∠KLM c ∠PQR e s° f t° g r° ˆ ˆ ˆF i AM K j WV Y k BD m ∠A n ∠M o ∠R q m° r y° s t° ˆN u ABˆ C v LM w OPˆ Q

222

Maths Dimensions 7

d h l p t x

∠HYT x° S Aˆ C ∠s a° RSˆ T

Types of angles

8B

When we talk about angles we can describe them according to their size. Angles may be classified as one of the following types: Type of angle

Diagram

Description

Acute angle

An angle between 0° and 90°

Right angle

90° or one quarter turn indicated by the little square

Obtuse angle

An angle between 90° and 180°

Straight angle

180° or one half turn

Reflex angle

An angle between 180° and 360°

Perigon or full circle

360° or one full revolution

Exercise 8B 1 What type of angle is each of the following? a b

c

d

e

f

g

h

i

j

k

l

m

n

o Chapter 8 Angles

223

8B 2 For each of the polygons shown below, classify their internal angles (those contained within the polygon) and their external angles (angles adjacent to the internal angles that add together to give 360°) as acute, right, obtuse etc.: a b c

d

e

3 From the diagram shown name: a two acute angles b two right angles c two obtuse angles d two straight angles e two reflex angles f a full circle or perigon

D I

E

O

F G

4 Identify the angles in the following photographs as acute, obtuse or reflex: a b c

d

224

Maths Dimensions 7

e

f

H

Measuring angles

8C

0 10 2 180 170 1 0 60

30 15 0 40 14 0

170 180 160 0 20 10 0 15 0 0 14 0 3 4

Angles are measured Read angle here by using an instrument 80 90 100 11 0 1 70 called a protractor. The 80 7 00 20 0 60 110 1 60 13 0 unit in which they are 50 0 12 50 0 13 measured is called the degree (°). Protractors are semicircular (i.e. Measure the shape of a half circle) angle from 0° and can measure any angle up to a straight line or 180°. They have two Vertex sets of numbers from Place baseline of protractor on arm of angle 0 to 180° so that you can measure angles from either direction. When you measure an angle with a protractor, place the centre of the protractor on the vertex and one arm on the baseline.

Example

Solution

1 Use a protractor to measure the following acute angle:

0 10 2 180 170 1 0 60

30 15 0 40 14 0

170 180 160 0 20 10 0 15 0 0 14 0 3 4

53°

80 90 100 11 0 1 70 80 7 00 20 0 60 110 1 60 13 0 50 0 12 50 0 13

Make sure that the vertex of the angle is at the centre point and that one arm of the angle is on the baseline of the protractor. The angle is 53°. For angles greater than 180°, measure the smaller angle on the other side of the angle and add this value to 180° to obtain the measure of the required angle.

Example

Solution

2 Use a protractor to measure the following reflex angle:

0 10 2 180 170 1 0 3 0 60 15 0 40 14 0

23°

170 180 160 0 20 10 0 15 0 0 14 0 3 4

80 90 100 11 0 1 70 80 7 00 20 0 60 110 1 60 13 0 2 50 0 1 50 0 13

23° + 180° = 203° The reflex angle is 203°. Chapter 8 Angles

225

8C Exercise 8C 1 Read off the size of each of the angles on the diagrams below: g

d

30 15 0 40 14 0 0 10 2 180 170 1 0 60

b a

h

80 90 100 11 0 1 70 80 7 00 20 0 60 110 1 60 13 0 50 0 12 50 0 13

2 Without measuring the angles below, estimate whether they are: i between 0° and 90° ii between 90° and 180° iii between 180° and 270° iv between 270° and 360° Measure the angles using a protractor and check your estimate.

226

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

Maths Dimensions 7

i j

170 180 160 0 20 10 0 15 0 0 14 0 3 4

170 180 160 0 20 10 0 15 0 0 14 0 3 4

c

e 80 90 100 11 0 1 70 80 7 00 20 0 60 110 1 60 13 0 2 0 1 5 0 50 0 13

0 10 2 180 170 1 0 3 0 60 15 0 40 14 0

f

k l

8C 3 Carefully consider the following pairs of angles: i Without measuring, estimate which angle is the larger, ∠Z or ∠M. ii Use your protractor to measure these two angles. iii Compare your answers to parts i and ii. How accurate were your estimations? a

b Z

M

Z

M

c

d Z

Z M

M

4 Using a protractor, measure the size of each of the angles below to the nearest degree: a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

Chapter 8 Angles

227

8C 5 Measure the size of the angles shown in the illustrations: a b

c

d

e

f

6 Draw a clockface showing the following times and then measure in a clockwise direction the size of the angle between the minute hand and the hour hand at: a 2 o’clock b 3 o’clock c 5 o’clock d 6 o’clock e 8 o’clock f 11 o’clock 7 What is the time when the minute hand is on 12 and the hour hand is at the following angles, measured in a clockwise direction? a 0° b 90° c 180° d 360° e 30° f 120° g 210° h 270°

228

Maths Dimensions 7

Using a protractor to draw angles Example

Solution Step 1: Draw a baseline and mark the vertex at one end with a dot. Step 2: Place the protractor on the baseline with its centre, 0, on the vertex dot. Step 3: Place another dot at the desired angle marked 65°. Step 4: Remove the protractor and draw a line between the vertex dot and the second dot you have drawn. Step 5: Mark in the desired angle with an arc.

170 180 160 0 20 10 0 15 0 0 14 0 3 4

65°

80 90 100 11 0 1 70 80 7 00 20 0 60 110 1 60 13 0 50 0 12 50 0 13

0 10 2 180 170 1 0 3 0 60 15 0 40 14 0

Use a protractor to draw the angle ∠M = 65°.

8D

vertex

Exercise 8D 1 Use your protractor to draw the following acute and obtuse angles. Ensure that they are correctly labelled: a ∠M = 35° b d = 42° c ∠XYZ = 55° d ∠F = 75° e h = 105° f ∠ABC = 158° g k = 167° h ∠AGY = 178° 2 Use your protractor to draw the following reflex angles. Ensure that they are correctly labelled: a ∠M = 185° b d = 197° c ∠XYZ = 205° d ∠F = 214° e h = 224° f ∠ABC = 239° g k = 275° h ∠AGY = 315° 3 Draw a house by using four 90° angles and three 60° angles. 4 Draw a kite which has two 125° angles, a 40° angle and a 70° angle. Use a protractor to draw accurate angles and decorate your kite with an interesting design. Chapter 8 Angles

229

8E

Complementary angles Two or more angles which are next to each other and have a common vertex are called adjacent angles. Adjacent angles are complementary if they add up to 90°.

L N

The sum of the angles ∠LMN and ∠NMO is 90°; therefore ∠LMN and ∠NMO are complementary.

Example

O

M

Solution

1 What is the complement of 63°?

The complement of 63° is 27° because 90° − 63° = 27°

2 Find the size of angle t in the diagram below:

t and 21° are complementary, so subtract 21° from 90° to find the complement. 90° − 21° = 69° t = 69° Angle t = 69°

t

21°

3 Find the size of angle t in the diagram below:

8t and t are complementary, so 8t and t add to give 90° 8t + t = 90° 9t = 90° 9 × t = 90° 9 × 10 = 90° so t = 10° Angle t = 10°

8t t

Exercise 8E 1 Find the size of angle a in each of the following: a

b

70°

c

12°

a

25°

a

a

d

e

f

a 47°

35° 53° a

g

a

h

a 61°

i a 68° a 52°

230

Maths Dimensions 7

8E 2 Find the size of a in each of the following: a

b

c

17°

34°

10°

36°

a

35° 25°

a

a

d

e

f

17°

5° 50°

a

25° a a 20°

17°

3 Find the size of a in each of the following: a

b

c

5a

d a

2a

a + 50° a

a

a

e

a + 30°

f

g

a + 10 a + 60

7a

h 2a

2a 2a

2a + 10

4a

4 Which of the following adjacent angles are complementary? a 25°, 73° b 34°, 66° c 43°, 47° d 52°, 28°

e 67°, 23°

5 Find the complement of: a 15° b 36°

e 89°

c 58°

d 71°

6 If ∠PBR = 35° and ∠PBC = 55°, are they complementary angles? 7 Give an angle ∠AMN that is complementary to ∠LMN = 75°. 8 Draw and label any two angles that are complementary. How do you know they are complementary? 9 a What time is showing on this clockface? b What is the size of the smaller angle between the two hands of the clock to the nearest degree? c What are two times when the two hands could make an angle of 90°? Chapter 8 Angles

231

8F

Supplementary angles Adjacent angles which join to form a straight angle, that is add up to 180°, are known as supplementary angles.

N

L

The angle sum of ∠LMN and ∠NMO is 180°, therefore ∠LMN and ∠NMO are supplementary.

Example

M

Solution

1 What is the supplement of 68°?

180° − 68° = 112° The supplement of 68° is 112°.

2 Find the angle t in the diagram below:

43°, 71° and angle t are supplementary, so subtract 43° and 71° from 180° to find the supplement. 180° − 43° − 71° = 66° t = 66° Angle t = 66°

71° t

43°

Exercise 8F 1 Find the size of angle a in each of the following: a b

a

a

143°

a

124°

125°

c

d a

e

a

135°

f

98°

a

2 Find the size of angle a in each of the following: a b 40°

36°

a

35°

125°

a

c

d 20° 25° a

232

Maths Dimensions 7

35°

a

87°

60°

O

8F 3 Find the size of angle a in each of the following: a b 140°

2a

a 30°

c

d 20°

25° a

a

100°

e

2a

f

53°

a

a

g

5a

60°

110°

a

h a a

a

2a

a

a

4 Which of the following sets of adjacent angles are supplementary? A 10°, 170° B 43°, 137° C 118°, 72° 5 Find the supplement of: a 15° b 56° d 111° e 159°

D 13°, 56°, 111°

c 98°

6 If ∠PQR = 165° and ∠PQC = 15° are they supplementary angles? 7 Give an angle ∠LBC that is supplementary to ∠LBN = 75°. 8 Draw and label any two angles that are supplementary. How do you know they are supplementary? 9 Draw and label any three angles that are supplementary. How do you know they are supplementary? 10 a Use a protractor to measure the two angles made by the crane’s arm and tower. b Check that the two angles are supplementary. 11 Write down a real-life situation in which supplementary angles are used.

Chapter 8 Angles

233

8G

Angles in a circle From the diagram below we can see that there are 360° in a full circle, or one complete revolution. Another name for one complete revolution is a perigon. Smaller angles can be added together to form 360°.

b

a

Angles a + b + c = 360°

c

Example

Solution

Find the size of angle y in each of the following:

a y

y and 276° add together to 360°. To obtain y, subtract 276° from 360°. y = 360° − 276° = 84° Angle y = 84°

276°

b

150°, 45° and y add together to 360°. Therefore to obtain y subtract the other two angles from 360°. y = 360° − 150° − 45° = 165° Angle y = 165°

150° y 45°

c

The sum of the angles shown is 360°. y + 2y + 3y = 360° 6y = 360° y = 360° ÷ 6 = 60° Angle y = 60°

y 3y

2y

Exercise 8G 1 Find the size of angle x in each of the following: a b x x

272° 288°

234

Maths Dimensions 7

8G c

d

289°

x 144°

x

e

f

x 92°

x 251°

2 Find the size of angle a in each of the following: a b

c a

a 88°

a

133°

127°

99°

221°

81°

136°

d

e

f

178°

a

109°

a

71°

8k

62° a 118°

68°

3 Find the size of k in each of the following: a b

46°

c k 2k

k

8k 8k 5k

d

e

f k 120° 2k

2k

k

k k

3k

4 If ∠PQR = 165° and ∠AQR = 15°, what is the measure of ∠AQP so that the angle sum is 360°? 5 Give an angle ∠AMN that joins with ∠LMN = 75° to form one complete revolution. 6 Draw and label any two adjacent angles whose angle sum is 360°. 7 Draw and label any three adjacent angles whose angle sum is 360°. Chapter 8 Angles

235

8H What’s your angle? Introduction Use a dynamic geometry environment to explore complimentary and supplementary angles.

Equipment TI-83Plus/TI-84Plus with Cabri® Jr App™. TIConnectV1.6™ and Cabri® Jr can be dowloaded from the Student CD in the back of this book. For further details see ‘Software downloads’ on the Companion Website.

Technology activity 8H Press the APPS key and start the Cabri Junior application.

Press F2 and select the Segment tool.

Use the arrow keys to draw a segment across the middle of the calculator screen. Press ENTER to start the line and ENTER to finish the line.

Press F2, select Point then Point on.

Place a point close to the middle of the original line segment and press ENTER to place the point.

Press F2 and select Point. Place a point above the original line segment.

236

Maths Dimensions 7

8H Press F2 and select Segment. Construct a segment joining the new point to the point on the original line segment (as shown). Note: The objects flicker when the drawing tool is close. This signifies the new construction will be attached to this object. In this case make sure the points flicker when the segment drawing tool is close, this ensures the segment joins the two points. Press F5, select Measure followed by Angle.

Three points must be selected in the correct sequence to measure an angle. The first point is a point on the line segment. The second point is the vertex of the angle. The third point is a point on the intersecting line segment. These points are labelled in the diagram. Select the points in the order indicated. (Press ENTER on points 1, 2 and then 3.) The angle will be displayed on the screen. Once the angle has been displayed on the screen, press CLEAR twice, then move the selection arrow to point 3 on the diagram. When the selection arrow is over point 3, press ALPHA to lock the arrow onto point 3. Use the arrow keys to rotate point 3 and see what happens to the size of the measured angle. Measure the angle between points 4, 2 and 3 shown on the diagram.

1 Construct the following angles using the line segment. Write down the second angle and comment on your findings: Angle a

Angle b

120° 30° 45°

b

a

10° 170° 90°

2 Construct a line parallel to the original line segment passing through point 3. Measure the four angles indicated in the diagram. Vary the angles and comment on your results.

d b

c

a

Chapter 8 Angles

237

Puzzles 1 Measure the angles below and match the letter to the correct angle to solve the riddle:

Why did the spider want to use a computer? E

A

B D

L

I

N

P

____ ____ 145° 75°

W

S

R

_____ ____ ____ ____ 240° 160° 100° 270°

____ ____ ____ ____ ____ 120° 75° 315° 300° 46°

O

T

____ 25°

____ ____ ____ ____ 120° 100° 46° 15°

____ ____ ____ 120° 15° 37°

2 Calculate the missing angle and match it to the correct letter below to solve the riddle:

What do you get if you cross a cocker spaniel, a poodle and a rooster? A

E

C 65°

63°

22°

100° K

D L L 78°

L

____ 27°

O

48°

P

43°

35° 62°

R T

____ _____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 115° 120° 115° 170° 68° 45° 67° 12° 12° 180° 30° 68°

____ ____ ____ 180° 12° 12°

238

R

65°

Maths Dimensions 7

Ch8 3 Decide whether each statement is true or false. Colour in the squares in the grid below that contain the numbers of the correct answers. You should find the name of a quadrilateral. True

False

An acute angle is between 180° and 360°.

10

1

A straight angle is 180°.

11

6

Complementary angles add to 90°.

9

7

Angles in a circle add to 180°.

2

19

20

3

150° and 30° are complementary.

5

12

Supplementary angles add to 360°.

8

4

A reflex angle is between 0 and 90°.

18

13

A right angle is 90°.

16

17

160° and 200° form a full circle.

15

14

Obtuse angles are between 90° and 180°.

15 2

5 13 10 15 3

1 19 20 3 20 15 1

19 3

4 10 18 4

3

8 20 6

7

6

7

4

8

5 11 2

5 16 1

9

16 17 12 8 17 9

6

7

13 12 18 7 1 14 6

2

9

9 17 2 13 14 10

9 14 16 10 14 11 5 18 19 4 12

The quadrilateral is a ______________________.

Chapter 8 Angles

239

Applications Arrow angles Trace onto thick paper the two arrows shown below then cut them out.

On one write the word ‘baseline’. Hold them together with a paper fastener.

a Start with the two strips together. Keep the baseline fixed and move the other arm, or ray, to create an acute angle, a right angle, an obtuse angle, a straight angle, a reflex angle and a full circle. b Choose an angle from those you’ve made in part a and show it to a partner. Ask them to classify your angle. Now get them to create an angle for you. Repeat this activity five times each.

Paper angles a Fold a piece of A4 paper lengthwise and label the ends of the fold LM. b Fold the paper in half widthwise. Label this line NP and the centre where the lines intersect mark O. c Fold the paper diagonally through the point O. Label these lines QR and ST. d From your piece of paper, name: i three acute angles ii three right angles iii three obtuse angles iv three straight angles

Pizza angles

A

B

C

D

Stephanie was cutting up her pizza when she noticed that she had formed four adjacent angles.

a Measure the four angles Stephanie has created. b What is their sum? c What do you notice about the opposite angles?

Light angles When submarine commanders want to spy on the enemy they put the periscope up out of the water. Use the Internet to research how a periscope works. Write down your findings or create a small presentation to share with your class.

240

Maths Dimensions 7

Ch8 Angles on a compass Different angles can be formed between the points of a compass. Begin at north (N) and continue clockwise around the compass estimating and then calculating the size of the angles formed. Angle formed between N and the compass point

Estimate of angle (in degrees)

Size of angle (in degrees)

N

NE E SE

E

W

S SW W NW

S

N

Angles in polygons i Find out the name of each of the polygon shapes shown below. ii Measure the internal angles in each of the shapes. iii Add up all internal angles and find the angle sum of each of the shapes. a

b

c

d

Golf angles Matt is intrigued by the mathematics involved in designing golfing equipment and golf courses. Matt has been measuring the loft angle of the ball and recording it. Iron Loft angle

3

5

7

9

25°

33°

42°

50°

Using a protractor and a ruler simulate the shots being played by Matt. Mark Matt’s ball on the fairway with a circle on your page. From this ‘ball’ draw a possible trajectory of the ball when it is hit by each of these irons. What do you notice? What would be the advice you give Matt for using each of these golf clubs?

Chapter 8 Angles

241

Enrichment 1 This diagram shows a number of angles. List angles of the following types: a 5 acute angles b 1 right angle c 5 obtuse angles d 1 straight angle

C

D

B E

A

O

F

2 NASA has a new satellite it wishes to launch into space. This satellite, which is shown here, will help scientists track ships at sea as a means of improving safety. a Measure the angles formed between the four panels shown on the satellite. b Identify pairs of supplementary angles.

3 Karen’s grandparents arrive home from their overseas trip with a new watch for her. Karen notices that the hands on her watch complete many revolutions each day. In one 24-hour period, how many revolutions does: a the hour hand make? b the minute hand make? c the second hand make? 4 Parallel lines travel in the same direction and are always the same distance apart. We indicate parallel lines by using arrows as shown below:

i List five different examples of parallel lines occurring in everyday life. ii Label the parallel lines in the following figures, using the pairs of arrows shown above: a b

c

d

e

iii List three examples of parallel lines you observe on your trip to school each day.

242

Maths Dimensions 7

Ch8 5 i Using your protractor measure the angles marked: a b C O O

C

The angles marked C are called corresponding angles.

The angles marked O are called vertically opposite angles.

c

d A

X A

X

The angles marked A are known The angles marked X are called as co-interior angles. alternate angles. ii Using your findings from part i, give a definition for corresponding angles, vertically opposite angles, co-interior angles and alternate angles.

6 Find the size of the marked angle for each of the following diagrams and choose one of the words to describe the relationship between the angles: alternate (X) co-interior (A) corresponding (C) vertically opposite (O) a b c

a

b

105°

70°

60°

c

d

e

f f

e d

118°

52° 58°

7 From this diagram calculate the size of these angles: a ∠AMC b ∠CMN c ∠MNE d ∠BNF e ∠FNM

A C

E

127°

D

M

F

N B

Chapter 8 Angles

243

Revision Exercise 8A 1 Give two names for the angle shown:

X Y

Z

2 Draw clearly labelled diagrams to represent the angles known as: a ∠ALM b ∠PMQ c ∠HTY ˆ ˆ e AF G f OM Q g RSˆ Y

d ∠RSB h LXˆ Y

Exercise 8B 3 For each of the following angles state its type: a b

c

d

e

f

g

h

i

Exercise 8C 4 Using your protractor, measure the size of the following angles: a b c

5 Kathy is a keen golfer and has been practising her game on the seventh hole. She can now reach the green in three shots. Measure the angle between each of her fairway shots.

Exercise 8D 6 Use your protractor to draw the following angles: a ∠A = 25° b a = 137° d ∠LPQ = 98° e ∠S = 123°

244

Maths Dimensions 7

Tee

c ∠UVW = 312° f h = 165°

Ch8 Exercise 8E 7 The angle sum of complementary angles is ______ degrees. 8 Find the size of angle a in each of the following: a b

c

a

a 56° a

69°

d

e

6°

f

3a 6a

4a + 10° a 4a

3a

Exercise 8F 9 The angle sum of supplementary angles is ______ degrees. 10 Find the value of h in each of the following: a b 37° h

46°

h

139°

h

d

c

h

h

e

f

94° 2h 48° h

4h

2h

Exercise 8G 11 One revolution is ______ degrees. 12 Find the value of the pronumerals in each case. a b

68° 68° p

c

m s

112° 147°

n

Chapter 8 Angles

245

CHAPTER

9 Polygons

Polygons Tessellations occur when an area is covered with shapes without any overlaps or gaps. Many tessellations occur in nature. Did you know that the honeycomb made by bees is a tessellation? A tessellated pavement is a flat area of rock marked with a mosaic pattern. This occurs when weathered rock dries out and shrinks. Cracks appear where the rock is weakest, as they do in mud or clay, creating a tessellated pattern.

This chapter covers the following skills: • Identifying a polygon • Defining polygons A polygon is a flat enclosed figure with straight sides • Identifying the relationship between the number of sides of regular and irregular polygons (3-, 4-, 5-, 6-sided) and the sum of their interior angles • Naming triangles according to their side or angle properties • Finding the angle sum of a triangle Angle sum = 180° • Identifying the interior and exterior angle properties of a triangle • Identifying special quadrilaterals and their properties • Finding the angle sum of a quadrilateral Angle sum = 360°

• Recognising angle relationships in parallel lines • Using angle relationships to work out angle size

9A

Triangles: Side properties Special symbols Symbol

Description Side lengths with a different number of dashes have different lengths.

Side lengths with the same number of dashes are the same length.

By marking an angle with a small square we indicate that the angle is a right angle or 90°.

Parallel lines, like tram tracks or train tracks, are always the same distance apart. Sides marked with an equal number of arrows are parallel.

There are many different types of triangles. We can name them according to their side properties or their angle properties. Triangles named according to their side properties are scalene, isosceles or equilateral.

Side properties Type of triangle

Diagram

Description

Scalene

In a scalene triangle all the side lengths are different.

Isosceles

In an isosceles triangle two of the sides are equal in length.

Equilateral

In an equilateral triangle all three sides are equal in length.

Example

Solution

Measure the lengths of the sides in triangle ABC and name it according to its side properties.

BC = 2·1 cm AC = 1·7 cm A

248

Maths Dimensions 7

AB = 2·1 cm

B

C

AB = BC Two sides are equal, therefore ΔABC is isosceles.

9A Exercise 9A 1 Use your protractor to measure the magnitude, or size, of each angle in the following isosceles triangles. Copy and complete the table and fill in the missing words in the sentence below: Triangle

Magnitude or size of angle ∠L

∠M

∠N

a b c d a

b L

L

M

N

N

c

M

d

L

M

L

N N M

An isosceles triangle has ____ sides of equal length and ____ angles of equal magnitude.

