HAESE
&
HARRIS PUBLICATIONS
Core Skills Mathematics
7 Helen Hall Sue Norris Cheryl Ross Mandy Spiers Wendy Stimson Chris Haines Stan Pulgies
CORE SKILLS MATHEMATICS 7 Helen Hall Sue Norris Cheryl Ross Mandy Spiers Wendy Stimson Chris Haines Stan Pulgies
B.Ed., Dip.T. B.Ed., Dip.T. B.Ed., Dip.T. Dip.T. B.Ed. B.Ed., Grad.Cert.Ed., Dip.T. M.Ed., B.Ed., Grad.Dip.T.
Haese & Harris Publications 3 Frank Collopy Court, Adelaide Airport SA 5950 Telephone: (08) 8355 9444, Fax: (08) 8355 9471 email:
[email protected] web: www.haeseandharris.com.au National Library of Australia Card Number & ISBN 1 876543 68 X © Haese & Harris Publications 2004 Published by Raksar Nominees Pty Ltd, 3 Frank Collopy Court, Adelaide Airport SA 5950 First Edition
2004
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This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese & Harris Publications. Copying for educational purposes: Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the Copyright Agency Limited. Acknowledgements: the Correlation Chart at the end of the book relates to the R-7 SACSA Mathematics Teaching Resource published by the Department of Education and Children’s Services. The Publishers also wish to acknowledge The Royal Agricultural & Historical Society of S.A. Inc. for permission to include the map of the Royal Adelaide Show.
FOREWORD
We have written this book to provide a sound course in mathematics that Year 7 students will find easy to read and understand. Our particular aim was to cover the core skills in a clear and readable way, so that every Year 7 student can be given a sound foundation in mathematics that will stand them in good stead as they begin their secondary-level education. Units are presented in easy-to-follow, double-page spreads. Attention has been paid to sentence length and page layout to ensure the book is easy to read. The content and order of the thirteen chapters parallels the content and order of the thirteen chapters in Mathematics for Year 7 (second edition) also published by Haese & Harris Publications and that book could be used by teachers seeking extension work for students at this level. Throughout this book, as appropriate, the main idea and an example are presented at the top of the left hand page; graded exercises and activities follow, and more challenging questions appear towards the foot of the right-hand page. With the support of the interactive Student CD, there is plenty of explanation, revision and practice. We hope that this book will help to give students a sound foundation in mathematics, but we also caution that no single book should be the sole resource for any classroom teacher. We welcome your feedback. Email:
[email protected] Web: www.haeseandharris.com.au HH SN CGR MS WS CAH SP
Active icons – for use with interactive student CD By clicking on the CD-link icon you can access a range of interactive features, including: ! spreadsheets ! video clips ! graphing and geometry software ! computer demonstrations and simulations.
CD LINK
TABLE OF CONTENTS
TABLE OF CONTENTS
Chapter 1 WHOLE NUMBERS
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7
Our number system Operations with whole numbers Problem solving with whole numbers Rounding and approximation One million and beyond Number opposites Review of chapter 1
8 10 12 14 16 18 20
Chapter 2 NUMBER PROPERTIES
Unit 8 Unit 9 Unit 10 Unit 11 Unit 12 Unit 13 Unit 14
Number operations and their order Factors of natural numbers Multiples and divisibility rules Powers of numbers Square and cube numbers Problem solving methods Review of chapter 2
22 24 26 28 30 32 34
Chapter 3 SHAPES AND SOLIDS
Unit 15 Unit 16 Unit 17 Unit 18 Unit 19 Unit 20 Unit 21 Unit 22 Unit 23 Unit 24
Points and lines Angles Angles of a triangle and quadrilateral Polygons Classifying triangles and quadrilaterals Constructing a triangle and bisecting angles 90° and 60° angles and Circles Polyhedra and nets of solids Drawing solids Review of chapter 3 Review of chapters 1, 2 and 3
36 38 40 42 44 46 48 50 52 54 56
Chapter 4 FRACTIONS
Unit 25 Unit 26 Unit 27 Unit 28 Unit 29 Unit 30 Unit 31 Unit 32
Representation of fractions Equivalent fractions and lowest terms Fractions of quantities Fraction sizes and types Adding and subtracting fractions Multiplying fractions Problem solving with fractions Review of chapter 4
58 60 62 64 66 68 70 72
Chapter 5 DECIMALS
Unit 33 Unit 34 Unit 35 Unit 36 Unit 37 Unit 38 Unit 39
Representing decimals Place value Rounding decimal numbers Ordering decimals Adding and subtracting decimals Multiplying and dividing by powers of 10 Multiplying decimal numbers
74 76 78 80 82 84 86
TEST YOURSELF
5
6
TABLE OF CONTENTS
Unit 40 Unit 41 Unit 42
Dividing decimals by whole numbers Fractions and decimal conversions Review of chapter 5
Unit 43 Unit 44 Unit 45 Unit 46 Unit 47 Unit 48 Unit 49 Unit 50 Unit 51