2 Use your protractor to measure the magnitude, or size, of each angle in the following equilateral triangles. Use your findings to complete the table and fill in the missing words in the sentence below: Triangle

Magnitude or size of angle ∠L

∠M

∠N

a b c d a

b

L

M

c L

L

M

d L

M

N

N M

N

N

An equilateral triangle has ____ sides of equal length and ____ angles of equal magnitude. Chapter 9 Polygons

249

9A 3 Use your ruler to measure the lengths of each side in the following triangles. Copy the table into your workbook and complete it as you go. Use your results to determine the type of triangle shown (scalene, isosceles, equilateral): Triangle

Length of side LM

Type of triangle

MN

LN

a b c d e f a

b

L

c

L

L M

N

M

d L

M M

N

N

e L

f L N

M N

M

N

4 Without measuring, state whether the following triangles are scalene, isosceles or equilateral triangles: a b c

250

d

e

g

h

Maths Dimensions 7

f

i

Triangles: Angle properties

9B

Triangles named according to their angle properties are acute, obtuse and right-angled. Type of triangle

Diagram

Description

Acute-angled

In an acute-angled triangle, all the angles are acute, that is, all the internal angles are between 0° and 90°.

85° 38°

57°

Right-angled

In a right-angled triangle, one of the internal angles is 90°.

Obtuse-angled

In an obtuse-angled triangle, one of the internal angles is obtuse, that is, between 90° and 180°.

35° 115°

30°

Example

Solution

Measure the magnitude of each angle and name this triangle according to its angle properties: B

A

∠A = 28° ∠B = 123° ∠C = 29° One angle is between 90° and 180° therefore, ΔABC is an obtuse-angled triangle.

C

Exercise 9B 1 Use your protractor to measure the angles in each of the following triangles. Copy and complete the table below, naming each triangle as acute-, right- or obtuse-angled: Triangle

Magnitude or size of angle ∠L

∠M

Type of triangle ∠N

a b c d e f

Chapter 9 Polygons

251

9B a

b

L

c

L

L M

N

M

d L

M M

N

N

e L

f L N

M N

M

N

2 Name the following triangles, giving each an angle name such as acute-, right- or obtuseangled triangle: a b c d

e

f

g

h

3 Name the triangles marked in the following photographs, giving each both a side and an angle name: a b

4 Copy each of the diagrams below and add in the third side to create a triangle. Measure the sides of each triangle and give it both a side and an angle name: a b c d

252

Maths Dimensions 7

Finding the third angle in a triangle If you draw a triangle with the angles 35°, 65° and 80°, and then tear off or fold down each corner, you can form a straight angle by placing each angle next to the other as shown. As there are 180° in a straight angle, we know that the sum of the interior angles in any triangle is 180°.

Example

65° 35°

9C

80°

Solution 125° + 30° = 155°. There are 180° in a triangle, so the size of the third angle is: 180° − 155° = 25° x = 25°

1 Consider a triangle in which we don’t know the size of one of the angles: 125° 30°

x

2 Consider an example where we don’t know the size of two of the angles in the triangle:

There are 180° in a triangle, so x + x + 42° = 180° 2x + 42° = 180° 2x = 180° − 42° = 138° x = 69°

x

x

42°

Exercise 9C 1 Calculate the size of the third angle in each of the following triangles: a b c

m 34°

25°

35°

68° x

x

77°

84°

d

e

126°

x 28°

t

40°

f

32°

32° h

g

h 32°

46° 76°

f

65°

g

Chapter 9 Polygons

253

9C i

j

k

57° 63°

47°

38° 93°

k f

b

l

m

n

p

r

132° 36°

82° m

25°

83°

2 Calculate the size of the pronumeral (letter) in each of the following triangles. Angles with the same pronumeral are the same size: a b c x x 47°

x

124° 93° 74°

x

b

d

e

f

126°

x

2h

3t

3t

32°

32°

x

g

h 3y

i

k

4t

k

t

k 2y

j

m

34°

k

l

r

3p 88°

m

m

n

2x

5y

2a

x

4y

254

Maths Dimensions 7

p

o

3a

80°

60°

Exterior angle properties of triangles Angle C is known as an exterior angle in the triangle shown.

9D

A

In any triangle the sum of the two interior opposite angles is equal to the size of the exterior angle. That is, ∠A + ∠B = ∠C

C B

Example

Solution x = 85° + 62° = 147°

Find the size of the unknown angle: 85°

62°

exterior angle property

x

Exercise 9D Using this property, find the size of the exterior angle marked in the following triangles:

a

b

c m

x 35° 84°

x

57°

34°

25°

d

68°

e

126°

g t

28°

40°

f

32°

32°

g

h 63°

h

i

76°

f

j b

93°

k

57°

46°

k

101° 42°

38°

c

f 47°

Chapter 9 Polygons

255

9E Triangle sums Introduction The purpose of this activity is to explore the properties of the angles contained in a triangle.

Equipment TI-83Plus/TI-84Plus with Cabri® Jr App™. TIConnectV1.6™ and Cabri® Jr can be downloaded from the Student CD in the back of this book. For further details see ‘Software downloads’ on the Companion Website.

Technology activity 9E Press the APPS key and start the Cabri Junior application.

Press F2 and use the arrow keys to select Triangle and press ENTER .

The cursor changes to a pencil and is ready to draw your triangle. Move the pencil to a point near the bottom left-hand corner of the calculator screen and press ENTER . This locates the first vertex (corner) of the triangle. Use the

key and drag the pencil towards the right of the screen.

Press ENTER to place the bottom right vertex of the triangle. Use the

and

arrows to place the top vertex of the triangle.

Press ENTER to locate the top vertex. Press F5, select Measure and then Angle. An angle is marked by the vertices of the triangle. The order in which the vertices are selected is important: A → B → C

To measure the angle at vertex A, press ENTER on point C, then A and finally point B. The angle measurement will automatically be moved adjacent to the angle being measured.

256

Maths Dimensions 7

9E Measure the remaining two angles: ∠ABC and ∠BCA.

1 a Write down the size of each of the angles in your triangle. b Add up your three angles and write down the answer. 2 a Move the vertices around to change the size of all the angles in your triangle. b Write down the size of all your new angles. c Add up your three angles and write down the answer. 3 Compare this new answer with the one obtained in Question 1. It is possible to use the calculator to automatically measure the sum of the angles in the triangle. Press F5 and select Calculate.

Press + and note that the indicator in the top left-hand corner of the screen locks onto a plus sign. Use the arrow keys to move around and press ENTER on two of the angle measurements. This new calculation is dynamic, just like the angle measurements. If the angles are changed, so too is the calculation. Add up all three angles then move the vertices of the triangle around to see how this affects the sum of all the angles in the triangle.

Note: It is only possible to add two angles at a time. If the original sum is ‘distracting’ then use F5 and Hide/Show to remove the original measurements.

4 Write a summary about what you found out about the sum of the angles in a triangle.

Further exploration 5 Adjust your triangle so that it contains an obtuse angle. Where is the longest side of the triangle in relation to the obtuse angle? 6 What is the position of the smallest angle on the triangle relative to the smallest side? 7 Based on your answers to questions 5 and 6, what do you think would happen to the lengths of the sides of the triangle if two angles were equal in size? 8 What would happen to the side lengths if all three angles in the triangle were the same length? Chapter 9 Polygons

257

9F

Quadrilaterals Quadrilaterals are figures with four corners and four sides. There are six main types of quadrilaterals that have special properties. Type of quadrilateral

Diagram

Description

Rectangle

In a rectangle opposite sides are equal, and all angles are equal and measure 90°.

Square

In a square all sides are equal, and all angles are equal and measure 90°.

Parallelogram

A parallelogram has two pairs of opposite sides that are equal and parallel.

Rhombus

A rhombus is a parallelogram with all sides equal.

Kite

A kite has two pairs of adjacent sides equal and one pair of opposite angles equal.

Trapezium

A trapezium has one pair of opposite sides that are parallel.

Exercise 9F 1 Name each of the following quadrilaterals: a b

258

c

d

e

f

g

h

i

Maths Dimensions 7

9F 2 Use the following pictures to identify as many quadrilaterals and triangles as you can: a

b

c

d

e

f

g

h

3 Copy each of the diagrams below and add in two additional sides to create a quadrilateral. From each diagram, can you make a square, a rectangle, a parallelogram, a rhombus, a kite and a trapezium? a b c d

4 How many squares are there on a checkers board?

Chapter 9 Polygons

259

9G

Angle sum of a quadrilateral We can find the angle sum of a quadrilateral with angles measuring 65°, 95°, 85° and 115°, as shown below: 95°

65°

65° 115° 85° 95°

85° 115°

When the corners are placed adjacent to one another, we see that they combine to form a perigon, or full circle. As there are 360° in a full circle, we know that the sum of the interior angles in any quadrilateral is 360°. We could also show that this quadrilateral can be split into two triangles, as shown. There are 180° in a triangle and as any quadrilateral can be shown to be made up of two triangles we can say that the sum of the angles in a quadrilateral is 2 × 180°, which is 360°.

Example

Solution

Find the missing angle: 133° 58°

43° t

The shape is a quadrilateral. There are 360° in a quadrilateral. 133° + 43° + 58° = 234° t = 360° − 234° = 126°

Exercise 9G 1 Use a protractor to measure all four angles in the quadrilaterals shown in the photographs below. Show that the angle sum of each quadrilateral is 360°. a b

260

Maths Dimensions 7

9G 2 a Use a protractor to measure the angles in the quadrilaterals below: i r ii iii r s

s

r

t

s

u

t

u

t

u

iv

r

s

t

u

v

r t

vi

s

r

s

t

u u

b Copy and complete the table using the quadrilaterals above: Shape

angle r

angle s

angle t

angle u

Angle sum

a b c d e f c Write a sentence (in words) describing the relationship between the four angles in any quadrilateral. 3 Find the size of the unknown angle in each of the following quadrilaterals: a b c b

70°

110°

a

c

30°

d

e 65°

e

62°

110°

f f

d

115°

118°

118°

65°

g

60° 120°

h

120°

i

130°

a 9g b

Chapter 9 Polygons

261

9G 4 Find the size of the unknown angle in the following quadrilaterals: a b c n 35° m 47°

35°

134°

d

36°

r

98°

e

154°

103°

96°

129°

f

g

85°

88°

46°

63°

56° r

87°

h

47° 92°

g

105°

h

156°

125°

x

40°

79°

j

k

m

95°

100°

88°

78°

j

i

k

100°

x

l

x

63°

y

120°

y

120°

63°

102°

m

b

n

b

60°

o c

a

110°

f 155°

e

28°

d

5 Find the pronumeral in each of the following quadrilaterals: a b a 3a a

3a

a

c 2a

3a

a

3a

3a

a

d

a

e

a 4a

f

2a

2a

2a 3a

262

Maths Dimensions 7

2a

a

a

Polygons

9H

The word polygon is taken from a Greek word meaning ‘many angles’. A polygon is a flat enclosed figure with straight sides.

•••••••••••

Polygons

Not polygons

not closed

not straight

In a regular polygon all the sides are of equal length and all the angles are equal in size. Number of sides

Shape

Name of polygon

3

Equilateral triangle

4

Square

5

Regular pentagon

6

Regular hexagon

7

Regular heptagon

8

Regular octagon

9

Regular nonagon

10

Regular decagon

Polygons with more than 10 sides do not always have a special name. An 11-sided polygon is called an 11-agon. A polygon with 35 sides is called a 35-agon. In general, a polygon with n-sides is called an n-agon. Chapter 9 Polygons

263

9H Exercise 9H 1 Identify the polygons in the photographs below: a b

d

e

f

2 Name each of the regular polygons below: a b

e

3

f

c

d

g

h

i Use a protractor to measure all internal angles in the regular polygons above. ii Copy and complete the table below. The first row has been done for you: Polygon

a

Name

Equilateral triangle

b c d e f g h

264

c

Maths Dimensions 7

Number of sides

Number of internal angles

Size of internal angles

Angle sum

3

3

60°

180°

Angle sum of polygons

9I

In section 9G we saw that the angle sum of any quadrilateral can be found by splitting the quadrilateral into triangles. We can use this approach to find the angle sum of any polygon. Name of polygon

Polygon split into triangles

Number of triangles

Angle sum of the polygon

Triangle

1

1 × 180° = 180°

Square

2

2 × 180° = 360°

Pentagon

3

3 × 180° = 540°

Hexagon

4

4 × 180° = 720°

Heptagon

5

5 × 180° = 900°

Octogon

6

6 × 180° = 1080°

We can extend the table to generate the angle sum of any polygon.

Example 1 Find the sum of the angles in a dodecagon (12 sides).

Solution There are 10 triangles. Sum of the angles = 10 × 180° = 1800°

When we have polygons with many sides it is not always practical to draw the shape and divide it into triangles. It can be useful to know a relationship between the number of sides and the number of triangles.

Chapter 9 Polygons

265

9I In an n-sided polygon there are n − 2 triangles, so the angle sum in any polygon is (n − 2) × 180°.

Example

Solution

2 Find the sum of the angles in a 21-agon.

There are 21 sides and 21 − 2 triangles. Sum of the angles = (21 − 2) × 180° = 3420°

3 Find the missing angle: a

123°

65°

b

There are 360° in a quadrilateral. ∠m = 360° − (123° + 23° + 65°) = 360° − 211° = 149°

23°

m 123°

83° 116°

65°

h

The shape is a pentagon. There are 540° in a pentagon. ∠h = 540° − (65° + 123° + 83° + 116°) = 540° − 387° = 153°

Exercise 9I 1 Name the following polygons and find their angle sum by first splitting the polygon into triangles: a b c d e

f

g

h

2 Find the total number of degrees, or the angle sum, of the following polygons without drawing them first: a 15-agon b 17-agon c 23-agon d 35-agon e 42-agon f 50-agon g 74-agon h 150-agon i 33-agon j 13-agon k 51-agon l 11-agon m 20-agon n 37-agon o 17-agon p 52-agon q 61-agon r 73-agon s 43-agon t 49-agon

266

Maths Dimensions 7

9I 3 Find the value of the missing angle in each of the following polygons: a b m 135° 123° 60°

124°

116°

105°

64°

y

115°

c

d

r 97°

92°

115°

113°

s

136°

128°

133°

137°

135° 144°

124°

e

f

153°

k

135°

80°

115°

y

91° 62°

150°

141°

117°

4 Find the value of the missing angle in each of the following polygons. Angles marked with the same pronumeral (letter) are the same size: a b 113° k j 52° 75° 71°

63°

130°

c

d

r 97°

92°

135°

119°

s s

e

h

119°

125°

r

113°

s

129°

f

h

76° h

h h

132°

136°

g

g

h

Chapter 9 Polygons

267

9J Exploring polygon constructions Learning task 9J Triangles and quadrilaterals can be constructed by using a compass and a straight edge.

Equilateral triangle

C

Side length of 4 cm ■ Draw a 4-cm baseline. Label the endpoints A and B. ■ Draw arcs from points A and B to intersect above the line. ■ Label the point C. ■ Use a straight edge to join the points A, B and C.

Right-angled scalene triangle Side lengths 3 cm, 4 cm, 5 cm ■ Rule a 5 cm baseline and label the endpoints A and B. ■ Open the compass to 4 cm and draw an arc from point A. ■ Open the compass to 3 cm and draw an arc from point B to intersect the other arc above the line. ■ Label this point C. ■ Use a straight edge to join the points A, B and C.

A

B

4 cm C

4 cm

A

3 cm

B

5 cm

Quadrilateral Trapezium with side lengths 2 cm, 2 cm, 3 cm, 5 cm ■ Rule a 5 cm baseline and label the endpoints A and B. 2 cm ■ Open the compass to 2 cm and draw arcs from points A and B. A ■ Ruler a line where the two arcs are exactly 3 cm apart. ■ Label the points of intersection C and D. ■ Use a straight edge to join the points A, B, C and D.

C

1 Construct the following polygons: a an equilateral triangle of side length 3 cm b a scalene triangle with side lengths 2 cm, 4 cm, 6 cm c an isosceles triangle with two sides 3·5 cm and one side 7 cm d a right-angled triangle with side lengths 4·5 cm, 6 cm and 7·5 cm e a square of side length 4 cm f a kite with two sides 4·5 cm and two sides 6 cm g a rectangle with two sides 2·5 cm and two sides 7 cm h a trapezium with side lengths 2·8 cm, 2·8 cm, 3·6 cm, 4·5 cm

268

Maths Dimensions 7

3 cm

D 2 cm

5 cm

B

9J Regular polygons can be drawn in a circle by using a ruler and a protractor to measure the required angles.

Pentagon ■ ■

■

Draw a circle. Using a protractor, mark points at 72° intervals around the circumference of the circle. Use a ruler or straight edge to join the five points.

72°

Hexagon ■ ■

■

Draw a circle. Using a protractor, mark points at 60° intervals around the circumference of the circle. Use a ruler or straight edge to join the six points.

60°

Octagon ■ ■

■

Draw a circle. Using a protractor, mark points at 45° intervals around the circumference of the circle. Use a ruler or straight edge to join the eight points.

45°

2 a Draw a pentagon in a circle of radius 4 cm. b Use the ruler to measure all side lengths. c Use the protractor to measure each of the five angles in the pentagon. d Explain why you divided the circle into 72° intervals. 3 a Draw a hexagon in a circle of radius 4 cm. b Measure the side lengths and internal angles and label the diagram. c Explain why you divided the circle into 60° intervals. 4 a Draw an octagon in a circle of radius 4 cm. b Measure the side lengths and internal angles and label the diagram. c Explain why you divided the circle into 45° intervals. 5 a Draw a heptagon in a circle of radius 4 cm. b Explain how you calculated the angle you used to divide the circle into intervals. 6 a Draw a decagon in a circle of radius 4 cm. b Explain how you calculated the angle you used to divide the circle into intervals.

Chapter 9 Polygons

269

9K Exploring geometric designs Geometric designs are used in sport, industry and commerce to create logos and icons. Many designs can be constructed using a ruler, compass and protractor.

Learning task 9K Disappearing triangles 1 Draw a circle of radius 8 centimetres. 2 Divide its circumference into angles of 60°. 3 Join every second point to produce two overlapping triangles. 4 Mark the midpoints of the six sides of the hexagon in the centre of the triangles. 5 Join the points to produce two more triangles. 6 Repeat the steps.

Kaleidoscope 1 Draw a circle of radius 8 centimetres. 2 Divide the circumference into angles of 36°. 3 Place the tip of the compass on one point and open the compass to reach the second point around the circumference. 4 Draw an arc to join two points. 5 Continue around the circle. 6 Colour the design.

Symmetrical tiles 1 Draw a circle of radius 8 centimetres. 2 Mark a point at the top of the circle. 3 Place the tip of the compass at the point and draw an arc through the centre of the circle and extending well outside the circle. 4 Repeat at 90° angles around the circle. 5 Join the points where the arcs intersect outside the circle to produce an enclosing square. 6 Rule in the two diagonals of the square. 7 Colour it in to create an interesting pattern.

270

Maths Dimensions 7

9K Triangle design 1 Draw a circle of radius 8 centimetres. 2 Divide the circumference into angles of 60°. 3 Join every second point to produce two overlapping triangles. 4 Colour it in to create the design shown. 5 Experiment with different ways to colour in the triangles to create different designs.

Flower petals 1 Draw a circle of radius 8 centimetres. 2 Mark a point at the top of the circle. 3 Keep the compass open at 8 centimetres. Place the tip of the compass on the point and draw an arc to intersect twice with the circumference. 4 Continue around the circle. 5 Colour the design to highlight two rows of petals. 6 Experiment with different ways to coloour the petals. 7 This design produces a flower with 12 petals. Can you construct a flower with 24 petals?

Woven hexagons 1 Draw a circle of radius 8 centimetres. 2 Divide the circumference using angles of 30°. 3 Join every second point to produce two overlapping hexagons. 4 Mark the midpoints of the six sides of the hexagons. 5 Join the points to produce two more hexagons. 6 Colour it in so that the hexagons appear to overlap. 7 Experiment with different angles to produce similar designs.

Chapter 9 Polygons

271

Puzzles 1 Answer the questions below and match the number to the correct letter to solve the riddle:

What did the doctor say to the man who thought he was a wigwam and a tee-pee? 65°

68°

50°

26° E

A

N

O

122°

T Y 40°

R

45°

U

S

78°

105°

33°

80° 75°

____ ____ ____ 50° 32° 45°

____ ____ ____ 60° 57° 40°

____ ____ ____ 25° 32° 32°

____ ____ ____ ____ ____ 25° 40° 47° 42° 40°

2 Find the missing angles in the problems below and match the letter to the correct angle to solve the riddle:

Why did the chicken cross the basketball court? A

C

132°

47° 85° 168°

D

126° 115° 98°

E

75° 72° H

U

L F

120° 105° 105°

95° 105°

T 122°

O W

76°

73° R

____ ____ ____ 72° 105° 60°

____ ____ ____ ____ ____ ____ ____ 30° 60° 75° 60° 30° 60° 60°

____ ____ ____ ____ ____ ____ 48° 54° 270° 270° 60° 132°

272

Maths Dimensions 7

____ 54°

____ ____ ____ ____ 75° 87° 288° 270°

Ch9 3 Below are 16 clues. Cross out the answers in the word search. The 18 letters which remain spell out the names of important equipment in geometry.

Names of these six shapes:

M U

I

Z

E

P

A

R

T

P

L

S

T

H

G

I

R

R

O

A

U

T

R

E

A

E

C

E

R

B

M U

S

S

C

E

T

E

M

T

O

U

E

T

L

U

T

O

R

X

T

C

A

G

C

A

H

O

I

B

M N

N

A

L

R

K

S

O

P

G

A

D

I

A

G

O

N

A

L

I

R

U

L

E

R

A

S

E

R

S

Q

X

E

L

F

E

R

T

S

E

L

E

C

S

O

S

I

A three-sided polygon Angle between 0° and 90° Angle between 180° and 360° Angle between 90° and 180° Used to draw a straight line An angle of 90° is a _______________ angle The angle _______________ of a triangle is 180° A hexagon has _______________ sides A _______________ joins opposite corners of a rectangle Initials for Eastern Standard Time

Chapter 9 Polygons

273

Applications Polygon patterns Some shapes will cover a surface completely. These shapes are said to tessellate. Some possible designs are shown below. Name the polygons used in each design:

On grid paper draw some tessellating patterns using these polygons:

Tangrams Tangrams are ancient Chinese puzzles. To make your tangrams, enlarge the grid below to a size of 10 cm × 10 cm. Use stiff cardboard, carefully cut out the seven tangram pieces. Using each of the seven pieces, can you make the silhouettes shown below? How may other shapes can you make?

274

Maths Dimensions 7

Ch9 Escher artwork Maurits Escher was born in The Netherlands in 1898. He was a traditionally trained artist who was skilled in graphic design. He based many of his designs on tessellating polygons.

To create an Escher-type tessellation, first draw a tessellation of a simple polygon such as a trapezium.

From each trapezium, cut out a simple piece from the top and add it to the bottom of the next shape.

Complicated tessellations such as this Escher design are based on repeating simple polygons. In this case, the basic repeating shape appears to be a trapezium.

1 Design your own Escher-type tessellation. 2 Prepare a PowerPoint presentation on the life and work of Maurits Escher. Include at least 10 slides and examples of at least three pieces of his artwork.

Chapter 9 Polygons

275

Enrichment 1 How many sides are there in a polygon whose angle sum is the following? a 540° b 720° c 1080° d 10 800° 2 Rob made a cylinder and a cube from a solid block of wood. He used a saw to make two cuts through the solids as shown below. Cut A was parallel to the base. If Rob looked down on the solid after cut A, what would the surface look like? a Draw a cross-section of the surface of the solid created by cut A. Cut B was at angle of about 45° to the base. b Draw a cross-section of the surface of the solid created by cut B. c What would happen if he cut the prisms using a vertical cut at right angles to the base? Draw the cross-section resulting from a vertical cut. d Which of the cross-sections are polygons? B B

A A

3 a Draw a triangular-based prism and a sphere. b Draw three lines A, B and C so that A cuts horizontally, B cuts at a 45° angle and C cuts vertically. c Draw each cross-section. d Which of the cross-sections is a polygon? 4 Rani made a cone and a pyramid from a solid block of wood. She then made three cuts A, B and C as shown below. a Draw each of the cross-sections A, B and C. b Describe the similarities and differences in the cross-sections. C

C B

B A

276

Maths Dimensions 7

A

Ch9 5 Draw an example of each of the polygons (triangle, square, pentagon, etc.). For each polygon, show all possible diagonals. Copy and complete the table to record your findings, and so find a rule to describe the pattern. The first few have been done for you. Name of polygon

Polygon

Number of diagonals

Rule

Triangle

0

3(3 – 3) -------------------- = 0 2

Square

2

4(4 – 3) -------------------- = 2 2

Pentagon

5

5(5 – 3) -------------------- = 5 2

Hexagon

Heptagon

Octagon

Nonagon

Decagon

From the table we can generate a rule for the number of diagonals for any polygon. Copy and complete this sentence in your workbook: In an n-sided polygon there are _____ diagonals.

6 Use your rule to find the number of diagonals for each of the following polygons: a pentagon b octagon c dodecagon d 18-agon e 200-agon 7 Use your rule to find the number of diagonals in a regular: a 14-agon b 25-agon c 40-agon 8 Using cardboard or straws make a model of a geodesic dome.

Chapter 9 Polygons

277

Revision Exercises 9A and 9B 1 Look at the triangles in the following diagrams. Give each triangle two names, one according to its side properties and the other according to its angle properties: a b c d

e

f

g

h

Exercise 9C 2 Find the size of the unknown angle in each of the following triangles: a b c 36° t

c

43°

52°

136°

89°

22° h

d

e

f

a

57°

y

63° k

56°

Exercise 9D 3 Find the size of the unknown angle: a b

c 80°

65°

45°

x 52°

x

42° 62°

x

d

125°

x

e

f x

28° x

278

Maths Dimensions 7

64°

Ch9 Exercise 9F 4 Write down the correct mathematical name for these shapes: a b c

e

f

d

g

h

Exercise 9G 5 Find the size of the unknown angle in each of the following quadrilaterals: a b c n 128° 113° 45°

y

57°

z

124°

d

123°

60°

36° 124°

98°

e 46°

f

42°

143°

43° 168°

59° m

w

43° t

44°

Exercise 9H 6 Write down the correct mathematical name for each of these shapes: a b c

d

e

f

Exercise 9I 7 Find the size of the unknown angle in each polygon: a b c 107° 115° p

84° p 107°

d

113°

89°

92°

95°

134° 118° 151°

166°

123°

85° 140°

w 125°

134°

114° h

145°

Chapter 9 Polygons

279

CHAPTER

10 Location

Location René Descartes, the 17th century French mathematician, devised the grid system that we now know as the Cartesian plane. It forms the basis of the grid used in street directories and by computers to determine the position of an object at any given time. Many years ago sailors relied on the stars and primitive instruments to chart their course across the seas.

This chapter covers the following skills: • Drawing and interpreting diagrams in two dimensions • Using directions in two dimensions • Locating points on a map • Using the Cartesian plane • Using true bearings A true bearing is represented by an angle measured clockwise from north • Using compass bearings A compass bearing gives the amount of turn east or west from the direction north or south, whichever is closer • Applying true and compass bearings • Using scales on maps

10A

Directions in two dimensions

In 1619 René Descartes, a French mathematician, invented the grid system that we use in mathematics today. He was lying in bed watching a fly crawl across the ceiling of his bedroom and he wanted to devise a way of describing the exact position of the fly on the ceiling. He began by picturing the ceiling covered with grid lines about 1 metre apart. If he started in the bottom left-hand corner, he could describe the position of the fly as ‘three across and two up’.

Up 6 5 4 3 2 1 0

2

1

3

4

5

6 Across

The starting point on the grid is known as the origin. It is from the origin that we start to count across (run) and then up (rise). To make the counting easier, the edges of the grid, known as axes, are numbered. The unit chosen for each axis depends on the information shown on the grid.

Exercise 10A 1 For each of the following describe the position of the centre of the flower from the origin: a A b B c C d D e E f F

Up 6 5

C

E

4 3

B

F

2

A

1 0

2 For each of the following describe the position of the centre of the sun from the origin: a A b B c C d D e E f F

Up 6 5

D 1

2

3

4

5

6 Across

B

F

4

E

3

D

A

2 C

1 0

282

Maths Dimensions 7

1

2

3

4

5

6 Across

10A 3 For each of the following describe the position of the centre of the ant from the origin: a A b B c C d D e E f F

Up 6

C B

5 4

A

E

3 F

2

D

1 0

4 For each of the following describe the position of the centre of the bone from the origin: a A b B c C d D e E f F

1

2

3

4

5

6 Across

Up 6 C

5

B

4 F

3 2

D A

E

1 0

5 For each of the following describe the position of Chloe from Sam : S

a A c C

b B d D

1

2

3

4

5

6 Across

4

5

6 Across

Up 6 C

5 4

B

3

D

2

A

1 0

1

2

3

6 Using the grid in Question 5, now describe the position of Sam from Chloe. 7 In your workbook draw a grid 6 up and 6 across and on it mark the following: a a sun at 3 across and 1 up b an ant at 5 across and 5 up c a flower at 2 across and 3 up d a dog at 1 across and 1 up e a bone at 1 across and 6 up f a moon at 6 across and 1 up

Chapter 10 Location

283

10B

The coordinate number plane

Maps are used to give accurate information as to where places are located. The horizontal axis is often called the x-axis and the vertical axis is known as the y-axis. To locate a point on a map we use coordinate points, written as two numbers in brackets e.g. (3, 5). This coordinate point means 3 units across then 5 units up and is called an ordered pair because the x-coordinate is written before the y-coordinate. y 5

L is 2 units across and 5 units up. We can write this as L(2, 5).

L

4 3 2 1 0

1

2

3

5 x

4

y 5

M is 3 units across and 1 unit up. We can write this as M(3, 1).

4 3

N is 1 unit across and 3 units up. We can write this as N(1, 3).

N

2 1 0

M

1

2

3

5 x

4

y 5

R is 3 units across and 4 units up. We can write this as R(3, 4).

R

4

S

3

S is 4 units across and 3 units up. We can write this as S(4, 3).

2 1 0

1

2

y 5

3

4

5 x

The ordered pair (2, 5) is not the same as the ordered pair (5, 2) because the order of the coordinates is different.

(2, 5)

4 3 (5, 2)

2 1 0

284

1

2

3

4

Maths Dimensions 7

5 x

Remember, for coordinate pairs (x, y), x comes before y, as in the alphabet.

10B Exercise 10B 1 Copy the set of axes shown onto grid paper. On your set of axes mark the point A(1, 2) as shown. Now mark the following points on the same set of axes: a B(4, 2) b C(4, 5) c D(2, 3) d E(4, 1) e F (5, 3) f G(5, 0) g H(0, 4) h I(3, 5)

y 5

4 3 A(1, 2)

2 1 0

2

1

3

5 x

4

2 On a sheet of graph paper draw a set of axes, with the x-axis and y-axis each marked from 0 to 6. On your axes mark the following points: a B(3, 2) b C(1, 5) c D(2, 5) d E(4, 4) e F(5, 1) f G(6, 2) g H(3, 0) h I(0, 6) 3 Which letter corresponds to each of the following points on the axes shown? a (2, 4) b (1, 6) c (0, 4) d (3, 1) e (5, 3) f (3, 4) g (6, 4) h (5, 2) i (0, 0) j (5, 6) k (1, 2) l (5, 0)

y 6

I

B

5 4

C

H

D

G E

3 L

F

2 A

1

J

K

0

1

2

3

4

5

6 x

4 On a sheet of graph paper draw a set of axes, with the x-axis and y-axis each marked from 0 to 6. On your axes mark the following points: a A(0, 4) b B(2, 4) c C(4, 4) d D(6, 4) e E(6, 0) f F(4, 0) g G(2, 0) h H(0, 0) Join the letters in alphabetical order (A to B to C etc. and then back to A). What shape do you see? 5 On a sheet of graph paper draw a set of axes, with the x-axis and y-axis each marked from 0 to 6. On your axes mark the following points: a A(0, 3) b B(1, 4) c C(3, 6) d D(3, 4) e E(6, 4) f F(6, 2) g G(3, 2) h H(3, 0) Join the letters in alphabetical order (A to B to C etc. and then back to A). What shape do you see? 6 On a sheet of graph paper draw a set of axes, with the x-axis and y-axis each marked from 0 to 6. On your axes mark the following points: a A(0, 3) b B(2, 4) c C(3, 6) d D(4, 4) e E(6, 3) f F(4, 2) g G(3, 0) h H(2, 2) Join the letters in alphabetical order (A to B to C etc. and then back to A). What shape do you see?

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10B 7 Write down the coordinates of the main parts of the picture and write instructions for a friend to follow. y 14

8 Write down the coordinates of the features of the campsite and write instructions for a friend to follow. Use the coordinates of the centre of each feature.

13 12

800

11 10

North (metres)

700

9 8 7 6 5 4

Bridge

Lookout

500

Cave

400 300

Obstacle course

200

2

Start

1

2

1

Dam

100 Camp 0

3

0

600

3

4 5

6

7

8

Toilet 100 200 300 400 500 600 700 East (metres)

9 10 11 12 x

9 Write the coordinates of each of the following animals in the form (x, y):

10 Write down the coordinates of the centre of pictures shown on the grid below:

a

b

c

d

a

b

c

d

e

f

g

h

e

f

g

h

y y

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

286

1

2

Maths Dimensions 7

3

4

5

6

x

0

1

2

3

4

5

6

7

8

x

10B 11 a Use the coordinates to plot and label the following points onto the graph: A(1, 7) B(5, 10) C(2, 1) D(4, 5) E(3, 2) F(5, 3) G(1, 6) H(7, 2) b Use a similar grid to plot these points. Join the points to form a picture: START (1, 2), (1, 8), (2, 8), (2, 6), (3, 6), (3, 8), (4, 8), (4, 2), (3, 2), (3, 4), (2, 4), (2, 2), (1, 2), STOP START (5, 2), (5, 8), (6, 8), (6, 2), (5, 2), STOP

y 10 9 8 7 6 5 4 3 2 1 0

12 a Join these coordinates on the grid to draw a picture: (1, 3), (2, 2), (3, 4), (4, 3), (3, 2), (7, 2), (8, 3), (4, 3), (8, 3), (10, 5), (8, 5), (10, 5), (8, 8), (4, 8), (3, 9), (7, 9), (8, 8), (4, 8), (3, 5), (2, 8), (1, 6), (2, 5), (1, 5), (2, 4), (1, 3) b Join these coordinates on a similar grid to draw a picture: (0, 6), (2, 6), (6, 6), (4, 1), (6, 2), (8, 6), (11, 6), (12, 5), (11, 7), (8, 8), (6, 10), (4, 10), (6, 8), (2, 8), (0, 10), (0, 6)

1

2

3

6

7

4

5

6

7 x

y 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

1

2

3

4 5

8

9 10 11 12 x

13 Draw a set of axes on graph paper and draw a picture using straight lines. Write down the coordinates for your picture, with instructions for another student to follow. 14 Draw a set of axes on graph paper and create your own geometric design using straight lines. Write down the coordinates for your picture, with instructions for another student to follow.

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10C Handy reflections Introduction Estimate and plot points then use a graphics calculator to draw a connected scatter plot.

Equipment TI-83/TI-83Plus/TI-84Plus

Technology activity 10C This purpose of this activity is to plot points in the first quadrant to draw your hand using a graphics calculator. The more points used, the greater the accuracy of the drawing. Reflecting the created image into the second quadrant draws the left hand.

Drawing your hand ■

Carefully trace around your right hand on the grid provided (see page 291).

■

Mark points along your trace that will become a ‘dot-to-dot’ representation of your hand. The more dots the better the final graph; however, the more dots you draw the longer it will take to write down and enter the coordinates.

■

Use the scale on the graph to determine the coordinates of each of your points. Write down the coordinates of each point next to the point on your graph.

■

Enter all your points in either L1 (abscissa) or L2 (ordinate) on the graphics calculator.

Press the MODE button then use the

arrow

keys and press ENTER on your selections to match those shown in the screen opposite.

Press Y= , use the

arrow keys to move up

and down through the equations and press CLEAR to remove any equations that may already be entered on the calculator. In the screen shot displayed, two statistical graphs have also been selected: Plot1 and Plot2. If your calculator has any plots switched on, use the

arrow keys to move the

cursor on top of each of the plots and press ENTER to switch them off. Press 2nd

ZOOM then use the

arrow

keys and press ENTER on your selections to match those shown in the screen opposite.

288

Maths Dimensions 7

10C Data may already be contained in the calculator. To remove any existing data press 2nd the lists.

+ and selection option 4 to clear all

Press ENTER to execute the command.

Finally, occasionally lists may be accidentally removed from the main editing screen, to chose a custom set-up press STAT and select option 5 SetUpEditor.

The calculator is now set up ready to accept data entry. Press STAT and select option 1 to edit or enter data. The headings L1, L2 … represent list 1, list 2 … These lists are similar to the columns on a spreadsheet. Use L1 to enter your x-coordinates (abscissa) and L2 to enter your y-coordinates (ordinates). When you have finished entering your data press 2nd

Y=

to enter the STAT PLOT menu. Press ENTER to select Plot 1. Use the shown opposite.

arrow keys to match the settings

Press ZOOM and select option 9 to view your graph. The window settings for the graphing screen can be viewed or changed by pressing the WINDOW button. Xmin = Minimum x value displayed. Xmax = Maximum x value displayed. Ymin = Minimum y value displayed. Ymax = Maximum y value displayed. Xscl = Tick mark frequency on x-axis. Yscl = Tick mark frequency on y-axis. Set the following: Xmin = –20 Ymin = 0

Xmax = 20 Ymax = 24

Xscl = 2 Yscl = 2

Press GRAPH to view the graph of your hand. Tip When plotting points from your sketch, constantly swap backwards and forwards from the data-entry screen to the calculator screen. This makes it easier to identify an error and also provides feedback on whether your points are too close together or too far away.

Chapter 10 Location

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10C Reflecting your hand—The y-axis ■

Suppose you wanted to draw a graph of your left hand. You could trace your hand and repeat all the measurements. A quicker solution is to imagine the y-axis as a mirror.

■

The left hand could be drawn by making a reflection of the right hand in the y-axis.

■

What would happen to your coordinates?

■

Write down some of the coordinates for your right hand and explain how the coordinates for your left hand were obtained.

■

Enter the new coordinates for your left hand into the calculator.

Note: This can be done by using a formula. Formulas can be entered at the top of a list.

Reflecting your hand—The x-axis ■

Suppose you wanted to draw an identical hand but upside down. Once again you could trace your hand and repeat all the measurements. A quicker solution is to imagine the x-axis as a mirror.

■

What would happen to your coordinates if you wanted to reflect your hand in the x-axis?

■

Write down some of your original right-hand coordinates and explain how the new coordinates were obtained.

■

Enter the new coordinates for your right hand into the calculator using a formula.

Double reflections—The x- and y-axis

290

Maths Dimensions 7

■

By now you should already know how to create reflections by changing your coordinates. How could you change your coordinates if you wanted to reflect your hand in the x-axis then in the y-axis?

■

Write down a selection of your coordinates and explain how the new coordinates were obtained.

■

Draw a graph of your right hand after it has been reflected in the x-axis and the y-axis. Include the original image in your graph.

10C 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

Chapter 10 Location

18

291

10D

Scale diagrams and maps

Maps are useful ways in which to record information about an area and to share that information with others. This can be useful when giving someone directions, or as a form of catalogue to observe the changes in marine life on a reef or wildlife in a park. A house plan used by many architects, builders and real estate agents is shown below. Ground floor

N Meals Kitchen P’try WO

F

Family

Family 1:100

L dry C

Store

Lounge Entry

Double Garage

Up

Study Porch

Included on the plan is the direction of north. From this we are able to work out the compass directions of east, west and south. The map also includes a scale of 1 centimetre for approximately every 1 metre. This means that every 1 centimetre on the map represents 1 metre in the actual house.

Example If 1 cm represents 1 m, write the scale as a ratio.

Solution 1 cm represents 1 m 1 cm represents 100 cm 1:100

Exercise 10D 1 Write the following scales as ratios: a 1 cm represents 5 m c 5 mm represents 1 m e 1 mm represents 1 m g 2 cm represents 550 m

b d f h

2 cm represents 5 m 2·5 cm represents 7·5 m 1 cm represents 4000 km 1 cm represents 1000 km

2 Find the distance represented by 1 centimetre on a map if the scale is written as: a 1:10 b 1:50 c 1:250 d 1:100 e 1:150 f 1:500 g 1:25 000 h 1:100 000

292

Maths Dimensions 7

10D 3 A map is drawn with a scale showing that 1 cm represents 10 km. If the actual distance from Melbourne to Ballarat is 110 km, how far will it be on the map? 4 The map of Australia shown below has a scale of 1:45 000 000. a Calculate the real distance represented by these distances on the map: i 1 mm ii 1 cm iii 10 cm iv 26.5 cm b Calculate the distance represented on the map by these real distances: i 90 km ii 900 km iii 2700 km iv 4050 km c Use the map to estimate the actual distance (as the crow flies) between the following pairs of capital cities: i Melbourne and Canberra ii Melbourne and Darwin iii Perth and Brisbane iv Adelaide and Hobart d The perimeter of Australia is estimated to be about 36 735 kilometres. Use the scale to estimate the perimeter of the map shown here. e The area of Australia is estimated to be about 7 686 850 square kilometres. Estimate the area of the map shown here. (Hint: 1 sq km = 1000 × 1000 = 1 000 000 sq m) f Estimate the shaded area on the map which contains: i rabbits ii cane toads Darwin

apricorn Tropic of C

Alice Springs Brisbane

Perth Adelaide Distribution of rabbits Distribution of cane toads

5 a Karen and Stephen want to lay modgrass on their tennis court. Choose a suitable scale and make a scale drawing of their tennis court given the information on their rough sketch. b Find out the exact measurements of the singles and doubles court to complete the drawing.

Sydney Canberra Melbourne Hobart

23·8 m

11·0 m

Chapter 10 Location

293

10E

Maps and bearings

Bearings are used in navigation to identify the position of an object at a given time. This was useful many years ago when sailors relied on the stars and primitive instruments to chart their course across the seas. Today bearings are used widely by many people ranging from sailors to bushwalkers. There are two possible ways of giving directions: true bearings and compass bearings.

True bearings True bearings are equivalent to the angle measured clockwise from north. The bearing of Karen from Stephen can be described as 063° in the first diagram. This is the angle measured clockwise from north. Three digits are used to describe a bearing. For example 5° is written as 005°.

In the second diagram, the bearing of Karen from Stephen can be described as 216°, which is the size of the angle measured clockwise from north.

N

N

63° Stephen

Karen

Stephen

E 216°

E Karen

063°

216°

Compass bearings Compass bearings always begin with north (N) or south (S), whichever is closest to the angle. The bearing of Karen from Stephen can be described as N63°E in the first diagram.

In the second diagram, the bearing of Karen from Stephen can be described as S36°W. Notice that Karen is 36° towards the west from south. 216° is 36° more than 180°.

N

N

63° Stephen

Karen

Stephen 36° Karen

N63°E

294

Maths Dimensions 7

E 216°

E

S36°W

10E Exercise 10E 1 Copy the table into your workbook and complete it by measuring angles in the sketches below with your protractor to find the bearing from the boy to the girl. Angle, as shown, from the boy to the girl

True bearing

Compass bearing

a b c d e f g h i j k l a

b

N

c

N

Amy

Cliff

Bianca

Alex

d

E

e

Duncan

Carly

f

N

Ken

E

h

Jamie

E

Emily

i

N Daniel E

E

Felicity

Kylie

N

N

Forbes

E

Danielle

g

E

E

Blake

N

N

N

Ryan

E Chloe

Jessie

Chapter 10 Location

295

10E j

k

N

Gareth

E

l

N

Louise

N

Luke

Jason

E

Jessie

E

Gail

2 Using your protractor to help you measure the angles clockwise from north, give the directions of each letter on the diagram both as a true bearing and as a compass bearing.

G

North

A B

F West

East

E D

C South

3 Use your protractor to draw diagrams to illustrate the following true bearings and compass bearings: a 025° b S15°W c N35°E d 166° e S78°E f 235° g N85°W h 315° 4 What are the true bearings of the points on the compass that are known as due north, due south, due west and due east?

5 The photograph above shows the Southbank Promenade in Melbourne. A street directory map of the area is shown on the next page.

296

Maths Dimensions 7

10E

N

Assume the scale of this map is 1:10 000. Use this map to answer the following questions. a Calculate the actual distance between: i Lonsdale Street and Flinders Street ii William Street and Swanston Street b Find the real dimensions of: i County Court in William Street ii GPO Building in Elizabeth Street c Find the bearing from the middle of the Rialto Observation Deck to: i Melbourne Town Hall in Swanston Street ii Melbourne Aquarium on the Yarra iii Former Mint in William Street Chapter 10 Location

297

Puzzles 1 Join the coordinates to form a picture. If you pass through any letters on the way, write them down in order below. You will find the answer to the riddle:

What do you get when you cross a vampire with a snowman? (2, 3), (0, 3), (0, 1), (20, 1), (20, 5), (16, 5), (16, 3), (2, 3), (2, 13), (16, 13), (16, 3), (15, 3), (15, 12), (3, 12), (3, 3), (4, 3), (4, 11), (14, 11), (14, 3), (13, 3), (13, 10), (5, 10), (5, 3), (6, 3), (6, 9), (12, 9), (12, 3), (11, 3), (11, 8), (7, 8), (7, 3), (8, 3), (8, 7), (10, 7), (10, 3), (9, 3), (9, 6) (9, 3), (16, 3), (16, 5), (18, 5), (18, 9), (18, 5), (20, 5), (20, 9), (20, 5).

C

20

H

18

A

16

W

Q

14 12 UP

10

P

S O

M T

8

B

6 N

T I

4

P

2 0

2

4

R R

F 6

8

10 12 14 16 18 20 ACROSS

____ ____ ____ ____ ____ ____ ____ ____ ____

298

Maths Dimensions 7

E

Ch10 2 Follow Old Tom’s instructions to the treasure.

N

✗

✗

Old Tom covers 100 cm in each pace. Scale map: 1 cm = 100 m

1 2 3 4

Start at Palm Cove.

5

At daybreak walk 300 paces on bearing of 045°T and collect eggs for breakfast.

6 7 8

Walk 250 paces on a bearing S20°E. Mind your step.

9 10

At daybreak walk 450 paces towards old shipwreck. Turn and walk due south for 200 paces through the Valley of Fear. At dusk walk approximately 700 paces towards the setting sun and camp for the night on the water’s edge.

Walk due east and camp for the night at Palm Cove. On a bearing of N35°W walk 400 paces keeping your feet dry and the bacon salty. Turn and walk 650 paces on a bearing S55°W to where the treasure is buried. Describe the location of the treasure.

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299

Applications Classroom plan Draw a seating plan for your classroom. Clearly label each desk with the person who sits there. Using the bottom left-hand corner of your page as the starting point, describe the position of your desk (e.g. three desks across and two desks up). Repeat this exercise for the desks of two friends.

Where do you relax? Draw a seating plan for your TV room at home. Clearly identify and label the seats for each member of your family. Your seating plan should include an accurate scale diagram.

Dream home a Design a floor plan of your dream home. Your home must include a kitchen, bathroom and toilet, sleeping area, living area and a laundry. It could be one or two storeys, but you must show the floor plan for each storey. Include an appropriate scale. b Draw an artistic impression of the front of your house.

Treasure map Construct a maths trail in your school grounds. Record it on an A4 piece of paper. Your map must have a start, a compass point showing the direction of north, a scale, a legend, and at least five steps or clues giving both a distance and a compass bearing leading to the ‘treasure’. Swap your trail with a friend and see if you can find each other’s treasure.

300

Maths Dimensions 7

Ch10 Orienteering On an orienteering camp, students were asked to use their compass, a map and a trundle wheel in order to navigate their way from place to place. The grid for the map the students were given is seen below.

The toilet block of the campsite is located at the origin (0, 0). The obstacle course is 200 m east and 300 m north of the campsite (200, 300). The cave is 700 m east and 400 m north of the campsite (700, 400). The lookout is 400 m east and 800 m north of the campsite (400, 800).

The bridge is 100 m east and 500 m north of the campsite (100, 500).

a Copy and complete the set of axes and mark all features on the map. b Use the scale to calculate the distance of the toilet block from: i the bridge ii the cave iii the lookout c Calculate the distance between the Dam and the bridge.

700 North (metres)

The dam is 600 m east and 200 m north of the campsite (600, 200).

800

600 500 400 300 200 100 0

100 200 300 400 500 600 700 East (metres)

Schoolyard orienteering a Using a map of your school create your orienteering course. It must have a north point, a key and up to 10 stations. b Swap maps with another student in your class and complete their course. c You could even ask class mates to collect information as they complete your orienteering course. Chapter 10 Location

301

Enrichment Points on the Cartesian plane need not necessarily be confined to the first quadrant. They may comprise points from all four quadrants by using negative numbers. Remember, the first coordinate means across and the second means up. The coordinates for L(−2, 5) are the ordered pair of numbers which describe its position on the Cartesian plane. For the point L, −2 is the x-coordinate, meaning move 2 units in the negative direction of the x-axis, and 5 is the y-coordinate, meaning 5 units in the direction of the y-axis. y 6 Quadrant 2

L

5 4

Quadrant 1

3 2 1

The coordinates for M and N are: M(3, −1) N(−4, −2)

–6 –5 –4 –3 –2 –1 –1 N –2

Quadrant 3

–3 –4 –5

1

2

3

M

4

5 6 x

Quadrant 4

–6

1 On a sheet of graph paper draw a set of axes, with the x-axis and y-axis each marked in order from −6 to 6. a On your axes mark the following points: i A(1, 2) ii B(3, 4) iii C(2, 3) iv D(4, 5) v E(0, 1) vi F(2, 6) vii G(4, 2) viii H(6, 1) b Join point A to B, then use your protractor to find the bearing of B from A. c Repeat part b for the points below: i C to D ii E to F iii G to H 2 On a sheet of graph paper draw a set of axes, with the x-axis and y-axis each marked in order from −6 to 6. a On your axes mark the following points: i A(2, 4) ii B(−2, 5) iii C(4, −4) iv D(−6, 4) v E(0, −1) vi F(2, −3) vii G(−4, 2) viii H(−6, 1) b Join point A to B then use your protractor to find the bearing of B from A. c Repeat part b for the points below: i C to D ii E to F iii G to H

302

Maths Dimensions 7

Ch10 3 Draw a map of the world and mark on it the major capital cities. Place a grid over your map so that the origin (0, 0) is located at Melbourne. Identify and record in your workbook as many capital cities and their coordinates as you can.

4 Locate your house in the Melway street directory. Imagine that your house is situated at the origin (0, 0). Give the coordinates of ten features in your neighbourhood, using all four quadrants. A good example would be a school, shops or your local park. 5 The following activity involves creating a Cartesian plan. Each group of students will need: ■ 2 pieces of string approximately 5 metres in length ■ a permanent marker ■ 50 pieces of card approximately 5 cm square. a Hand out a piece of card to each member of the group. b Mark X, Y, (0, 0) and two sets of the numbers 1 to 10 on the cards. c Use the cards and string to lay out the first quadrant of a Cartesian plane on a large flat space such as a basketball court or asphalt playground. Place (0, 0) at the origin, and label the axes from 1 to 10 with the remaining cards. d Write a coordinate point on one of the remaining cards and take turns to stand at that point. e Extend the axes to include all four quadrants, so the x-axis and the y-axis extend from –10 to 10. f Write a new coordinate point from quadrants 2 to 4 and take turns to stand at that point. g Prepare a group presentation of your Cartesian plane to the rest of the class. h Give each student a card with a coordinate point from one of the four quadrants and ask them to place it in the correct position on the Cartesian plane. i Evaluate your presentation based on how well the students carry out their tasks.

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Revision Exercise 10A 1 In each of the following describe the position of the centre Up of the sun from the origin: 6 F a A b B c C 5 d D e E f F

B

4

E

3

C

2

D

A

1 0

1

2

3

4

5

6 Across

Exercise 10B 2 On a sheet of graph paper draw a set of axes with the x- and y-axes each marked from 0 to 6. On your axes mark the following points: a B(3, 2) b C(1, 5) c D(2, 5) d E(4, 4) e F(5, 1) f G(6, 2) g H(3, 0) h I(0, 6) 3 Give the coordinates of the points marked on the Cartesian plane: a A b B c C d D e E f F

y 6 E

5

D

4 3

C

2 F

B

1

A

0

4 Determine the coordinates of the centre of each animal shown: a kangaroo b platypus c crocodile d goanna e numbat f dingo g redback spider h emu i wombat

304

Maths Dimensions 7

y 12 11 10 9 8 7 6 5 4 3 2 1 0

1

2

3

4

5

6 x

1 2 3 4 5 6 7 8 9 10 11 12 x

Ch10 Exercise 10D 5 Write these scales as ratios: a 1 cm represents 5 m c 10 mm represents 2·5 m

b 1 cm represents 5 km d 0·5 cm represents 2 km

6 What would 1 cm on a map represent if the scale was: a 1:1000? b 1:500? c 1:80?

d 1:10 000?

Exercise 10E 7 Using your protractor to help you, draw diagrams to illustrate the following true bearings or compass bearings: a 075° b 205° c 345° d 126° e S71°E f S35°W g N63°W h N39°E 8 The illustration below shows the Melbourne Sports and Aquatic Centre. The map has been drawn to scale.

Facility Plan

Melbourne Sports and Aquatic Centre Level 1 1 2 3 4 5

6 75 metre 10 lane swimming pool with spectator seating for 2000 7 14 board diving facility 8 multi purpose pool 9 25 metre lap pool 10 water slide 11 wave pool & toddler’s play area 12 cafe and bar 13 main reception area & foyer 14 sports shop 15 gallery 16 creche & pirate’s cave fun activity centre

2000 seat show court 8 court basketball hall 12 court badminton hall 27 court table tennis/volleyball hall 10 court squash centre including championship glass court & spectator seating for 300

2 3 1

4

5

15 16

Level 2 • fitness centre • personal training services • function and meeting rooms • centre management • sports associations – Squash – Table Tennis – Badminton

14 13 12

Level 3

6 11

9 10

7 8

• sports medicine centre • sports associations – Diving – Swimming – Basketball – Water Polo

a Calculate the scale, using the 25 m lap pool (9) or the 75 m lap pool (6 and 7). b Use the scale to find the dimensions of the 8-court basketball hall (2). c Use the scale to find the dimensions of the 2000-seat show court (1). Chapter 10 Location

305

CHAPTER

11 Algebra Symbols

Algebra Symbols Algebra was originally devised in Ancient Babylon. The word actually derives from the Arabic term al-jabr which means ‘the reunion of broken parts’, and one of the earliest algebra problems was documented on the Rhind Papyrus (1650 BC) in ancient Egypt. The first treatise on algebra was written in the third century AD by Diophantus of Alexandria. The Rhind Papyrus, now in the British Museum, poses the following problem: ‘Divide 100 loaves among 10 men including a boatman, a foreman and a doorkeeper, who receive double portions. What is the (exact) share of each?’ The ancient Egyptians could solve this problem using their early notions of algebra.

This chapter covers the following skills: • Simplifying expressions and collecting like terms • Working with symbols and pronumerals • Translating verbal statements into mathematical expressions • Substituting into expressions • Writing algebraic rules • Applying the distributive law a(b + c) = ab + ac • Using a graphics calculator to evaluate expressions and store values • Exploring triangular, square and pentagonal numbers • Applying simple formulas • Deducing a formula in a modelling situation and testing its validity

11A

Writing expressions

Algebra is the language of mathematics. To solve problems, we can translate information into mathematical sentences called expressions. We use symbols or letters to represent unknown numbers. For example, Morgan has some birds and cats. We can use the symbol for the number of cats and

to stand

to represent the number of birds. The total number of birds

and cats is represented by the expression

Example

+

.

Solution

1 I think of two different numbers and . Write an expression representing the sum of plus twice .

+ 2 represents the sum of twice .

2 It costs $5 per week to feed a cat and $2 per week to feed a bird. Write an expression for the total cost of pet food for a week.

The cost for a week = 5

3 Write an expression for the perimeter of this rectangle.

2L + 2W represents the perimeter.

+2

plus

.

L W

W L

Exercise 11A 1 Write expressions for each of the following statements: a the sum of and four b three less than c one quarter of d five times the value of e divided by five f five divided by g the product of and h the term 3 subtracted from the term 5 i multiplied by 3 then add 2 j minus 4 times then add three • •• • • • • • • • ••

• •• • • • • • • • ••

2 Write an expression for the following statements: a A number n is multiplied by five then four is added. b A number x is divided by three then two is subtracted. c A number y is multiplied by 3 and six is added. d A number p is halved and thirty is added. e Two is added to the number q and the result is divided by y. f Three is added to the number x and the result is divided by 4. g A number y is multiplied by four and then 10 is subtracted. h A number x is multiplied by 3 and the result is divided by 5. i A number x is multiplied by itself and then 7 is added.

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Maths Dimensions 7

11A 3 Write algebraic expressions which represent the perimeters of these particular figures: a b c l a a

a

x

l

x b

a

4 The school ski club has girls and boys. Write an expression for each of the following: a the total number of students in the club b the number of boys if three more boys join c the number of girls if two girls leave d the new total number of club members 5 Adult tickets to the school concert cost $d and a program costs $p. Write expressions for the cost of: a five adult tickets b a half-priced ticket for a child c three half-priced tickets d two adult tickets and one program e two half-priced tickets and two programs 6 The cost of travelling by taxi is f (dollars) for the ‘flag fall’ (the cost of engaging the taxi) plus d (dollars) for each kilometre travelled. Write an expression for the cost of: a travelling 2 kilometres b travelling 10 kilometres c travelling n kilometres d each person travelling n kilometres when the cost is shared equally among four people 7 Ice creams at the school canteen cost x (dollars) and drinks cost y (dollars). Write an expression for the cost of: a four ice creams and three drinks b three ice creams and four drinks c m ice creams and n drinks 8 An aquarium has f fish, r rays and s sharks. Write an expression for the following: a the total number of marine animals b the number of fish if they double the number c the number of rays if three are eaten by sharks d the number of sharks if half of them are removed 9 After another m dives, a scuba diver will have made n dives in total. a Write an expression for the current number of dives. b Write an expression for the number of dives made p dives ago. Chapter 11 Algebra Symbols

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11B

Pronumerals

Letters that are used to represent unknown numbers in expressions are called pronumerals or variables. To simplify expressions with pronumerals add or subtract like terms. This is sometimes called collecting the like terms. We usually write the pronumerals in a term in alphabetical order.

Example

Solution

Simplify the following expressions: +

a b 4

+

+

+3

You can only add like terms: + + =3 +

+

4 +3 =7

c x+x+x

x+x+x = 3x

d 9abc + 3bac − cba

9abc + 3bac − cba = 9abc + 3abc − abc = 11abc

Exercise 11B 1 Simplify the following collections: a zebra, gorilla, zebra, bobcat, bobcat, zebra, gorilla, zebra, bobcat b apple, cherry, banana, apple, apple, orange, cherry, apple, banana, apple c pencil, pencil, eraser, pen, pen, pen, pen, pencil, scissors, pencil, pen d caravan, car, car, truck, scooter, car, car, scooter, truck, car, car, truck 2 Write down the pairs of like terms: 3 ,4 ,2 ,6 , 10 , 3 , 5

, 10

3 Write down the pairs of like terms: 8f, 10ab, 8kz, 3pq, 2f, 9jg, 12qp, 5kz, 2gj, 6b, 3ba, b

4 Simplify the following expressions: a + + + c 2★ + 3★ − 2★ e 10★ + 10★ + 10 g 4▲ + 3▲ − ▲ + ■ i 3♥ + ♥ + ▲ + 2▲ k 4■ − ▲ + 4▲ m 2★ + 5★ + 3▲ + 2▲

310

Maths Dimensions 7

b d f h j l n

+ 4 +4 −2 + 3 +4 +2 6■ + 4● − 2■ 5 +6 −●+ 6♥ − 3♥ + 2● 5■ + 2■ + 3▲ + 2▲

11B 5 Simplify the following expressions: a m+m+m b x+x+x+x d a+b+a+a+b+a e 10x + 5x g 12x − 7x h 15y − 8y j 25p − 17p k 9x − 8x 6 Simplify the following like terms: a 2xy + 6xy c 12ab − 5ab e 5pq + 8pq + 9pq g 18mn + 16mn − 28mn i 23xy − 15xy − 6xy k 13wx − 12wx

b d f h j l

c f i l

x+x+x+y+y 5a + 2a 27t + 18t 8y − 8y

7xy + 9yx 18ab − 6ba 7pq + 2pq + 4qp 15mn + 17nm − 5nm 28xy − 16yx − 9xy 6rs + 2rs − 8rs

7 Simplify the following expressions by combining like terms: a 4x + 5x + y b 5m − 2m + 3n c 6xy + 2xy + zy d 6pq − 3q + 2qp e 2x + 3y + 4x + 5y f 3x + 2 + 4x + 5 g a + 5 + 2a − 2 h 4 + 3x − 2 − x i 6x + 3y + 2 + 4x + 5y + 1 j 8x + 2y − 3 − 6x + y + 7 8 Simplify the following expressions by combining like terms: a 2a + 3b − c − a + b + 2c + 3a − b + c b 6ab + 3bc − ca − 2ab + 4bc − ab + ca c 7xy + 5yz − 2zy + 3zx + 4xy − 3yz + 6zx 9 Copy and complete the following table: +

3

5y

2x

5xy

2 3y 4x 2xy

10 Copy and complete the following table, taking away the black terms from the green ones: −

6

8a

10b

7ab

4 3a 5b 2ab

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11C

Multiplying and dividing pronumerals

Tom has some rabbits. The initial number of rabbits is represented by the letter r. The rabbits breed until there are three times as many rabbits or 3 × r or 3r. To simplify expressions, leave out the multiplication sign. For example, 3a means 3 lots of a, or 3 multiplied by a.

Example 1 Simplify these expressions: a 6×z×y

Solution 6×z×y = 6zy

b 5y × 7y

5y × 7y =5×7×y×y = 35y2

c 2×a×b

2×a×b = 2ab

r 1 If the rabbits are divided equally between 2 people, each person would have r ÷ 2 or --- or --- r. 2 2 To simplify expressions with division, write as a fraction and cancel common factors.

Example

Solution

2 Simplify: --------a 16x 12

b 14x ÷ 2y

4

16x 4x --------- = -----12 3 3

7

14x 7x --------- = -----2y y 1

Exercise 11C 1 Choose a symbol to represent the original number of guinea pigs and write an expression for the following situations: a The guinea pigs breed until the population doubles in size. b The guinea pigs breed until there is five times the original number. c A disease wipes out half of the original number of guinea pigs. d The guinea pigs are divided equally among three different people. e The guinea pigs are divided equally among six different people. f The guinea pigs breed until the population doubles in size and then four are given away. g The guinea pigs breed until the population trebles, then five more are added.

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Maths Dimensions 7

11C 2 Simplify the following products as far as possible: a 4×y b 5×y d b×6 e a×b g 12 × y × z h m×n×9 j y×6×5 k y×8×6 m 3×a×7×b n 3×y×8×z p 8×z×9×z q 4 × y2 × z × 9

c f i l o r

a × 10 m×n a×b×c 4×7×p a×b×b 3×m×n×m

3 Simplify the following products as far as possible: a 5 × 2x b 6 × 8y d y × 5x e m2 × 6n g 5y × 6z h 7p × 9q j 4y × 5y k 2a × 4b × 7 m 3p × 5q × 2r n 3a × 6b × 7c

c f i l o

3a × 7 7c × 5b 6z × 6z 6m × 2n × 9p 4y × 3z × 5y

4 Express the following quotients in fraction form: a x÷3 b m÷6 d 5÷x e 7 ÷ 2x

c p ÷ 10 f 3x ÷ 5y

5 Simplify the following algebraic fractions by cancelling: -----a 3a 15

f

21 ------7m

12x k -------10

-----b 7a 14

-----c 2y 8

12 d ----6x

8 g --------24 p

5 h -------20q

i

18y --------12

32 m -------12x

12 n -------30b

l

18 e ----9y

16 -----------64xy

j

10x --------8

21 o --------35w

6 Copy and complete the following table: ×

3

5y

2x

5xy

2 3y 4x 2xy

7 Copy and complete the following table. Divide the terms in green by the terms in black: ÷

24

48a

72b

96ab

3 4a 2b 8ab

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11D

The distributive law: Expanding brackets

The perimeter of a rectangle is twice the length plus twice the width: Perimeter = 2L + 2W

W

Alternatively, we could add the length L to the width W and take twice the total:

L

L

Perimeter = 2 × (L + W ). The two expressions are the same, and so

W

2 × (L + W ) = 2L + 2W

This illustrates the distributive law, in which the term outside the bracket multiplies each term inside the bracket, when the brackets are removed: a(b + c) = ab + ac

and

a(b – c) = ab – ac

The term a can also come after the bracket so that we would have: (b + c)a = ba + ca

and

(b – c)a = ba – ca

Example

Solution

Expand the following expressions (i.e. remove the brackets):

a 5(x + 4)

5(x + 4) =5×x+5×4 = 5x + 20

b 6x(2x – 5)

6x(2x – 5) = 6x × 2x – 6x × 5 = 12x2 – 30x

c 7x + 5(x + 2)

7x + 5(x + 2) = 7x + 5x + 10 = 12x + 10

Exercise 11D

314

1 Complete the following: a 4 × (7 + 3) =4× = c 8 × (12 – 4) =8× =

b 4×7+4×3 = + = d 8 × 12 – 8 × 4 = + =

2 Expand each of the following: a 3(x + y) b 7(a + b) e 6(y + 7) f 10(p + 8) i (b + 9)5 j (6 + p)6 m (x + 5)3 n (a + b)7

c g k o

Maths Dimensions 7

7(m + n) (a + 4)8 (12 + q)7 (m + n)10

d h l p

9(x + 4) (c + 5)9 (5 + n)8 7(r + s)

11D 3 Expand each of the following: a 5(x – y) d 8(x – 7) g 2(13 – m) j (7 – x)8

b e h k

7(a – b) 3(x – 9) 8(7 – q) (18 – b)3

c f i l

9(m – n) 9(x – 12) 4(9 – p) (15 – m)4

4 Expand each of the following: a x(y + z) d r(s – t) g m(n – 8) j c(9 – b)

b e h k

m(p + q) p(q – r) a(b – 12) p(14 – q)

c f i l

a(b + c) l(m – n) z(y – 8) m(9 – n)

5 Expand each of the following: a 8(3x + 2) d 5(7q – 6) g 4(2x + 5y) j (3a – 7b)6

b e h k

5(4y + 6) 9(6p – 12) 5(6m + 9n) (8m – 6n)11

c f i l

7(9b + 4) 8(3s – 7) 7(8a + 12b) (2m – 3n)9

6 Expand each of the following and simplify as far as possible: a 3(2x + 6) + 9 b 7(7y – 1) + 10 c 4 + 5(3m + 6) d 30 + 9(b – 2) e 4(2a + 3b + 4c) + 5b f 4x(3x + 2) g 2x(3x +2) + 3(3x + 2) h 6y(5y – 8) i 3x(2x + 5) + 4x(3x – 2) j 5p(4p + 2m) – 20p2 – 8mp 7 Write an algebraic expression for the following using brackets, then remove the brackets. a I think of a number n, add 3 and then multiply the total by 6. b I think of a number n, subtract 4, then multiply the result by 7. c I think of a number n, double it, add 3, then multiply the result by 4. d I think of a number n, multiply it by 4, subtract 6, then multiply the result by 3. e I think of a number n, halve it, subtract 3, then multiply the result by 4. 8 On a particular day a adults and c children attend a swimming pool: a Write an expression for the total number of people attending the pool. b If adults pay $3 each to enter and children pay $1·50, write an expression for the total amount collected by the pool. c Assuming the attendance pattern is the same each day, write an expression using brackets for the pool’s takings over a 5-day period. d Remove the brackets from your expression in part c. Chapter 11 Algebra Symbols

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11E

Substituting into expressions

Each pronumeral or variable represents an unknown number. Numbers can be substituted into pronumerals and a result is worked out to evaluate the expression. Remember to use the correct order of operations or BODMAS.

Example

Solution

1 Find the value of 3 + 2 if = 4 and = 5.

3 +2 =3×4+2×5 = 12 + 10 = 22

2 Find the values of the following expressions, where x = 3 and y = 4: a 8x − 2 y

8x − 2y =8×3−2×4 = 16

2x + y b -------------5

2x + y --------------5 2×3+4 = --------------------5 10 = -----5 =2

Exercise 11E 1 Find the values of the following expressions where ♣ = 4: a 2♣

---c ♣ 2

b ♣+5

2 Find the values of the following expressions where a 3 b 7 c 1---

d ♣−3 = 8:

d

2

3 Find the values of the following where x = 5 and y = 4: a 6xy b 2xy − 1 d 2x + 6y e 9x − 8y

c 1--- xy − 1 4 f 4y + 3x

4 If a = 5, b = 6, c = 3, evaluate the following expressions: -----a ab c

2ac b -------b

b+4 c ----------a

b+c d ----------c

+2 --------------e 2a b

f

bc – 8 --------------a

i l o r

2c(a + b) (2a)2 (a + b)2 (3c)3

g j m p

316

a(2c + 4) 5c2 2a2 b2 – c2

Maths Dimensions 7

h k n q

a(b + 4) b2 − 2c a 2 + b2 (b − c)2

+9

11E 5 Complete the following tables by substituting each value of x into the given expression: a

x

1

2

3

4

5

b

2x + 1

c

x

x

1

2

3

4

5

1

2

3

4

5

x+2

1

2

3

4

5

8−x

d

x 3x − 2

6 Sam has c CDs and d DVDs in his music collection: a Write an expression for the total number of disks (n) in his collection. b If c = 24 and d = 16, find n. 7 Tarli has m magazines and b books on horse riding in her collection: a Write an expression for the total number (n) in her collection. b If Tarli has 5 magazines and 12 books, find n. 8 CDs cost $15 each and DVDs cost $20 each: a Write an expression for the cost of x CDs and y DVDs. b By substituting 5 for x and 3 for y, find the cost of 5 CDs and 3 DVDs. c If $120 was spent in total, how many CDs and DVDs were purchased? Can you find more than one answer here? 9 At the movies popcorn costs $2 per container and drinks cost $3 each: a Write an expression for the cost (c) of x containers of popcorn and y drinks. b By substituting 3 for x and 4 for y, find the cost of three containers of popcorn and four drinks. c If $24 was spent in total, how many containers of popcorn and how many drinks were possibly purchased? Can you find more than one answer here? 10 Hot dogs cost $2·50 each and hamburgers cost $4·00 each: a Write an expression for the cost (c) of x hot dogs and y hamburgers. b By substituting 2 for x and 5 for y, find the cost of two hot dogs and five hamburgers. c If $27 was spent in total, how many hot dogs and hamburgers was it possible to buy?

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11F Calculator subs Introduction The aim of this activity is to investigate algebraic rules numerically via substitution in order to gain an insight into ‘like terms’.

Equipment TI-83Plus/TI-84Plus

Technology activity 11F The purpose of this activity is to explore the properties of simple algebraic expressions. The two expressions, 2a + 5b + 3a – b and 5a + 4b, are going to be compared using the calculator. Start on the calculator Home screen. To access each of the letters on the calculator press ALPHA followed by the corresponding key (letter). Choose a value for a. To store this number in a, type in the number then press STO

ALPHA

A . (A is located above the MATH key.)

Choose a value for b. To store this number in b, type in the number then press STO

ALPHA

B . (B is located above the APPS key.)

Note: Make sure your values for a and b are different. Type the expression 2a + 5b + 3a – b into the calculator and press ENTER . Note: Access the letter keys as before. A multiplication sign between the coefficient and the variable is not required. Type the expression 5a + 4b into the calculator and press ENTER .

318

Maths Dimensions 7

11F 1 a Write down the value given for the two expressions used in the example above: 2a + 5b + 3a – b and 5a + 4b. b Did the two expressions give the same answer for your values? 2 a Try two different values for a and b. b Write down the two answers the calculator gives for the two expressions, 2a + 5b + 3a – b and 5a + 4b. c Comment on the result. 3 a Try two more values for a and b. b Write down the two answers the calculator gives for the two expressions, 2a + 5b + 3a – b and 5a + 4b. c Comment on the result. d Are these expressions the same for all values of a and b? 4 a Enter two new values for a and b and two new expressions, 5a + 2a − 6b + 9b and 7a + 3b. b Try three different sets of numbers in each expression. c Comment on your results. 5 a Enter two new values for a and b and two new expressions, 5a + a − 3b + 2b and 5a + b. b Try three different sets of numbers in each equation. c Comment on your results. 6 Study the sets of expressions carefully, and explain what you have observed about each pair of equations. 7 a Enter three new values a = 3, b = 4, c = 5 and evaluate the expressions a2 + b2 and c2. b What do you notice about the results? c Can you find any other number combinations this works for?

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11G Exploring dot patterns Learning task 11G Triangular numbers 1 Dots are arranged in the shape of a triangle and counted to form ‘triangular numbers’. The first four terms are drawn below:

1 dot

3 dots

6 dots

10 dots

a Draw the next two shapes in this pattern. b Write down the number of dots in each of the first 10 triangular numbers. c Describe the pattern for triangular numbers.

Square numbers 2 Dots can make the shape of a square and are called ‘square numbers’. The first three square numbers are drawn below:

1 dot

4 dots

9 dots

16 dots

a Draw the next two shapes in this pattern. b Write down the number of dots in each of the first 10 square numbers. c Describe the pattern for square numbers.

Pentagonal numbers 3 The first three pentagonal numbers are drawn below:

1 dot

a b c d

5 dots

9 dots

Explain why these are called ‘pentagonal numbers’. Draw the next two shapes in this pattern. Write down the number of dots in each of the first 10 pentagonal numbers. Describe the pattern or rule for pentagonal numbers.

4 a Find two numbers that are greater than 1 and are both pentagonal and square. b Find a number that is greater than 1 and is both triangular and square.

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Maths Dimensions 7

Exploring match patterns 11H

11I

Whole numbers produce some interesting patterns when the four basic operations, +, −, ×, ÷, are applied to them. Patterns are a part of our everyday life and giving them rules can save us time and help us to make predictions.

Learning task 11H 1 Use matches to make the following shapes:

3 matches

5 matches

7 matches

a b c d

Draw the next three shapes by continuing the pattern of matches. Complete this sequence 3, 5, 7, ____, ____, ____, ____, ____, ____. Explain how you found each number. Predict how many matches would be needed to make the 15th shape (term) in this pattern. e Test your prediction for the 15th term, by making it with matches.

2 Build these first three shapes with matches:

4 matches

a b c d

7 matches

10 matches

Predict what the next term in this pattern might be. Make the fourth term of this sequence with matches. Was your prediction correct? Explain your answer. Complete the statement: ‘The rule for this pattern is: We start with 4 matches, then add __________ matches each time’.

3 Build these first three shapes with matches:

6 matches

a b c d

12 matches

18 matches

Predict what the next term in this pattern might be. Make the fourth term of this sequence with matches. Was your prediction correct? Explain your answer. Copy and complete the statement: ‘The rule for this pattern is: We start with __________ matches, then add __________ matches each time’. Chapter 11 Algebra Symbols

321

11I

Rules and formulas

When we use an identifying symbol and an equals sign before an expression, the result becomes a formula. A formula explains the process for working out a mathematical result. For example, we have developed 2L + 2W as the expression for finding the perimeter of a rectangle. P = 2L + 2W is the formula that works out the perimeter of a rectangle. P (perimeter) is the subject of the formula. The pronumerals in any formula are often referred to as variables, because the values we substitute for them usually vary, depending on the situation.

Example

Solution

1 Use the formula to find the perimeter of this rectangle:

3m 4m

P = 2L + 2W P=2×4+2×3 P=8+6 P = 14 metres

2 The cost C ($) of hiring a surfboard for t hours is given by the formula: C = 10 + 3t where there is a $10 initial charge, plus a charge of $3 for each hour the board is in use. Find the cost of hiring the board for: a 1 hour

C = 10 + 3 × 1 = $13 It costs $13 to hire the board for 1 hour.

b 4 hours

C = 10 + 3 × 4 = $22 It costs $22 to hire the board for 4 hours.

Exercise 11I 1 The perimeter of the isosceles triangle shown is given by the formula P = 2x + y. a Find the perimeter where x = 5 cm and y = 3 cm. b Find the perimeter if x = 8 m and y = 4 m. 2 The area A of a rectangle of length L and width W is given by the formula A = LW. a Find the area of a rectangle where L = 12 cm and W = 9 cm. b Find the area of a rectangle where L = 6 m and W = 7 m.

x

x

y

3 The area of a triangle of base b and height h is given by the formula A = 1--- × b × h. Find the area of a triangle with base 2 8 cm and height 6 cm.

h

b

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Maths Dimensions 7

11I 4 The number of points P gained by a football team is given by the formula P = 6g + b, where g is the number of goals kicked and b is the number of behinds kicked. a Find the points score for two teams that have scored: i 2 goals and 10 behinds ii 3 goals and 4 behinds b Which team won the match? 5 The number of matches N needed to tessellate T equilateral triangles in a straight line is given by N = 1 + 2T: a How many matches are needed to tessellate 5 triangles? b How many matches are needed to tessellate 20 triangles? c How many triangles can be tessellated with 33 matches? 6 The diagram shows a linear tessellation of rhombi. a How many matches are needed to tessellate: i four rhombi? ii five rhombi? b List as a sequence the number of matches needed for 1, 2, 3 … 10 rhombi. c Can you find the formula that works out N the number of matches needed to tessellate R rhombi? Test your formula using your results to part a above. 7 The diagram shows a linear tessellation of an arrow. a How many matches are needed to tessellate: i four arrows? ii five arrows? b List as a sequence the number of matches needed for 1, 2, 3 … 10 arrows. c Can you find the formula that works out N the number of matches needed to tessellate A arrows? Test your formula using your results to part a above. 8 The cost in dollars of printing programs for the school production is made up of a fixed ‘set up’ cost of $50 plus $0·50 for each program printed. a Find the cost of printing: i 10 programs ii 25 programs iii 50 programs iv n programs b Write a formula to work out the cost C of printing n programs. c Check your formula by using it to find the cost of printing 10 programs. d Use your formula to find the cost of printing 150 programs. 9 A student starts saving for a new skateboard. She opens an account with a $30 birthday gift and adds $5 to this each week from her pocket money. a How much has she saved after: i 2 weeks? ii 5 weeks? iii 8 weeks? iv n weeks? b Write a formula to find A, the amount saved after n weeks. c Check your formula by using it to find A when n equals 5. d How long will it take her to save $135, the cost of the skateboard?

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Puzzles 1 Simplify these expressions, then substitute the letter for the answer below to solve the riddle:

What do nuclear scientists eat for lunch?

A F L P

m + m + 3m 10m − 5m + 5 12a − a 3x + 4x

C H N R

D I O S

5m − m + 3m 5a + 2a 4a + 12 − 4a 8x − 8x

_____ _____ _____ _____ _____ _____ _____ 5m + 5 a + 7 4x 4x a + 7 2x + 2 12

4m + 3 − 3m + 4 4a − 3a + 7 3x − x + 2 7x − 4x + x

_____ _____ _____ _____ _____ 7m 7a a+7 7x 4x

2 Expand the brackets in each of the following expressions, then match the letter to the correct answer below to solve the riddle:

What did the water say to the boat?

A G J S V

3(x + y) 2(x − 4) 2(4 − a) s(s + 2) 3(2a +1) _____ _____ _____ 15 − 5b 2 − 2m x2 + x

D H N T W _____ 3a + 6

_____ _____ _____ _____ 8 − 2a y2 − 3y s2 + 2s x2 + x

324

Maths Dimensions 7

2(a + b) 3(a + 2) 5(3 − b) x(x + 1) 4(3m − 1) _____ _____ _____ 5b − 5 15 − 5b 2x − 8

E I O U Y

4(m + n) 5(b − 1) 2(1 − m) y(y − 3) 2(2b + 2) _____ 5b − 5

_____ x2 + x

_____ _____ _____ _____ _____ 12m − 4 3x + 3y 6a + 3 4m + 4n 2a + 2b

Ch11 3 Simplify the expressions below, then match the letters to the answers that appear below to answer the riddle:

What did the spider do on the computer?

A D I N W

4×a×b 2×p×3×q 2m × 3n 3p ÷ 6 3a -----15

B E K S

5×a×a

X

4b -----12

_____ 6mn

_____

_____

a -----2b

m ---6

_____

_____ a2b

a --5

_____ 5a2

_____ a -----3b

C G M T

a×b×a 4p × 3q × 2q 3a ÷ 9b

_____ 4ab

_____ 6pq

_____ a2b

_____ 6mn

_____

_____ a2b

a -----2b

3×m×n×m 4×a×2×b m÷6 2a ÷ 4b

_____ 4ab

4 Use your calculator to help complete the cross-number puzzle. If a = 6, b = 4, c = 5, find:

Across 1. 10a + 9c 1.

2.

3. 7a + 3b + 15c 4. 3b2c + 2 6. a + 3bc2

3.

4.

5.

8. abc 9. 9ac

6.

7.

8.

Down 1. b3 + c3 2. 2(a + c) + 110c 3. 3(a2 + c)

9.

5. 10c2 7. a + 6(a + b)2 − b 8. 5ab + 4c

Chapter 11 Algebra Symbols

325

Applications Body mass index The body mass index of a person is worked out by dividing his/her mass in kilograms by the square of his/her height, measured in metres. As a formula this is expressed as m I = ----- . 2 h For example, a student of mass m = 70 kg and height h = 1·75 metres would have a body 70 mass index calculated by I = ------------- , which 2 1·75 works out to 22·9. A person of normal weight should have a body mass index in the range 20 to 25. An index outside this range indicates the person is either underweight or overweight.

1 Calculate your own body mass index. Is it in the preferred range? 2 Calculate the body mass indices for other people, perhaps for members of your family or classmates. m The formula I = ----- can be rearranged to work out a range of preferred weights from a 2 h person’s height. The formula mmin = 20h2 will work out the preferred minimum weight (mass), and mmax = 25h2 will work out the preferred maximum weight. For someone whose height is 1·75 metres, the range of preferred weights is given by: mmin = 20 × 1·752 and mmax = 25 × 1·752, which results in the range 61·25 kg to 76·3 kg.

3 Using your own height, work out your preferred weight range. 4 Complete the following table, which lists a person’s possible height h and their preferred minimum and maximum weights. h (m) 20 ×

1·2

1·3

1·4

1·6

1·7

1·8

1·9

2·1

2·2

2·3

h2

25 × h2

5 Draw up a set of graphical axes of your own, with height h on the horizontal axis and mass m on the vertical axis. Carefully scale and label your axes. Plot the middle row of the table (mmin) and join the plots with a smooth curve. Next plot the third row (mmax) and join these plots with a smooth curve as well. Colour in the region between the two curves. 6 What can you say about the preferred weight range for people who are taller?

326

Maths Dimensions 7

Ch11 Energy expended during physical activity The amount of energy expended each hour when you perform different types of physical activity can be worked out by using the metabolic rate for that particular activity. One unit of metabolic rate, called a MET, is 1 calorie expended per kilogram of body mass per hour. For someone at rest (sitting still), the rate is 1. So the energy expended by a student of mass 60 kg sitting in class for an hour and a half would be given (approximately) by 1 × 60 × 1·5 = 90 calories. This answer can be multiplied by 4·18 to convert it to kilojoules, so 90 calories = 376·2 kilojoules. As a formula, the calculation can be expressed as: E = rmt where E is the energy expended in calories, r is the metabolic rate for the particular activity, m is body mass (weight) in kilograms, and t is time measured in hours. If you jog or walk at a speed of 6 kilometres per hour, the metabolic rate is 6. In fact, for this type of walking or running activity the rate is just the speed in km/h. For example, a student of mass 65 kg who jogs at 6 km/h for 45 minutes would expend: E = 6 × 65 × 0·75 = 292·5 calories, which is 1223 kilojoules (292·5 × 4·18).

1 Using your own body mass, work out the energy you expend sitting in class during a day at school. Take the rate as 2, to include the activity of writing etc. 2 The metabolic rates for a variety of sports are listed: Golf 3–4 Volleyball 3–6 Tennis (singles) 4·25–6·5 Dancing 3·5–5 Tennis (doubles) 3·5–4·5 Cricket 2–5·5 Swimming 4–12 Cycling 5–9 Basketball 5–8 Squash 9–12 Water skiing 5–7 Football 5–10 Select a sport that you play and from the given range estimate the appropriate metabolic rate, depending on how vigorously you participate. State how long the activity usually lasts and then calculate the energy you expend in calories as well as kilojoules. 3 Select another sport you play (or would like to play), choose a time and calculate the energy expended. 4 If a slice of bread supplies 100 calories (418 kilojoules) how many slices of bread would you need to eat to replace the energy expended in questions 1, 2 and 3? Chapter 11 Algebra Symbols

327

Enrichment 1 Use the numbers in the table below to discover the rule relating the values of Q to those of P. The first is done for you: Q = 2P – 1

b

P

1

2

3

4

Q

1

3

5

7

P

5

6

7

8

Q

12

15

18

21

a

c

P

10

20

30

40

Q

6

11

16

21

P

1

2

3

4

Q

0

2

6

12

2 Copy and complete the Table Tangle using the clues given: m

9

1

2

8

10

m+1

5m

25

m2 m ---2

3m + 1

22

2m2 (2m)2

64

17 – m m3

27

3 Try this problem using different starting numbers: Think of a number, double it, add 3, multiply the result by 5, subtract 15, then divide by 10. What answer do you end up with? Can you explain what is happening? To assist, use n for the number first thought of and write an algebraic expression which shows the steps performed, then simplify it. Make up your own number puzzle with different mathematical steps. Show your teacher the number puzzles you devise. Reversing an expansion or reinserting the brackets is called factorisation. 4 Copy and complete the following: a 8x + 12y = 4( x+ y) c 6x2 + 9x = 3x( + ) e 3x2 − 6x = 3x( − ) 2 g 5x − 10 = ( − ) 5 Factorise the following: a 14x + 21y d 8a − 12b g 6x2 − 12x

328

Maths Dimensions 7

b 16x + 20y e x2 + 5x h 4x2 − 6x

b d f h

10m + 15n = x2 + 6xy = x2 − 4xy = 2x2 + 8x =

( ( (

(2m + 3n) + ) − ) + )

c 25a + 30b f x2 − 7x i 6x2 − 14xy

Ch11 6 The speed s kilometres per hour attained by a freefalling parachutist can be found approximately using the formula s = 36t, where t seconds is the time for which the parachutist falls before the parachute opens. a Find the speed of the parachutist after: i 1 second ii 2 seconds b Complete the following table which shows the speed of the free-falling parachutist over the first 5 seconds: t (seconds)

0

s (km/h)

0

1

2

3

4

5 180

c Plot the points on a coordinate plane. d Join up the points with a straight line and label your graph with the formula s = 36t. e In reality, due to air resistance the speed of a free-falling parachutist levels out at a maximum of 180 km/h and this is called the terminal velocity. With this information continue the graph in part d over the first 10 seconds. 7 The distance p metres, covered by another free-falling parachutist while falling for the first 5 seconds is given approximately by the formula p = 5t2. a Complete the following table to show the distance the parachutist falls: t (seconds)

0

p (metres)

0

1

2

3

4

20

5 125

b Plot the points on a coordinate plane. c Join the points to form a smooth curve and label your graph. From 5 seconds onwards, after the parachutist has reached terminal velocity, the distance fallen is given by the formula d = 125 + 50(t − 5). d Use this formula to complete the following table: t (seconds)

5

d (metres)

125

6

7

8

9

10

275

e Plot these values on the axes in part b and join up the points with a straight line. f Find how long it takes the parachutist to fall: i 80 metres ii 400 metres iii 600 metres g What is the terminal velocity of the parachutist, in metres per second? Chapter 11 Algebra Symbols

329

Revision Exercise 11A 1 Write expressions for the following statements: a the sum of ♣ and 9 b 12 less than c one-fifth of ♥ d three times the value of e divided by 7 f the sum of + 2 Write the following algebraic expressions in words: x+2 a c + 10 b 8a c --yd ----------9 3

-----e 4b 3

f

(x + 6) × 3

Exercise 11B 3 Simplify the following: a 8 +9 d 19 − 4 g 7 + 3 + 2■ + 4■

b 6♥ + 5♥ e 7 +2 h 4 + 3 + 6★ + 2★

c 13 − 5 f + + i 5 − 2 + 8■ − 6■

4 Simplify the following expressions by combining like terms: a 8a + 4b − c − 2a + 3b + 5c + 3a − b + c b 4x + 3y – 2x – y + 1 + 3 c 7ab + 5bc − ca − 3ab + 4bc − ab + 2ca d 5x + 4y + 7 – 2x + y − 3 e 8xy + 9yz − 2zy + 3zx + 4xy − 3yz + 5zx f 9a + 7b + 6c – 6a − 6b − 5c

Exercise 11C 5 Simplify the following products as far as possible: a 7×a b 9×y d b×8 e a×c g 16 × y × z h m×n×6 j p×8×9 k y×4×2

c f i l

6 Express the following quotients in simplest fraction form: a x÷5 b m÷9 d 5×y÷z e 9 × y ÷ 4x g 5x ÷ 15 h 16 ÷ 32x

c p ÷ 12 f 5x ÷ 7y i 28 ÷ 35w

a × 15 m×p a×p×q 8×7×p

Exercise 11D 7 Expand (remove brackets) each of the following and simplify: a 4(x + y) b 9(a + b) c d y(x + 4) e y(y + 7) f g (a – 4)5 h (c + 5)4 i j (6 + p)p k (12 – q)3q l m x(x + 1) n x(x − 3) o p 6y(5y − 2) q 5q(2q – 4) r 2 s x(2x − 1) + 2x(x + 1) t 3x(x + 5) – 3x u

330

Maths Dimensions 7

6(m + n) p(p + 8) (b – 9)5 (5 – n)8n x(2x + 5) 2x(x + 1) + x(x + 1) mp(m – p)

Ch11 Exercise 11E 8 The perimeter of the isosceles triangle shown is given by the formula P = 2x + y. a Find the perimeter where x = 7 cm and y = 4 cm. b Find the perimeter if x = 4 cm and y = 3 cm. c If the perimeter of the isosceles triangle is 13 cm, and side y is 1 cm longer than side x, find the sides x and y.

x

y

9 Find the values of the following where x = 6 and y = 8: a 8xy

b 5xy – 1

c

1 --- xy 4

d 6x + 2y g 5(2x + 1) + y j 6 + y2

e 10x – 7y h 3(2x – 1) + 2y k 2y2 + 3y + x

f i l

4y + 3x 2(2x – 4) +3y 2x2 ÷ y

–2

10 At the cinema popcorn costs $3 per container and drinks cost $2 each: a Write an expression for the cost of x containers of popcorn and y drinks. b By substituting 3 for x and 5 for y, find the cost of three containers of popcorn and five drinks. c If $24 was spent in total, give some possibilities for the number of drinks and containers of popcorn that were purchased. If twice as many containers of popcorn as drinks were purchased, how many of each were bought?

Learning task 11G 11 Shapes made of dots can form interesting patterns, like this star: a Draw the next star shape in this pattern. b Write down the number of dots in each of the first 10 star numbers. c Describe the pattern or rule for star numbers.

dots

Exercise 11I 12 The area A of a rectangle of length L and width W is given by the formula A = LW: a Find the area of a rectangle where L = 12 cm and W = 8 cm. b Find the area of a rectangle where L = 16 cm and W = 4 cm. c Find the perimeter of each of the rectangles in parts a and b. d If the dimensions of the rectangle in part b are halved, what is the new area? How does this area relate to the previous area? Chapter 11 Algebra Symbols

331

CHAPTER

12 Equations and Inequations

Equations and Inequations Some water rides offered at Wet ’n Wild Water World on the Gold Coast require participants to satisfy a minimum height or a maximum weight requirement for safety reasons. Some rides such as the Double Screamer require people to be at least 110 centimetres tall. Using mathematical symbols, the height requirement could be represented as h ≥ 110, where the height, h, is measured in centimetres. The Speedcoaster ride requires that the combined weight, w, of the two people taking that ride not exceed 220 kilograms, or w ≤ 220.

This chapter covers the following skills: • Solving equations by inspection • Solving equations by flow charts • Solving equations by inverse operations • Using equations to model real-world problems • Solving equations on a CAS calculator • Solving inequations • Exploring the coordinate plane

12A

Solving equations by inspection

Equations can be written using pronumerals to represent unknown numbers. For example, 5x + 4 = 14 is an equation and true only for x = 2. Simple equations can be solved by inspection.

Example

Solution

1 Solve x + 7 = 11.

x + 7 = 11 4 + 7 = 11 so x = 4

2 Solve y − 2 = 3.

y−2=3 5−2=3 so y = 5

3 Solve a + a + a = 12.

a + a + a = 12 3a = 12 3 × 4 = 12 so a = 4

4 Solve 2y = 14.

2y = 14 2 × 7 = 14 so y = 7

5 Solve --p- = 8. 4

p --- = 8 4 32 ------ = 8 4 so p = 32

Exercise 12A 1 Determine the value of the symbol in each of the following: a a + 15 = 20 b b−9=8 a=? b=? d d + 21 = 30 e e − 2 = 19 d=? e=? g g + 154 = 181 h h − 57 = 26 g=? h=?

c c − 10 = 26 c=? f f + 37 = 50 f=? i i + 219 = 302 i=?

2 Determine the value of the symbol in each of the following: a k + k + k = 21 b m + m + m + m = 36 d r + r + r + r + r = 55 e t + t + t + t + t = 60 g x + x + x = 21 h y + y + y + y + y = 30

c p + p + p + p + p = 10 f v + v + v + v + v + v = 42 i m + m + m + m + m = 25

3 Solve the following equations by inspection: a x + 5 = 11 b y + 12 = 21 e r + 12 = 17 f t+5=9 i x + 7 = 19 j s + 6 = 11 m m + 3 = 16 n a + 80 = 220 q x + 19 = 35 r y + 16 = 31

334

Maths Dimensions 7

c g k o s

7 + m = 13 10 + q = 18 b+2=8 e + 45 = 156 p + 21 = 45

d h l p t

9 + a = 17 22 + p = 30 y + 13 = 17 f + 9 = 110 25 + x = 46

12A 4 Solve the following equations by inspection: a x−7=9 b s − 8 = 11 e m − 13 = 14 f a − 18 = 20 i 10 − b = 8 j 13 − c = 6 m 39 − m = 19 n 45 − n = 22

c g k o

b−5=8 e − 24 = 6 9−d=7 33 − p = 8

d h l p

y − 13 = 7 f − 9 = 11 20 − a = 14 50 − q = 32

5 Solve the following equations by inspection: a 6x = 12 b 9y = 27 e 12 × m = 60 f 8 × n = 56 i 17x = 34 j 65s = 195 m 30 × m = 900 n 7 × p = 280

c g k o

8a = 32 p × 4 = 12 2b = 88 e × 45 = 135

d h l p

h × 5 = 30 q × 15 = 45 13y = 52 f × 19 = 114

6 Solve the following equations by inspection: a x ÷ 3 = 12 b y ÷ 2 = 26 e 12 ÷ m = 6 f z ÷ 12 = 7

c a ÷ 9 = 12 g 135 ÷ p = 27

d h ÷ 12 = 30 h 77 ÷ q = 7

x --- = 2 7

m -=2 k ----10

l

n q--- = 3 9

r -=4 o ----11

t -=6 p ----12

i

y --- = 3 6

m --p- = 4 5

j

n --- = 7 8

7 Write an equation for each of the following statements, then solve it: a I think of a number x then add twelve and the result is twenty. b I think of a number y then add sixteen and the result is thirty. c I think of a number a then subtract eight and the result is six. d I think of a number b then subtract eighteen and the result is fifteen. 8 Write an equation to represent the information, then determine the answer to each of the following: a Six more than a number is eight. What is the number? b Eleven less than a number is twenty. What is the number? c One-half of a number is fifty. What is the number? d Eight more than four times a number is twenty. What is the number? e Five less than three times a number is sixteen. What is the number? f The square root of a number is five. What is the number? 9 For each equation, determine by substitution whether the given solution in brackets is correct: a x + 15 = 38 [x = 22] b y − 12 = 7 [y = 9] c 5m − 15 = 10 [m = 4] d 2n + 3 = 15 [n = 6]

e b--- = 2 [b = 12] 6

f

50 ------ = 10 p

g q--- + 8 = 10 [q = 10] 5

h 100 --------- – 1 = 4 [b = 25] b

i

3x + 1 --------------- = 7 [x = 9] 4

1 k m ---- + --- = 4 [m = 11] 3 3

l

n 1 = 1 [n = 9] --- – --8 8

j

4x – 2 --------------- = 7 [x = 4] 3

[p = 10]

Chapter 12 Equations and Inequations

335

12B

Solving equations with flow charts

This method displays the steps used to compile an equation. It is used where the unknown pronumeral appears only once in the equation. The solution is obtained by reversing the steps.

Example

Solution

Solve the following:

a x+5=8

Display the flow chart: +5

x

x+5

Reverse the flow and use ‘inverse’ or opposite operations

= −5

3

8

The solution is then x = 3, which is easily checked by inspection.

b x − 9 = 16

Display the flow chart: −9

x

x−9

Reverse the flow and use ‘inverse’ or opposite operations

= +9

25

16

The solution is then x = 25, which is easily checked by inspection.

Exercise 12B For questions 1–3, complete the details to solve the equations.

1 Solve 9x = 63. Complete the flow chart: x

×9

9x =

÷9

63

The solution is then x =

336

Maths Dimensions 7

Reverse the flow and use ‘inverse’ or opposite operations.

2 Solve --y- = 3. Complete the flow chart: 8 y

÷8 = ×8

3

The solution is then y =

12B 3 Solve 4x + 9 = 57. Complete the flow chart: x

4

4x

+9

4x + 9 =

÷4

−9

57

The solution is then x =

4 Solve each of the following equations by first drawing a flow chart and then reversing the flow: a p + 8 = 19 b 10 + a = 26 (Hint: Rewrite as a + 10 = 26.) c x + 10 = 17 d y + 15 = 31 e r − 9 = 13 f a + 8 = 20 g b − 14 = 10 h m−6=9 5 Solve each of the following equations by drawing a flow chart and then reversing the flow: a x × 6 = 18 b y × 8 = 40 c m × 12 = 108 d 9a = 54 e 7b = 35 f 8c = 32 g --p- = 4 7

h q--- = 3 9

i

r ------ = 5 12

m ---- = 4 9

k n--- = 13 3

l

p --- = 12 8

m --c- = 3 4

n a--- = 11 5

o --p- = 13 7

j

6 Solve each of the following equations, which have several steps, by drawing a flow chart and reversing the flow: a 2x + 9 = 15 b 4y + 6 = 30 c 7a + 4 = 67 d 3m − 5 = 13 e 8n − 10 = 78 f 9p − 8 = 28 g 2x + 4 = 10 h 2y + 6 = 0 i 6a + 5 = 17 j 5m + 1 = 11 k 2n − 10 = 12 l 9p − 2 = 34 m 2x + 5 = 7 n 4y + 6 = 46 o 7a + 4 = 39 7 Write equations for the following statements and then solve them by using a flow chart: a I think of a number x, multiply it by 5 and the result is 20. b I think of a number y, divide it by 8 and the result is 6. c I think of a number n, multiply it by 8, then add 5 to get a result of 37. d I think of a number q, multiply it by 9 then subtract 8 and get a result of 55. e I think of a number m, multiply it by 6, then subtract 2 to get a result of 40. f I think of a number s, multiply it by 9 then subtract 1 and get a result of 80. g I think of a number x, divide it by 5, then subtract 3 to get a result of 2. h I think of a number x, multiply it by 3, then divide by 4 and finally add 5 to get a result of 11. Chapter 12 Equations and Inequations

337

12C

Solving equations by inverse operation

The inverse operations used in solving an equation when using a flow chart can also be applied directly to solve it using algebra.

Example

Solution

1 I think of a number x and add 5 to get a result of 11. The equation is x + 5 = 11.

We subtract 5 from both sides to keep the ‘balance’: x + 5 = 11 −5 −5 Solution is x = 6.

2 I think of a number n and subtract 7 to get a result of 13. The equation is n − 7 = 13.

We add 7 to both sides: n − 7 = 13 +7 +7 Solution is n = 20.

3 A certain number y is multiplied by 8 to get a result of 32. The equation is 8y = 32.

We divide both sides by 8: 8y 32 ------ = -----8 8 Solution is y = 4.

4 A certain number q divided by 12 gives a result of 8.

We multiply both sides by 12: q ------ × 12 = 8 × 12 12

q The equation is ------ = 8. 12

Solution is q = 96.

Exercise 12C 1 Solve the following equations algebraically, showing the inverse operation steps presented in the worked examples: a x + 4 = 16 b 13 + y = 24 (Hint: Rewrite as y + 13 = 24.) c m − 7 = 15 d n − 19 = 6 e 18 = p − 5 1 1 f p − --- = 3 --g q + 2·5 = 4·75 h t − 3·25 = 1·75 2

4

3 --4

= 1 1---

i

m+

l

p + 3·5 = 8·75

2

j

q − 2 1--- = 1 3--2

4

m s + 2·25 = 3·75

k x − 1 2--- = 3 1--3

3

n t + 3·5 = 6·75

2 Solve the following equations algebraically, showing the inverse operation steps presented in the worked examples. Note that part g onwards will involve answers which are simple fractions or decimals: a 4x = 12 b 6y = 72 c 9n = 45 d 3g = 33 e 11p = 44 f 5m = 60 g 2r = 5 h 3s = 7 i 7t = 11 j 4z = 10 k 8q = 44 l 6h = 22 m 10x = 5 n 12y = 8 o 24z = 18 p 6m = 15 q 2n = 1·8 r 3t = 2·7 s 5a = 2·5 t 0·5x = 1·5

338

Maths Dimensions 7

12C 3 Solve the following equations algebraically, showing the inverse operation steps presented in the worked examples: a --x- = 8 9

a -=3 b ----13

c b--- = 12 7

d --p- = 16 6

q -=7 e ----15

f

r ------ = 3 18

1 ---- = 2 --g m 4 2

h n--- = 1·5 3

i

p 3 --- = 2 --4 4

4 Six popcorns at Luna Park cost me $19·50. a Let p be the cost of one popcorn, then write an equation and solve it for p. b Let c be the change obtained from $50, then write an equation and solve it for c.

5 When the price of a movie ticket P is divided by 11 to calculate the amount of GST paid, the result is $1·05. Write an equation for P and solve it to find the price of the movie ticket. 6 Using the perimeters of the following figures, write equations and solve to find x. a

b

x x

x x

x

x

x x

8 cm

Perimeter is 216 cm

Perimeter is 38 cm Chapter 12 Equations and Inequations

339

12D

Solving two- and three-step equations

Equations with more than one operation can be solved by using the same method of inverse (opposite) operations.

Example

Solution

1 Solve the equation 3x + 2 = 14. Subtract 2 from both sides: Divide both sides by 3:

2 I think of a number y , multiply it by 3, then divide the result by 4 to get an answer of 6. 3y The equation is ------ = 6 4 3 or alternatively --- y = 6. 4

3y ------ = 6 4 Multiply both sides by 4:

Divide both sides by 3:

3 I think of a number x, multiply it by 5, divide the result by 4 and finally add 6 to get an answer of 16. 5x The equation is ------ + 6 = 16 4 5 or alternatively --- x + 6 = 16. 4

3x + 2 = 14 −2 −2 3x = 12 3x 12 ------ = -----3 3 x=4

3y ------ × 4 = 6 × 4 4 3y = 24 3y 24 ------ = -----3 3 y=8

5x ------ + 6 = 16 4 Subtract 6 from both sides: −6 −6 5x ------ = 10 4 5x Multiply both sides by 4: ------ × 4 = 10 × 4 4 5x = 40 5x 40 ------ = -----Divide both sides by 5: 5 5 x=8

Exercise 12D 1 Solve the equations below: a 3x + 4 = 31 b 8y − 9 = 31 e 10 + 5q = 35 f 15 + 12a = 39 i 5x − 7 = 33 j 2r + 3 = 4 2 Solve these two-step equations: ------ = 10 ------ = 9 a 5x b 3y 4 4

340

c 9m − 3 = 60 g 18 + 6b = 36 k 6s + 5 = 14

d 7 + 3n = 19 h 45 + 5c = 80 l 7a − 6 = 14 ----- = 4 c 2z 3

d 3--- a = 6 5

e 2--- b = 8 9

f

5 --- c = 10 2

g 2--- m = 5 3

h 4--- n = 2 5

i

3p ------ = 4 2

Maths Dimensions 7

12D 3 Solve the following equations: a --x- + 5 = 7 6

b --y- + 9 = 12 7

c --z- − 6 = 8 2

d 1--- m + 7 = 9 8

e 1--- n − 2 = 4 4

f

1 ------ p − 1 = 5 12

g q--- + 3--- = 2 2 4

h 1--- r − 1--- = 1 4 2

i

1 2 --- s + 2 = 3 --3 3

2x -+1=5 a ----3

------ + 2 = 8 b 3a 4

------ + 3 = 15 c 4n 3

------ − 1 = 9 d 5n 3

e 4------p + 1 = 9 5

f

2q ------ − 3 = 1 7

g 5--- m − 7 = 3 6

h 2--- z + 4 = 10 5

i

3 --- t − 4 = 2 8

4 Solve these equations:

5 Write an equation for each of the following statements and then solve it: a I think of a number x, multiply it by 5, then subtract 15 to get a result of 50. b Four times a certain number x divided by 5 gives a result of 8. c A certain number y is divided by 5, then 7 is added to give a result of 12. d Three times a certain number y is divided by 9, then 3 is subtracted to give a result of 2. 6 Ski-pass day tickets on Mount Buller normally sell for d dollars each. On a particularly bleak day, the company decided to give a school group of 10 students a group discount of $75. If the total amount paid is $375, write an equation involving d and solve it to find the normal price d of a ski ticket.

7 At the kiosk near the ski run I bought a drink, which cost $3 dollars, and four doughnuts for a total of $8. If p is the price of a doughnut, write an equation involving p and solve it to find the cost of a single doughnut. 8 For a group of students to stay at a ski lodge, a special deal is negotiated so that the group pays only three-quarters of the usual price $P. If the group pays $126, write an equation involving P and solve it to find the usual price. 9 At the bottom of the ski slope there are 20 people in the queue waiting to catch the ski tow. Each minute the tow clears 12 people from the queue and 15 people join it. If t is the time required for the queue to grow to 71 skiers, write an equation involving t and solve it.

Chapter 12 Equations and Inequations

341

12E

Inequations

An inequation occurs where two algebraic expressions are related by one of the following four inequality signs: < meaning ‘less than’ > meaning ‘greater than’

≤ meaning ‘less than or equal to’ ≥ meaning ‘greater than or equal to’

Some simple inequation statements are 3 < 7 and 8 > 6. Note that the inequality signs always point to the smaller of the two quantities, or equivalently the wider end opens up beside the larger of the two quantities.

Example 1 If x is the age at which a person can obtain a driving licence in Victoria, write an inequation for x and illustrate the solution on a number line. 2 If x is the age at which you are called a teenager, write an inequation for x and show the solution on a number line.

Solution x ≥ 18 17 18 19

x

20 21 22

Note that a ‘filled in’ or ‘closed’ end circle indicates that the end number is included. x has to be 13 or more but less than 20. Reading from the centre to the left and then to the right, we have 13 ≤ x < 20 12 13 14

15 16 17

18

x

17 20 21 22

Note that an ‘empty’ or ‘open’ end circle indicates that the end number is not included.

3 A number y is multiplied by 3, then 2 is subtracted to give a result which is 10 or more. Write the inequation and solve it, showing the solution on a number line.

We apply the previously covered algebraic steps: Start with 3y − 2 ≥ 10 +2≥+2 3y ≥ 12 3y 12 ------ ≥ -----3 3 y≥4 That is, any number which is 4 or more satisfies the inequation. 0

1

2

3

4

Exercise 12E 1 Use the symbols or = to correctly complete the following statements: a 15 + 12 2×5 b 3×6 40 ÷ 2 c 4+8 6×2 d 16 − 5 24 ÷ 3 e 4 × 0·5 20 ÷ 10 f 36 + 27 6×7 2 g 9 × 12 216 ÷ 2 h 4 +1 20 − 2 2 2 2 2 i 5 + 12 13 j (2 + 3) 2 2 + 32

342

Maths Dimensions 7

5

6

x

12E 2 Draw a number line to represent the set of possible values for x: a x>3 b x < −1 c 3<x≤6

d −1 ≤ x < 5

3 Write down the correct inequality for x which is represented by the following number lines: a b x x −1

c e

0

1

2

3

4

−2 −1

0

1

2

3

−2 −1

0

1

2

3

x

x

d f

−2 −1

0

1

2

3

−2 −1

0

1

2

3

−2 −1

0

1

2

3

x

x

4 Write each of the following statements as an inequation, and show the solution on a number line: a x is less than 10 b y is more than 8 c z is 5 or less d m is 7 or more e x is greater than 3 but less than 6 f x is between 4 and 8 g x is 7 or more but less than 10 h y is greater than 5 but less than or equal to 11 i The price of petrol p varies from 128 cents per litre to 148 cents per litre inclusive. j In NSW the age a at which you can get a driving licence is 17 years or more. k The age a of the people attending the blue light disco on a particular Saturday night was more than 13 but less than or equal to 20. l The time taken to get to school t varies from 20 to 30 minutes inclusive. m The speed s at which I can ride my bike varies from zero to 35 kilometres per hour inclusive. n The speeds at which I may drive my car in the suburbs is any speed greater than zero but not more than 50 kilometres per hour. o The discount d I receive when I pay cash can be anything from 5 to 10 per cent inclusive. 5 Solve the following inequations: a x+5 10 e n + 19 < 21 f p − 17 ≤ 3 i 6z < 33 j 9m ≤ 36

c z − 15 ≥ 16 g 4x ≥ 20 k 5p < 45

d m + 7 ≤ 12 h 12y > 72 l 10q > 55

z ->4 o ----15

p 1--- m < 6 4

m --x- ≥ 5 7

y -≤4 n ----13

q 1--- n ≥ 9 2

r

1 --- p < 8 3

2x - ≤ 12 s ----3

t

----- < 8 u 4z 3

v 2--- a ≤ 4 5

w 7--- b < 14 6

x 3--- c ≥ 6 4

3y ------ > 10 5

Chapter 12 Equations and Inequations

343

12E 6 Solve the following inequations by using the setting out of earlier sections: a 4x + 2 ≥ 8 b 5 + 2y < 13 c 6z − 2 ≤ 22 d 2a + 5 < 12 e 10b + 9 ≥ 24 f 5c − 8 < 15 ---- + 5 ≥ 8 g m 7

h n--- − 9 < 2 6

i

p ------ − 3 ≥ 2 15

k 1--- y − 9 ≥ 1 8

l

1 ------ z − 5 ≤ 4 10

------ + 4 ≤ 14 m 2x 3

------ − 2 > 7 n 3y 5

----- − 7 < 5 o 4z 3

p 2--- a + 3 ≤ 5 5

q 7--- b − 2 < 12 6

r

j

1 --- x + 2 < 3 4

3 --- c − 9 ≥ 3 4

7 Write an inequation for each of the following statements and then solve it. Illustrate your solution on a number line: a I think of a number x, multiply it by 5 and then subtract 12 to get a result greater than 18. b Four times a certain number x divided by 5 gives a result less than 16. c A certain number y is divided by 4, then 7 is added to give a result of 9 or more. d Three times a certain number y is divided by 4, and 1 is subtracted to give a result greater than 8. 8 When I withdraw $100 from my bank account, the balance is still in excess of $175. Letting b be the original balance, write an inequation for b and solve it. 9 I have d dollars in my pocket and my friend has 6 dollars more than me. Together we have less than 16 dollars. Write an inequation for d and solve it. 10 The time t minutes taken for me to get to school is usually greater than 15 minutes, but is no more than 25 minutes. Write an inequation for t and show its solution on a number line. 11 Free tickets for the carnival ride are given to people who are either less than 15 years of age or at least 60 years of age. Let a be a person’s age in years, and write an inequality for a. Display the solution on a number line.

344

Maths Dimensions 7

Exploring the coordinate plane 12F

1F

In Chapter 10, the first quadrant of the coordinate plane was introduced. If two number lines are used, a coordinate plane with four quadrants is produced.

Learning task 12F 1 Coordinates can be joined together to form pictures using all four quadrants of the coordinate plane. a Copy the set axes below into your workbook. b Join the points to form a picture. c Colour in the design.

Join these points

y 6 4 3

→ →

(0, 3) (0, −3)

Now join these points

2

1 −1

2

3

4

5

6

→ → stop → → → → →

(6, 0) (−6, 0) (6, 0) (5, 5) (−5, 5) (−5, −5) (5, −5) (5, 5)

1 −6 −5 −4 −3 −2 −1

→ → stop

(3, 0) (−3, 0) (3, 0)

5

x

−2 −3 −4 −5

(0, 6) (0, −6)

→ →

(0, 2) (−2, 0) (0, −2) (2, 0) stop

→ → → →

−6

2 Construct a coordinate plane which has the x- and y-axes numbered from −6 to 6. Connect the points in the order given to the draw the picture: START (2 1--- , 1), (3, 3), (2 1--- , 3 1--- ), (4, 4), (4, 5), (3, 5 1--- ), (2, 5), (0, 5 1--- ), (−2, 5), (−3, 5 1--- ), 2

2

(−4, 5), (−4, 4),

(−2 1--- , 2

2

3 1--- ), 2

2

(−3, 3),

(−2 1--- , 2

1), (−1,

2

1 --- ), 2

(1,

1 --- ), 2

(2 1--- , 2

2

1),

(3 1--- , 2

0), (4, −1),

(4, −2), (3, −2 1--- ), (2 1--- , −4), (0, −5), (−2, −4), (−2 1--- , −2 1--- ), (−3 1--- , −2), (−3 1--- , −1), (−3, 0), 2

(−2 1--- , 2

2

2

2

2

2

1) STOP

START (4, −2), (5, −2 1--- ), (5 1--- , −4), (5, −5), (4, −5 1--- ), (3, −5), (2 1--- , −4) STOP 2

2

2

2

START (−2, −4), (−2 1--- , −5), (−3 --1- , −5 1--- ), (−4 1--- , −5), (−5, −4), (−4 1--- , −2 1--- ), (−3 1--- , −2) STOP 2

2

2

2

2

2

2

START (−1, −1 1--- ), (0, −1), (1, −1 1--- ) STOP 2

START (−1, START (0, JOIN (3,

1 1--- ), 2

2

(− 1--- , 2

1 1--- ), ( 1--- , 2 2

−2 1--- ) TO 2

PUT A DOT AT

1),

2 1--- ), 2

(2 1--- , 2

( 1--- , 2

(− 1--- , 2

−1 1--- ) 2

1), (1, 1 1--- ) STOP 2

2 1--- ), 2

(0, 1 1--- ) JOIN (−2 1--- , −2 1--- ) TO (−2 1--- , −1 1--- ) 2

JOIN (2, 5)

2

1 TO (2 --- , 2

3 1--- ) 2

2

2

2

JOIN (−2, 5) TO (−2 1--- , 3 1--- ) 2

2

(−1, 3) AND (1, 3). Chapter 12 Equations and Inequations

345

12G Just CAS Introduction The aim of this activity is to learn how to solve basic linear algebra problems in an environment that allows for exploration. This form of technology doesn’t allow you to make errors in algebraic processes. For example, for 2x = 10, if you subtract 2 from both sides (under the misapprehension that subtracting 2 will remove the x-coefficient) you won’t get x = 8. The application will only give the mathematically correct result 2x – 2 = 8.

Equipment Voyage200 or Titanium calculator and Symbolic Maths Guide SMG App™. Symbolic Maths Guide files Problem Set 1 and Problem Set 2 and other software, such as TIConnectV1.6™ and Cabri® Jr, can be downloaded from the Student CD in the back of this book. Note: The symbolic maths guide files on the resource CD are for Symbolic Maths Guide Version 2. The calculator needs operating systems Version 3.00 or higher to use this version of the application.

Technology activity 12G (Keystrokes shown for the Titanium or Voyage200) Switch the calculator on and use the arrow navigation keys to locate the Symbolic Maths Guide application. Note: The symbol must be the same as the one shown opposite.

Press ENTER on the application and a window will be displayed prompting for the location of the file. Selection option 2.

Use the arrow navigation keys to locate Variable: probset1 Press ENTER to open Problem Set 1.

The first problem to solve is 2x + 5 = 15. What should you do first? Press F4 to see what ‘transformations’ are possible. Use the arrow keys to scroll down through the possible options. For this problem, choose option 2: Subtract ? from each side.

346

Maths Dimensions 7

12G Notice the equation is displayed. Enter 5 in the box then press ENTER .

Notice that 5 has been subtracted from both sides of the equation. This keeps the equation balanced. Press ENTER again to see the result.

The next step is to divide both sides of the equation by 2. This can be done through the ‘transformations’ menu as before, alternatively press the ÷ sign. (This is a short cut.) Enter 2 in the box then press ENTER .

Notice that both sides of the equation have been divided by 2. Press ENTER once again to see the result.

To check your answer, press F8 . Press ESCAPE to return to the problem.

The calculator has helped solve this algebraic problem, with your assistance of course. The solution process looks like this: 2x + 5 = 15 2x + 5 – 5 = 15 – 5 2x = 10 x=5 Each of the problems in Exercise 12C Question 1 is located in Problem Set 2. Open Problem Set 2 by pressing F1 – Open Problem Set. Select probset2. When you have finished, show your teacher the working out on your calculator and check your answers against those given in the textbook for Exercise 12C Question 1.

Chapter 12 Equations and Inequations

347

Puzzles 1 Determine the value of each symbol, then place the corresponding letters in the answers below to solve the riddle:

Why do seagulls fly over the sea?

A − 20 = 52 D × 4 = 44 G − 7 = 43 J ÷ 3 = 10 M + 6 = 22 P × 2 = 70 S − 25 = 3 V ÷ 10 = 2 ____ ____ 22 15

B × 3 = 24 E÷6=3 H ÷ 4 = 11 K × 5 = 45 N + 4 = 25 Q × 3 = 51 T × 2 = 64 W×1=5 ____ ____ ____ ____ 32 44 18 52

____ ____ ____ ____ 15 23 18 5

____ ____ ____ ____ 13 20 18 2

____ ____ ____ 32 44 18

____ ____ ____ ____ 32 44 18 52

____ ____ ____ ____ ____ 5 13 27 23 11

____ ____ ____ ____ ____ ____ 8 72 50 18 23 28

348

C÷3=1 F + 4 = 19 I ÷ 11 = 2 L + 8 = 31 O + 1 = 14 R + 9 = 11 U − 7 = 20 Y − 48 = 4

Maths Dimensions 7

____ ____ ____, 8 72 52 ____ ____ 8 18

Ch12 2 Solve the equations, then match the letter to the answer below to find the answer to the riddle:

What did the alien say to the garden? a --- + 1 = 5 3

D 3d + 1 = 4

E

3e − 10 = 11

K

k --- − 1 = 4 2

M 4m − 10 = 6

O 4o − 5 = 15

R

1 --- r + 1 = 5 2

T

3t − 24 = 3

U

W 1--- w + 1 = 4

Y

1 --- y + 1 = 2 3

Z

3z − 3 = 13

A

3u + 1 = 7

2

____ ____ ____ ____ 9 12 10 7

____ ____ 4 7

____ ____ 9 5

____ ____ ____ ____ 3 5 2 8

____ ____ ____ ____ ____ ____ 6 7 7 1 7 8

3 Number pyramids 3

7

8

10

15

10

Number pyramids are problems which look like upsidedown pyramids and involve finding the sum (adding) of two numbers.

18

25

They are completed by adding two numbers, which are next to each other, then placing this total into the box below. This continues down until the last box is completed.

33 58

3+7

8 + 10 25 + 33

a Copy and complete the following number pyramids: i

2

9 11

3 12

5

ii

1

8

10

9

2

11

b Determine the value of the missing symbol in each of the following number pyramids: i

1

7

5

40

ii

9

5

11

44 Chapter 12 Equations and Inequations

349

Applications In this section we will solve equations using ‘balance diagrams’. Balances of this type have been used in banks to manually count large numbers of coins.

Example

Solution

1 The equation n + 4 = 8 can be represented by a lightweight container of n dollar coins plus four separate coins on the left, balanced by eight coins on the right. Solve for n.

Subtracting 4 (coins) from both sides, just as you would with the equation, gives the solution n = 4. n

n

2 The equation 4n = 12 can be represented with four lightweight containers of n dollar coins on the left, balanced by 12 coins on the right. Solve for n.

Dividing both sides into groups of 4, just as you would divide both sides of an equation by 4, gives the solution n = 3. n

n

n

n

3 The equation 3n + 5 = 14 can be repesented by the following balance diagram. Solve for n. n

n

n

n

n

n

Subtracting 5 from both sides: n

n

n

n =

3n 3n + 5

=

14

and then dividing each side into groups of 3: n

n

n

gives the solution n = 3.

350

Maths Dimensions 7

9

Ch12 1 Solve the following equations by drawing balance diagrams: a n + 7 = 13 b n + 8 = 12 c 6 + x = 15 d 4y = 16 e 5x = 35 f 8n = 24

2 Solve the following equations by drawing balance diagrams: a 3x + 4 = 22 b 2n + 3 = 17 d 5 + 2n = 11 e 9 + 4x = 13

c 6n + 4 = 16 f 10 + 5n = 20

3 Write equations to represent the following sequence of balance diagrams, then solve the equation 2n + 6 = 4n + 2. Take 2 from both sides: n

n

2n + 6

n

=

n

n

n

n

n

n

n

n

n

4n + 2

Take 2n from both sides:

Divide both sides into groups of 2: n

n

n

4 Solve the following equations using balance diagrams: a n + 6 = 3n + 2 b 2n + 1 = n + 5 d 4n + 6 = 2n + 12 e 4n + 8 = 6n

c 2n + 6 = 3n + 2 f 6n = 5n + 6

5 Write an equation for each of the following statements and solve it: a Ten more than a certain number is equal to 3 times that number. b If I add 8 to a certain number the value is equal to 5 times that number. c I think of a number n, multiply it by 7 and then add 5. The answer is equivalent to 8 times the number I thought of plus 2. d Six times a certain number x plus 5 gives the same answer as 3 times the same number plus 17. e Three times a certain number x plus 6 equals the same number plus 10. f Five times a certain number x plus 9 is equal to 3 times the same number plus 17.

Chapter 12 Equations and Inequations

351

Enrichment In this section we will cover extended applications of flow charts to solve more complicated linear equations. The only condition is that the unknown pronumeral (symbol) must appear only once.

Example

Solution

1 Solve 10 – m = 4.

Display the flow chart: subtract from 10

m

10 − m =

subtract from 10

6

4

This is a special case in which the pronumeral is substracted from a number; the inverse operation when flowing back is the same operation. The solution is x = 6. ------ = 9. 2 Solve 36 n

Display the flow chart: divide into 36

n

36 -----n =

divide into 36

4

9

This is the other special case in which the pronumeral is divided into a number; the inverse operation when flowing back is the same operation. The solution is x = 4.

Extending the basic ideas 1 I think of a number x, multiply it by two and subtract the result from twenty-six to get an answer of ten. Write an equation that models this process and solve it to find x, by completing the following flow chart. The equation is 26 – 2x = . x

×2

2x

subtract from 26

26 − 2x =

÷2

The solution is x =

352

Maths Dimensions 7

subtract from 26

.

10

Ch12 2 A certain number x is multiplied by two and the answer is subtracted from eight to give a result that when divided into twenty-four gives a final answer of twelve. Write an equation that models this process and solve it to find x, by completing the following flow chart. 24 The equation is --------------- = 8 – 2x x

×2

. 2x

subtract from 8

8 − 2x

divide into 24

24 --------------8 – 2x

= ÷

Solution is x =

subtract from 8

. Verify by substituting: 8 – 2x

= divide into 24

gives 2, divided into 24 gives 12.

3 Use a flow diagram to solve the following equations and then verify your solutions by substitution: a 34 – a = 20 b 23 – b = 10 c 16 – m = 9 d 10 – 3x = 4 e 45 – 2y = 9 f 52 – 8m = 12 g 45 – 4n = 35 h 29 – 6p = 15 i 125 – 12q = 74 4 Use a flow diagram to solve the following equations and then verify your solutions by substitution: ------ = 9 a 36 x

------ = 27 b 81 y

------ = 4 c 64 z

16 -=2 d ----------------20 – 3x

18 e -------------=3 12 – y

f

44 ----------------- = 2 33 – 2z

18 g -----------------=6 12 – 3 p

27 -=9 h ----------------18 – 5q

i

50 17 – ------------------- = 12 15 – 2m

5 Write equations for the following statements and solve them to find the unknown pronumerals: a A certain number x is multiplied by seven and the result is subtracted from fifty to give a result of eight. b A certain number y is multiplied by four and the result is subtracted from eighteen to give a result of ten. c A certain number p is multiplied by six and the result is subtracted from twenty-five to give a result of seven. d A certain number y is multiplied by seven and the result subtracted from twenty. When this answer is divided into twenty-four, a final result of four is obtained. e A certain number q is multiplied by three and the result is subtracted from nineteen. This answer is divided into fourteen to get a final answer of two. f A certain number m is multiplied by four and the result is subtracted from eighteen. This answer is divided into twelve to give a final result of 1·5.

Chapter 12 Equations and Inequations

353

Revision Exercise 12A 1 Solve the following equations by inspection: a x + 6 = 13 b 9 + y = 14 d m − 9 = 21 e 16 − q = 10 g 7m = 56 j

q --- = 8 7

c z − 7 = 11 f 20 − p = 12

h n × 5 = 45

i

p --- = 3 4

------ = 3 k 36 r

l

28 ------ = 4 s

2 Check by substitution whether the given solutions to the following equations are correct: b --y- + 5 = 10 [y = 20] 4

a 3x − 4 = 11 [x = 5]

Exercise 12B 3 Solve the following equations by using a flow chart: a m + 6 = 11 b p − 13 = 7 d --t- = 3 7

e 4q + 3 = 19

c 8r = 48 f

6p − 7 = 41

Exercise 12C 4 Solve the following equations, this time setting out the appropriate algebraic steps: a x + 12 = 21 b y − 17 = 18 c 9z = 72 d 8m = 44

---- = 4 e m 5

f

n 1 --- = --8 2

Exercise 12D 5 Solve the following equations setting out the appropriate algebraic steps: a 6x + 2 = 50 b 20 + 3y = 32 c 8z − 4 = 68 ------ = 10 d 5x 2

------ = 6 e 2y 3

f

3 --- q = 6 4

g --z- + 5 = 7 3

h 1--- m + 7 = 9 5

i

1 --- n − 1 = 3 4

k 4--- q + 5 = 13 5

l

3 --- r − 2 = 4 7

j

3p ------ + 8 = 14 5

6 Write an equation for each of the following statements, then solve it: a I think of a number x, multiply it by 7, and add 4 to get a result of 60. b Three times a certain number y divided by 4 gives a result of 6. c A certain number y divided by 2 then added to 5 gives a result of 14. d From three times a certain number n divided by 2, 3 is subtracted to give result of 12. e Three times a certain number q divided by 4, minus 5 gives a result of 1.

354

Maths Dimensions 7

Ch12 7 At the cinema, a group of 10 students was given a total discount of $20 when they each bought an ice cream. If they paid a total of $25, find the price of an individual ice cream before any discount was applied.

Exercise 12E 8 Use the symbols , or = to correctly complete the following statements: a 16 + 13 4×6 b 4×5 80 ÷ 2 2 c 12 × 6 144 ÷ 2 d 5 −4 6×4 9 Write each of the following statements as an inequation and show the solution on a number line: a x is less than 13 b z is greater than 8 c y is 10 or more but less than 15 d m is between 20 and 30 e x is greater than 5 but less than 8 f x is either less than 4 or greater than 10 10 Write each of the following statements as an inequation: a The number of goals g by which I expect my football team to win next weekend is at least 5 but no more than 8. b The mark m that I expect to get on my next Mathematics test is more than 60% but no more than 100%. c Students in Year 7 are at least 11 but no more than 13 years old. d To get a discount ski ticket, a person must be less than 15 or greater than 70 years of age. 11 Solve the following inequations by using the appropriate setting out: a x + 9 < 12 b z − 13 ≤ 8 c 5z > 15 d 7m > 42

e --p- ≥ 4 5

f

1 --- q ≤ 11 4

------ ≥ 6 g 3n 4

h 5--- t ≤ 10 3

i

2 --- s ≥ 4 5

12 Solve the following inequations, showing appropriate setting out: a 3x + 2 > 17

b 5y − 7 ≤ 8

c --z- − 4 ≥ 0 5

d 1--- m + 3 > 9 4

e 2------p + 9 < 13 5

f

3 --- q − 8 ≤ 7 4

13 Write an inequation for each of the following statements and then solve it. Illustrate your solution on a number line: a I think of a number n, multiply it by 6, then add 2 to get a result greater than 50. b Four times a certain number y divided by 12 gives a result greater than 3. c A certain number q divided by 9, plus 8 gives a result of 10 or more. d Twice a certain number p divided by 8, minus 1 gives a result of 1 or less. 14 When I add d dollars to my bank account, which had a balance of $100, the new balance is between $125 and $148. Write an inequation for d and solve it. Show your solution on a number line. Chapter 12 Equations and Inequations

355

CHAPTER

13 Probability

Probability Forecasting is used today to make predictions of future events. Weather forecasters analyse data collected over a number of years and use this information to predict the likelihood of rain, cyclones or drought. Computers are used today to process information and make predictions about, for example, the likely winners of the AFL grand final. Using the data for the season, the odds of each team winning the premiership flag can be calculated.

This chapter covers the following skills: • Using the language of chance in everyday situations • Comparing probabilities using language • Using language to estimate a probability by using the results of simple experiments • Finding the probability of a simple event • Predicting and testing probabilities using spinners • Measuring probabilities using percentages, decimals or fractions • Calculating theoretical probabilities using the rule probability of an event number of favourable outcomes = ------------------------------------------------------------------------------------total number of outcomes

13A

The language of chance

The likelihood of an event occurring is called probability or chance. Every day we use terms such as ‘unlikely’, ‘definitely’, ‘even chance’, ‘probably’ and ‘impossible’ to describe whether or not a particular event will happen. Some events are impossible and will not happen. For example, this book will not grow legs and run away. Some events are certain. All other events are called chance events. Is it likely or unlikely that you will leave class early today? To be able to compare probabilities, we can use words or percentages, fractions or decimals. ■

If an event is impossible we say it has a zero chance of occurring.

■

If an event will definitely happen we say it is certain or it has a 100% chance.

■

If an event is unlikely to happen we can say it has little chance of occurring.

■

When you throw a coin there is an even chance that the coin will land on ‘heads’. 0

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

▲ Impossible

▲ Coin will land heads

▲ Certain

Exercise 13A 1 Write down a list of all the words or expressions relating to chance and probability that you know. Combine these with the expressions of other students in your class. 2 Rewrite the class list in order, beginning with impossible. 3 You have some control of your future, but which of the following are impossible (I), definite (D), or chance (C) events? a You will play sport on the weekend. b You will marry before your 21st birthday. c You will blink within the next 30 minutes. d You will watch the TV tonight. e You will go to the movies sometime this month. f You will leave your classroom today. g You will wake up on Mars tomorrow. h You will win a gold medal in an Olympic competition. i You will earn $1 million in your lifetime. j The Sun will rise tomorrow.

358

Maths Dimensions 7

13A 4 Which of the chance events in Question 3 is i most likely and ii least likely to happen to you? Explain your answer. 5 Write down an event that is impossible and an event that will definitely happen. 6 Write down three other chance events. Choose a word from the class list to match the chance of each occurring. 7 Choose an appropriate word to describe the likelihood of the following: a A woman will become the Australian Prime Minister in your lifetime. b It will rain at least once in Melbourne during the month of April. c Fremantle Dockers will win the next AFL premiership. d Australia will become a republic within a decade. e Australia will win the Soccer World Cup. f The Sun will set in Melbourne tonight. 8 Copy the number line below into your workbook. Mark on your number line where you think the probabilities of the following would lie: 0

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Impossible

a b c d e f

Certain

A woman will become the Australian Prime Minister next year. It will rain at least once in Melbourne during the month of April. Carlton will win the next AFL premiership. Australia will become a republic within 100 years. Australia will win the Hopman Cup in tennis. It will snow in your town.

9 Write down a possible event for each of these probabilities: a impossible b maybe d possible e more than likely

c certain

10 Use the following expressions to describe each event below. Use each expression once. Event

Approximate probability

Throwing a five or a six on a die

Small chance

A mother planning three children will have three boys

Likely

Throwing different numbers on two dice

Even chance

Randomly selecting a red card from a pack of 52 playing cards

Negligible chance

Randomly selecting a red lolly from a bag containing 24 red lollies and 1 black lolly

More than half a chance

Randomly choosing a girl from a class of 15 girls and 13 boys

Unlikely

Guessing correctly the numbers for a first division win at Lotto

Almost certain Chapter 13 Probability

359

13B

Theoretical probability

Probabilities may be expressed as fractions in their simplest form. 1 --4

0

1 --2

3 --4

Impossible

1 Certain

You can find probabilities by considering all the equally likely possibilities. To work out the probability of an event you need to consider all the possible outcomes, and find what fraction gives a success. Usually these probabilities are written as fractions in their simplest form.

••••

the number of successes The probability of an event = ---------------------------------------------------------------------------------total number of possible outcomes

Example

Solution There are six possible outcomes: 1, 2, 3, 4, 5 and 6 There is only one success: 4 The probability of throwing a four = 1--- .

What is the probability of getting a 4 when you throw a fair six-sided die?

6

Exercise 13B 1 A fair six-sided die is rolled. What is the probability of: a throwing a 6? b not throwing a 6? c throwing an even number? d throwing a number less than 4? 2 Dawson has a set of 10 cards numbered from 1 to 10. He asks a friend to randomly select a card. What is the probability that she will choose a card with a number: a 7? b below 4? c above 3? d that is even? e that is divisible by 3? f that is not divisible by 3? 3 Display the probabilities in Question 2 on a number line like the one below. You may like to convert each fraction into a decimal first.

0

360

Maths Dimensions 7

0·5

1·0

13B 4 You have a bag containing 12 lollies, 3 of them are red, 3 of them are black and 6 are yellow. a When a lolly is chosen at random, what is the most likely colour? b What is the probability that a friend would randomly select a yellow lolly? c What is the probability that a friend would randomly select a black lolly? d What is the probability that a friend would not select a red lolly? e What is the probability that a friend would not select a green lolly? 5 Draw a number line like the one in Question 3. Use percentages to show the probabilities in Question 4 on the number line. 6 In a class of 30 students there are 12 boys and 18 girls. a Is a girl or a boy more likely to be chosen at random? b Does each boy have more chance of being chosen than each girl? c What is the probability that a girl is chosen at random to be the class representative on a committee? d What is the probability that a boy is chosen at random to be the class representative on a committee? 7 It is believed that E is the most commonly used letter in the alphabet. What is the probability of randomly choosing the letter E from the following place names? a PERTH b BRISBANE c HOBART d MELBOURNE e SYDNEY f CANBERRA 8 What is the probability of randomly choosing the letter R from the names above? 9 Write a sentence to explain which you think is more likely: ■ choosing a boy from a class with 12 boys and 8 girls, or ■ choosing a boy from a class with 14 boys and 10 girls. 10 The probability that a heart is randomly selected from a pack of cards is 1--- . What is 4 the probability that a randomly selected card is not a heart? 11 What is the chance of randomly choosing a person with a hat from this group of people?

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13C Exploring simple experiments Experiments can be used to estimate the probability of outcomes. It is important to repeat the experiment many times. The more trials you perform, the more accurate the results.

Learning task 13C 1 Coin toss a If you throw a coin 50 times, how many times do you expect it to land on tails? b Throw the coin 50 times. Copy and complete the table below. Result

My guess

Tally

Total

Head Tail

c Did your results match your guess? d Would you change your guess next time? Explain your comments.

e Throw two coins 50 times. Copy and complete the table below. No. of heads

Guess

Tally

Total

0 1 2

f

Is this what you expected?

2 Counters You need a bag containing 20 counters: 5 red and 15 of another colour. a Take a counter from the bag without looking. b Replace it, then perform a total of 100 trials. c Copy and complete the table below. Result

Tally

Red counter Other colour

d How many times did you get a red counter? e How many times did you get a counter of another colour? f Based on these results, estimate the probability of getting a red counter: number of red counters Probability = ------------------------------------------------------total number of trials g Is this what you expected? h Choose a word from your list to describe the chance of choosing a red counter.

362

Maths Dimensions 7

Total

13C 3 Numbered cards You need 10 cards numbered from 1 to 10. a Get a friend to shuffle the cards. Without looking at the number, take a card from your friend. b Record the result in the table below. Replace it. Repeat this for 50 trials. c Complete the table below. Result

Tally

Total

Prime number 2, 3, 5, 7 Not a prime number 1, 4, 6, 8, 9, 10

d e f g h

How many times did you get a prime number? How many times did you get a non-prime number? Choose a word to describe the chance of choosing a prime number. Based on these results, estimate the probability of getting a prime number. Is this what you expected?

4 Jelly beans You need a bag containing 20 jelly beans prepared by your teacher with unknown quantities of two different colours. Do not look in the bag. a Take a jelly bean from the bag without looking. b Replace it, then perform a total of 50 trials. c Copy and complete the table below.

Result

Tally

Total

Colour 1 Colour 2

d e f g

Use your results to predict the number of jelly beans of each colour in the bag of 20. Open the bag and test your prediction. How many times did you get colour 1? Based on these results, estimate the probability of getting colour 1: number of colour 1 Probability = -------------------------------------------------total number of trials

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13D Exploring spinners Learning task 13D 1 Spinner 1 a Make a spinner like the one shown. b Estimate how often it would land on a yellow edge. c Spin it 100 times and record the colour of the edge on which it lands. Result

Tally

Total

Red Blue Yellow

d Did you get the results you expected? e Compare your results with those of other students in your class. 2 Spinner 2 a Make a spinner like the one shown. b Estimate how often it would land on the red section. c Spin it 100 times and record the colour of the edge on which it lands. Result

Tally

Total

Red White

d Is this what you expected? e Compare your results with those of other students in the class. 3 a Design a spinner to give the results shown in each table below. i

Result

Total

ii

Result

Total

Red

20

Red

38

White

30

White

12

Blue

50

Blue

50

b Make the two spinners and spin each 100 times to check your predictions. 4 a Design a spinner of your own using at least two colours. b Ask another student to estimate the probability of landing on each section. c Spin it 100 times and check the predictions.

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Maths Dimensions 7

Exploring games of chance 13E Learning task 13E 1 Two-up In the game of two-up people bet on two coins that are thrown in the air. They usually bet on ‘heads’, ‘tails’ or ‘odds’. Heads: both coins land with heads facing up Tails: both coins land with tails facing up Odds: one coin lands tails and the other heads a A person who bets on heads is hoping that both coins will land with the Queen’s head facing upwards. Make a prediction about the chance that this will occur. b Take two coins and throw them in the air 50 times. Record the result in the table below. Result

Tally

Total

Chances out of 50

Heads (HH) Odds (HT) (TH) Tails (TT)

c How many throws landed heads? What was the total number of throws? So what is the chance of a throw landing heads? d What is the probability that a person who bets on heads will win? 2 Craps In the game of craps, people bet on the scores of two dice added together. You will win if the total score on the dice is 7 or 11. a With a partner, roll two dice 100 times. b Record the total score for each roll in the table below. Result

Tally

Total

Sum is 7 Sum is 11 Sum is neither 7 nor 11

c Use your results to determine the chance: i of a total of 7 iii that you will win

ii of a total of 11 iv that you will lose

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365

13F Exploring card games Learning task 13F Use a pack of 52 playing cards to answer the following questions.

1 How many cards are of the following type: a Hearts b Diamonds e Black cards f Red cards i Picture cards: kings, queens, jacks

c Spades g Aces j Numbered cards

d Clubs h Twos

2 a Use the formula number of particular card Pr(card) = ------------------------------------------------------------------------------total number of cards in the pack to calculate the probability of dealing the ace of spades from a well-shuffled pack of 52 cards. b If you deal one card from the pack 50 times, replacing the card each time, how many times would you expect to deal the ace of spades? c Perform an experiment to test the accuracy of your prediction. Deal one card from a well-shuffled pack of 52 playing cards. Record the card then replace it in the pack. Repeat this a total of 50 times. d Copy and complete the table below: No. of times ace of spades

No. of times other card

Total 50

e Compare your findings with your prediction and comment on your results. 3 a Use the formula number of particular card Pr(card) = ------------------------------------------------------------------------------total number of cards in the pack to calculate the probability of dealing a red card from a well-shuffled pack of 52 cards. b If you deal one card from the pack 50 times, replacing the card each time, how many times would you expect to deal a red card? c Perform an experiment to test the accuracy of your prediction. Deal one card from a well-shuffled pack of 52 playing cards. Record the card then replace in the pack. Repeat this a total of 50 times. d Copy and complete the table below: No. of times red cards

No. of times black cards

Total 50

e Compare your findings with your prediction and comment on your results.

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Maths Dimensions 7

13F 4 Use the formula number of particular card Pr(card) = ------------------------------------------------------------------------------total number of cards in the pack to calculate the probability of dealing the following cards: a Hearts b Diamonds c Spades d Clubs e Black cards f Red cards g Aces h Picture cards: kings, queens, jacks i Numbered cards 5 Most card games are played with a hand of cards. Poker is played with a hand of five cards. a Calculate the probability of dealing one heart from a well-shuffled deck of 52 cards. b If a heart is dealt and not replaced, how many cards remain in the pack in total? c How many hearts remain in the pack? d Calculate the probability of dealing a heart from the deck now. To find the probability of the two events, we can multiply the two probabilities. For example: Pr(two hearts) = =

1 4 --- × -----4 17 1 -----17

e Use this process to find the probability of dealing three hearts in a row without replacement. f Now find the probability of dealing four hearts in a row without replacement. g Now find the probability of a hand of five hearts. h Compare your calculations with your prediction. 6 a Many card games can be found on computers. Choose one game, such as Hearts or Solitaire, and estimate the probability of winning a game. b Play the game a number of times and record in a table how many times you win and lose. Win

Lose

Total

c Compare your findings with your prediction and comment on your results.

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13G

Spinners

Spinners can also be used to replace dice in board games, because they can be divided up into sectors. They also make great games at fundraising fetes and on television shows. The name of a prize can be written on each segment. Each of the sectors is equally likely, so the probability of a prize can be calculated using the formula below.

••••

number of prize sectors The probability of a prize = -------------------------------------------------------total number of sectors

Car Holiday Computer TV CD player

The spinner above is similar to the prize wheel on a TV show.

Example 1 a In the spinner above are you more likely to win a car or a CD player? Explain. b What is the probability of winning a holiday?

2 What is the probability that the pointer from the spinner lands on the blue section of this spinner?

Solution There are two equally likely ways to win a car and nine ways for the CD player, so a CD player is more likely. Probability of winning a holiday = 3 chances out of 24 1 = --8 Half of the spinner is blue so there is an even chance of the pointer landing on blue. Probability that the pointer lands in the blue section = 1--- . 2

Exercise 13G 1 How many sections are there in the spinner above? a How many sections are labelled with: i CD player? ii car? iii fridge?

368

Maths Dimensions 7

iv computer?

13G number of successes -, b If the probability of an event = --------------------------------------------------------------------number of possible outcomes work out the chance of winning each of the prizes in part a.

2 Calculate the probability of each spinner landing on red then choose a word to describe the probability: a

b

c

d

3 Design then draw a spinner for which the probability of landing on a red edge is: a certain b unlikely c probable d even chance 4 Design a spinner for which the chance of landing on blue is most likely, red is unlikely, and yellow and black are highly unlikely. 5 What is the probability that the pointer of these spinners lands in the blue section? a

b

c

d

6 Design a spinner for which the probability of the pointer landing in the blue section is: a half b less than half c three times as likely as on red 7 Laura and Zac were doing some experiments to test their spinners. Unfortunately they mixed up their results. Match each of these spinners with its results. Spinner

Results

Spinner

Results

a

40 red and 20 green

b

10 red and 40 green

c

30 red and 60 green

d

4 red and 12 green

e

60 red and 40 green

f

10 red and 15 green

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13H Spinner simulations Introduction The aim of this activity is to investigate the relationship between the number of segments on a spinner and the frequency with which each segment occurs.

Equipment TI-83Plus/TI-84Plus with Probability Simulation App™. TIConnectV1.6™ and Cabri® Jr can be downloaded from the Student CD in the back of this book. For information on other applications, such as Probability Simulation, see ‘Software Downloads’ on the Companion Website.

Technology activity 13H A hand-constructed spinner is a good hands-on activity, but it is very time consuming to collect, say, 100 trials or more. The Probability Simulation application can generate large samples in a few seconds. Press the APPS key and start the Probability Simulation application.

In this activity you will use a spinner. Select option 4 by either pressing 4 on the calculator or by using the arrow keys and pressing ENTER . In this case you could use the arrow keys and Y= , as OK is written above the Y= key. Press the ‘spin’ button WINDOW to see a simulated spin. The graph displays a column representing frequency. As this is the first spin, no other numbers are displayed. The scale on the graph automatically changes to suit the columns. Press the ‘+1’ button WINDOW several times. Each time it is pressed the graph is updated to represent the total number of trials. Once you have finished, use the GRAPH key (Soft Clear key) to clear all the spins. Spin the spinner 50 times by pressing TRACE . Your graph may look different to the one shown opposite.

When the calculator has finished rotating the spinner, use the key to find out how many times each number came up.

370

Maths Dimensions 7

13H 1 Complete the frequency table below after 50 trials: Number

1

2

3

4

Frequency

2 Spin the spinner 50 more times and compete table once more affter 100 trials: Number

1

2

3

4

Frequency

3 How many times would you expect each number to turn up? Set the number of segments to 5. To do this, press the ZOOM key, located underneath the word ‘Set’.

Use the arrow keys to move down to Sections and change the number of sections to 5. Press ‘OK’ once you have set the sections to 5 GRAPH .

4 If you generate 100 spins when there are five possibilities on the spinner, how many times do you think a 2 should turn up? It is possible to change the ‘weighting’ of a section of the spinner. For example, if you set the weighting of section 1 to be 2 and all other sections are set to 1, this means the 1 is twice as likely to turn up as the other numbers. Change the number of sections back to 4. Under the settings menu there is an advanced option, ADV. Change the weighting of the number 1 to be equal to 2. Sections 2–5 should still have a weighting of 1.

5 Sketch a graph of 100 spins when the number 1 is biased. The number 1 is twice as likely to turn up as any other number. 6 If 500 spins are generated how many 1s would be likely to turn up? 7 Suppose there are four segments on a spinner. The spinner has been spun 400 times and produced a table as shown below: Number Frequency

1

2

3

147

52

48

4

a How many 4s turned up? b Draw a column graph of the frequency of each number. c If the spinner was biased, suggest the possible weighting for each of the numbers.

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13I

Using statistics to find probabilities

When you attempt to estimate the probability of an event, you usually think back to previous experiences. ‘I often go to the movies on the weekend, so it is highly likely that I will go this weekend. Collingwood hasn’t won many games this season. I won’t pick them in the footy tipping as they probably won’t win.’ Weather forecasters often predict the chance of rain by using data collected over a period of time. The 2000 Olympic Games were held in Sydney. The International Olympic Committee had to choose between three cities: Sydney, Manchester in England and Beijing in China. People from Manchester believe that Sydney was chosen because it was less likely to rain in Sydney. Collect some data on the rainfall of these three cities over a number of years, by using the library or the Internet, and test this theory.

Example

Solution

Records show that in Sydney in one particular week in September it rained in 5 years out of 20. Predict the chance of rain using the data.

number of weeks of rain Probability of rain = ---------------------------------------------------------total number of weeks 5 = -----20 = 0·25 = 25%

Exercise 13I 1 In a netball game a player had 32 attempts at shooting and scored 7 goals. What is the probability that she will score a goal with her next shot? 2 In a football game a player had 24 shots at goal and scored 5 goals and 4 points. What is the probability that he will score: a a goal with his next shot? b a point with his next shot? c with his next shot? 3 The table below lists reasons why Australians travelled overseas in one year. Reason for travel

Number of travellers

Work

611800

Visiting family or friends

644700

Holiday Education Total

1140400 38200 2435100

Use the data in the table to estimate the probability that an Australian travelling overseas: a is going to visit family or friends b is going overseas to study c is not going for work reasons d is not having a holiday

372

Maths Dimensions 7

13I 4 The table gives the average number of rainy days for each month in five Australian cities. J

F

M

A

M

J

J

A

S

O

N

D

Adelaide

4

4

6

9

13

15

16

16

13

11

8

6

Brisbane

13

14

15

11

10

8

7

7

7

9

10

12

Canberra

7

5

7

7

8

9

10

11

10

9

10

12

Melbourne

8

7

9

12

14

14

15

16

15

14

12

11

Perth

3

3

4

8

14

17

18

17

14

11

7

4

a b c d e f g

Which city is most likely to have a rainy day in January? Which city is most likely to have a rainy day in June? Which city is least likely to have a rainy day in September? What is the probability of rain in Canberra on a particular day in March? What is the probability that it doesn’t rain in Melbourne on a particular day in April? Overall, which city is most likely to have a rainy day? Overall, which city is least likely to have a rainy day?

5 At the end of an AFL 22-week season, the ladder looked like this: Team Carlton Geelong Richmond Essendon West Coast North Melbourne Footscray Brisbane Melbourne Collingwood Adelaide Sydney Fremantle St Kilda Hawthorn Port Adelaide

Won

Lost

Drew

20 16 15 14 14 14 11 10 9 8 9 8 8 8 7 2

2 6 6 6 8 8 10 12 13 12 13 14 14 14 15 20

– – 1 2 – – 1 – – 2 – – – – – –

a Using these statistics, estimate the following probabilities: i Collingwood wins its next game ii Brisbane wins its next game iii Richmond loses its next game iv Adelaide loses its next game b The first round of the next season had the following games: Brisbane v Footscray Collingwood v Carlton Fremantle v West Coast Geelong v Melbourne Essendon v Richmond Hawthorn v Port Adelaide St Kilda v North Melbourne Adelaide v Sydney Using only the information from the previous season, tip the winners for this round. Chapter 13 Probability

373

Puzzles 1 The words in the list below are hidden in the word search. Can you find them? Words can go up or down as well as forwards and backwards. Certain Chance Counter Definite Diagram Dice Even Head Impossible Likely Odds Prime Probability Raffle Table Tails Tree Venn

374

C

E

R

T

A

I

N

E

R

E

S

E

L

F

F

A

R

S

H

E

A

D

L

I

K

E

L

Y

E

V

E

L

I

B

P

A M

E

E

C

I

D

V

A

I

D

D

I

C

E

N

N

E

C

G

S

E

E

R

T

C

A

E

F

O

R

S

R

D

P

F

T

H

V

I

U

A

O

D

D

S

T

A

C

E

N

N M

P

R

O

B

A

B

I

L

I

T

Y

M

I

T

A

I

L

S

T

T

E

P

I

E

A

N

N

E

V

R

E

R

O

Maths Dimensions 7

Ch13 2 For each spinner, calculate the probability of landing on red or blue. Match each letter to the correct probability for each colour below to solve the riddle:

Why did the coach give lighters to the football team? A

H

I

S

U

O

L

B

M

C

Y

R

T

E

____ ____ ____ ____ ____ ____ ____ 1 --2

1 --8

3 --8

1 --4

3 --4

____ ____ ____ ____ 2 1 5 ------0 3

6

1 --6

____ ____ ____ ____ 5 1 7 ------1

1 --8

8

____ ____ ____ 1 --4

8

2 --3

8

8

____ ____ ____ ____ ____ 5 1 1 5 --------1

2 --3

8

8

3

7

____ ____ ____ ____ ____ ____ ____ 2 1 5 3 1 1 ------------1 7

4

8

8

8

6

3 Calculate the probability of randomly selecting the cards from a deck of 52 playing cards. Match the letters to the correct probability to solve the riddle:

Why did the farmer plough his fields with a steamroller?

A E H N P S W

Heart

♥

D Jack of hearts ♥ G Red card M Red or black card O Even-numbered diamond R 13 of clubs ♣ T Odd-numbered red card

Picture card JQK Ace Black picture card Not an ace King or Queen of spades Not a heart

♥

____ ____ 1 -----13

♠

(not including aces) ____ ____ ____ ____ ____ ____

3 -----13

3 --4

1 --4

____ ____ ____ ____ 1 5 3 ---------0 2

52

♦

3 -----26

2 -----13

3 -----13

____ ____

1 -----52

2 -----13

5 -----52

____ ____ ____ ____ ____ ____ 1 1 1 3 1 ----------------------1

4

4

26

13

13

52

____ ____ ____ ____ ____ ____ ____ ____ 12 -----13

5 -----52

2 -----13

1 --4

2 -----13

5 -----52

3 -----13

1 -----26

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Applications Surveys Sometimes when we want to find the probability of an event or characteristic occurring it is impossible to do an experiment. We may choose to do a survey instead. Have you ever been questioned in a supermarket or shopping centre? For example, a big company may survey the public to estimate the likelihood that their product would sell. The results are more likely to be accurate if a large group is surveyed. Estimate the probability of these chance events in your class: ■ a student is left handed ■ a student is a twin ■ a student is born in February ■ a student is male ■ a student has blue eyes ■ a student travels to school by bus ■ a student is from a family of three or more children.

a Choose some of the events above and survey the class. b Based on the data collected estimate the probability of each of the chance events. c Would these results be different if you surveyed a different class or a different school? Discuss.

Thumbtacks You will need a thumbtack. Work with a partner so that one person can throw while the other records the results. Throw the thumbtack to see which way it lands. If the point is touching the desk, record DOWN. If the point is straight up, record UP. Result

Tally

Total

UP DOWN

a What is the chance of your thumbtack landing safely DOWN? b Compare your results with those of other groups in your class. Are the results similar?

A gamble Australians love to gamble people will risk money: ■ horse racing ■ ■ lotto ■ ■ scratchies ■

on chance events. Listed below are some of the things on which pokies card games at the casino footy tipping

■ ■

roulette wheel at the casino dog racing

a Which of these are you most likely to win or lose money on? b Try to rank these activities in order from most likely to least likely to win money.

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Maths Dimensions 7

Ch13 Letters You will need a book or magazine for this exercise. ■ Randomly choose a page of your a book or magazine. ■ Count the number of times each letter occurs in the first 200 words of the page. ■ Record your results in a table like this. Letter

Tally

Number

Probability

Percentage = probability × 100

A B C · · · · Z Total

1

a Which letters were more likely to occur? b Did any letters not occur? Does this mean that there is a zero probability of these letters occurring? Explain your answer. c Compare your results with those of a friend. Explain any possible differences. d In the English language the probability of each of the letters of the alphabet occurring is as listed in this table: Letters

Probability

E

13%

T

9%

I, N, R

8%

A, O

7%

S

6%

D, L, G

4%

C, F, H, M, P, U

3%

B, V, W, Y, K

1%

J, Q, X, Z

Less than 1%

e Compare these with your results.

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Enrichment 1 Four students, Amy, Brenton, Caroline and David, sit in a row on a bench. a If we use letters to represent their names, write down all the different possible arrangements. ABCD, ACDB, DCAB … You many like to write A, B, C and D and rearrange them until you have all the possible arrangements. Assume that they sit down randomly. b How many of your arrangements have David sitting next to Caroline? What is the probability that Caroline sits next to David? c Brenton does not like David. How many of your arrangements have David sitting next to Brenton? What is the probability that they do not sit next to each other? d What is the probability that Amy sits between Brenton and David? e Repeat the exercise, this time add a fifth student, Elizabeth. f What if the students sit in a circle? 2 The adjacent cardboard nets are folded and glued to form a lettered die. List the probability of each letter coming up when each die is rolled: a a net of a cube 2 faces A, 2 faces B, 2 faces C b a net of a cube 1 face A, 4 faces B, 1 face C c a net of a tetrahedron 2 faces A, 1 face B, 1 face C

B A

A

C

C

B

3 Copy the net and use it to design a die that has: a an equal chance of rolling black, blue, red, yellow, white and green b an equal chance of rolling yellow, green and white c an even chance of throwing a 6 and more chance of throwing a 2 than a 1 d more chance of an A than a B and more chance of a C than an A 4 Design a spinner that will produce the results shown in each of the tables below: a

378

b

Colour

Chance

Red

1 --2

A

Black

1 --4

Gold

1 --4

Maths Dimensions 7

Symbol

Probability

c

Symbol

Probability

1 --8

A

1 --5

B

1 --2

B

2 --5

C

1 --4

C

2 --5

D

1 --8

Ch13 5 At a wedding reception the following courses were available and served randomly to the guests. Starter: Pumpkin soup or French onion soup Main: Steak, peppercorn chicken or vegetable lasagne Dessert: Cheesecake or chocolate mousse These choices could be displayed in a tree diagram: Starter

Main Steak

Pumpkin soup

French onion soup

a b c d

Dessert Cheesecake Chocolate mousse

Chicken

Cheesecake Chocolate mousse

Lasagne

Cheesecake Chocolate mousse

Steak

Cheesecake Chocolate mousse

Chicken

Cheesecake Chocolate mousse

Lasagne

Cheesecake Chocolate mousse

List the different meals available. How many possible arrangements are available? How many arrangements have mousse as a dessert? If one person is chosen at random, what is the probability that the person has: i chocolate mousse as a dessert? ii steak as a main? iii pumpkin soup as a starter? iv cheesecake as a dessert and fish as a main? v pumpkin soup or lasagne?

6 Repeat Question 5a–d, but now the options are: Starter: Pumpkin soup or French onion soup Main: Steak, fish or vegetable lasagne Dessert: Cheesecake, chocolate mousse or strawberry torte Use a tree like this one:

Starter

Main

Dessert

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Revision Exercise 13A 1 Use an appropriate word to describe the following chance events: a You will travel to Europe. b You will play tennis on the weekend. c People will have holidays on the Moon in your lifetime. d You will get homework tonight. e It will snow on Uluru in February. 2 Place the events described in Question 1 on the number line below. 0 Impossible

50%

100% Certain

Exercise 13B 3 A bag contains 30 marbles. Some of the marbles are red, the rest are black. Kyle wanted to estimate the probability of getting a red marble when one is taken from the bag without looking. He decided to do an experiment in which he took a marble from the bag, recorded its colour and replaced it. He repeated this 50 times. These are his results: Colour

Number

Red

32

Black

18 50

Probability as a fraction

Probability as a percentage

100%

a Copy the table into your workbook and fill in the missing information. You may use a calculator to find the probability as a percentage. b What is the probability of Kyle getting a red marble? c What is the probability of Kyle getting a black marble? d Estimate the number of red marbles in the bag. 4 The bag in Question 3 actually contains 20 red and 10 black marbles. What is the probability that a randomly selected marble: a is black? b is green? c is red? d is not black? 5 What is the probability that an odd number shows when a normal six-sided die is rolled? 6 What is the probability that a girl is randomly selected from a group of 8 girls and 12 boys? 7 A card is randomly chosen from a set of cards numbered from 1 to 12. a What is the probability that the number on the card is a prime number? b What is the probability that the number on the card is not a prime number? 8 The probability of having three boys in a family of three children is 1--- . What is the 8 probability that a three-child family does not have three boys?

380

Maths Dimensions 7

Ch13 9 In a class of 25 students there are 8 with blue eyes, 3 with hazel eyes and 14 with brown eyes. Use these results to estimate the probability that: a a randomly selected person has blue eyes b a randomly selected person does not have brown eyes

Exercise 13G 10 These are the results of an experiment to test a spinner: Number of spins Red

50

White

30

Blue

20

Total

100

11 What is the probability that the spinners shown will land with a red edge touching the surface? a

b

a Explain what the spinner could look like. b Draw a possible sketch of the spinner.

Exercise 13I 12 The table gives the number of fatal road accidents and the number of registered motor vehicles for each state or territory. State or territory

Road accidents

Registered motor vehicles

NSW

563

3 332 500

Vic

371

2 869 900

Qld

408

2 012 900

SA

163

962 800

WA

194

1 175 500

Tas

53

319 900

NT

56

90 400

ACT

14

183 800

Total

1822

10 947 500

a Based on these statistics, for each of the states and territories calculate the probability that a vehicle is in a fatal accident. b In which state or territory is a registered vehicle most likely to be involved in a fatal accident?

Chapter 13 Probability

381

CHAPTER

14 Statistics

Statistics Data about the Australian community is collected in a census every 5 years. Personal information such as race, age, gender and salary is collected from every household on one night of the year. The information is collected by the Australian Bureau of Statistics and used by governments and businesses to plan for the future.

This chapter covers the following skills: • Recognising the different types of data

• Finding summary statistics

Discrete numerical data is numerical data that involves distinct values

The mode is the most frequently occurring value

Continuous numerical data is numerical data where every number on a scale has meaning

The mean = sum of the numerical values --------------------------------------------------------------------------number of values

• Collecting and organising different types of data • Designing a simple database to collect data • Representing numerical data in histograms, dot plots and stemplots • Finding the angle in a pie graph frequency of the category -------------------------------------------------------------------- × 360 = 60° total frequency

The median is the middle value when the results are written in order. If there is an even number of results, it is the average of the middle pair The range is a measure of the spread of the data. It is the difference between the highest and the lowest result • Using means to compare two data sets • Using Venn diagrams and two-way tables to display data • Interpreting line graphs

14A

Collecting data

The study of statistics begins with the collecting of information or data. The Australian Bureau of Statistics (ABS) regularly collects data about Australians. Some of this data can also be obtained from other sources such as the Australian Tax Office. Every 5 years the ABS surveys every household to collect information about how families live. This is known as a census. Generally, however, information is collected by surveying a smaller group of people (a sample). For example, when a company wants to know if its products are well known, it may choose to do a survey, or a political party may survey a sample of people to see how popular it is. Three common ways of surveying are: ■

questionnaires in magazines

■

phone polls

■

street surveys.

There are different types of data. Type of data

Example

Discrete numerical data: This is data that involves distinct values. These are often whole numbers, but not always. If collecting data involves counting then it will be discrete numerical data. When there is a large range of numbers, this data may need to be grouped into classes so that it can be analysed.

■

The number of children in a family: 1, 2, 3, 4, 5

Continuous numerical data: This is numerical data where every number on a scale has meaning. If collecting data involves measuring then it is probably continuous numerical data. This type of data always needs to be grouped into classes so that it can be analysed.

Heights in cm: 150–