Percentages and fractions Percentage, decimal and fraction conversions Percentages on display and being used Representing percentages Quantities and percentages Money and problem solving Discount and GST Simple interest and other money problems Review of chapter 6 Review of chapters 4, 5 and 6
94 96 98 100 102 104 106 108 110 112
Chapter 7 MEASUREMENT (LENGTH AND MASS)
Unit 52 Unit 53 Unit 54 Unit 55 Unit 56 Unit 57 Unit 58
Reading scales Units and length conversions Perimeter Scale diagrams Mass Problem solving Review of chapter 7
114 116 118 120 122 124 126
Chapter 8 MEASUREMENT (AREA AND VOLUME)
Unit 59 Unit 60 Unit 61 Unit 62 Unit 63 Unit 64 Unit 65
Area (square units) Area of a rectangle Area of a triangle Units of volume and capacity Volume formulae Problem solving Review of chapter 8
128 130 132 134 136 138 140
Chapter 9 DATA COLLECTION AND REPRESENTATION
Unit 66 Unit 67 Unit 68 Unit 69 Unit 70 Unit 71
Samples and population Collecting and interpreting data Interpreting graphs Mean and median Line graphs Review of chapter 9
142 144 146 148 150 152
Chapter 10 TIME AND TEMPERATURE
Unit 72 Unit 73 Unit 74 Unit 75 Unit 76 Unit 77 Unit 78
Units of time Differences in time Reading clocks and timelines Timetables Time zones Average speed and temperature Review of chapter 10
154 156 158 160 162 164 166
Review of chapters 7, 8, 9 and 10
168
Chapter 6 PERCENTAGES
TEST YOURSELF
TEST YOURSELF
88 90 92
TABLE OF CONTENTS
Chapter 11 ALGEBRA
Unit 79 Unit 80 Unit 81 Unit 82 Unit 83
Geometric and number patterns Formulae and variables Practical problems and linear graphs Solving equations Review of chapter 11
170 172 174 176 178
Chapter 12 TRANSFORMATION AND LOCATION
Unit 84 Unit 85 Unit 86 Unit 87 Unit 88 Unit 89 Unit 90 Unit 91 Unit 92
Number planes Transformations and reflections Rotations and rotational symmetry Translations and tessellations Enlargements and reductions Using ratios Bearings and directions Distance and bearings Review of chapter 12
180 182 184 186 188 190 192 194 196
Chapter 13 CHANCE
Unit 93 Unit 94 Unit 95 Unit 96 Unit 97
Describing chance Defining probability Tree diagrams and probability Expectation Review of chapter 13
198 200 202 204 206
Review of chapters 11, 12 and 13
208
TEST YOURSELF ANSWERS
210
Correlation chart: R-7 SACSA Mathematics Teaching Resource
239
INDEX
243
7
CHAPTER 1
8
WHOLE NUMBERS
Unit 1
Our number system
Numbers less than one million The chart shows the place value of each digit in a number. Thousands Hundreds Tens Units 1 2 0
Units Hundreds Tens 9 9
This digit represents 900.
The number shown is one hundred and twenty thousand, nine hundred and ninety three.
Units 3 This digit represents 90.
Exercise 1 1 What number is represented by the digit 8 in the following? a 38 b 81 c 458 e 1981 f 8247 g 2861 i 60 008 j 84 019 k 78 794
d h l
847 28 902 189 964
2 What is the place value of the digit 7 in the following? a 497 b 37 482 c 856 784
d
755 846
3 Write down the place value of the 3, the 5 and the 8 in each of the following: a 53 486 b 3580 c 50 083 d
805 340
4
a Use the digits 6, 4 and 8 once only to make the largest number you can. b Write the largest number you can using the digits 4, 1, 0, 7, 2 and 9 once only. c What is the largest 6 digit numeral you can write using each of the digits 2, 7 and 9 twice? d How many different numbers can you write using the digits 3, 4 and 5 once only?
5 Put the following numbers in ascending order (smallest first): a c e
57, 8, 75, 16, 54, 19 1080, 1808, 1800, 1008, 1880 236 705, 227 635, 207 653, 265 703
b d f
660, 60, 600, 6, 606 45 061, 46 510, 40 561, 46 051, 46 501 554 922, 594 522, 545 922, 595 242
6 Write the following numbers in descending order (largest first): a c
361, 136, 163, 613, 316, 631 498 231, 428 931, 492 813, 428 391, 498 321
b d
7789, 7987, 9787, 8779, 8977, 7897, 9877 563 074, 576 304, 675 034, 607 543, 673 540
7 Write the numeral for: a 8 £ 10 + 6 £ 1 b 6 £ 100 + 7 £ 10 + 4 £ 1 c 9 £ 1000 + 6 £ 100 + 3 £ 10 + 8 £ 1 d 5 £ 10 000 + 2 £ 100 + 4 £ 10 e 2 £ 10 000 + 7 £ 1000 + 3 £ 1 f 2 £ 100 + 7 £ 10 000 + 3 £ 1000 + 9 £ 10 + 8 £ 1 g 3 £ 100 + 5 £ 100 000 + 7 £ 10 + 5 £ 1 h 8 £ 100 000 + 9 £ 1000 + 3 £ 100 + 2 £ 1
The numbers in question 7 are in expanded form.
DEMO
WHOLE NUMBERS (CHAPTER 1)
8 Write in expanded form: a 975 e 56 742
b f
c g
680 75 007
d h
3874 600 829
9083 354 718
9 Write the following in numeral form: a twenty seven b eighty c six hundred and eight d one thousand and sixteen e eight thousand two hundred f nineteen thousand five hundred and thirty eight g seventy five thousand four hundred and three h six hundred and two thousand eight hundred and eighteen. 10 What number is: a one less than eight b two greater than eleven c four more than seventeen d one less than three hundred e seven greater than four thousand f 3 less than 10 000 g four more than four hundred thousand h 26 greater than two hundred and nine thousand?
= + or ¼ >
or < : a 5268 ¡ 3179 ¢ 4169 b c 672 + 762 ¢ 1444 d e 20 £ 80 ¢ 160 f g 5649 + 7205 ¢ 12 844 h
29 £ 30 ¢ 900 720 ¥ 80 ¢ 8 700 £ 80 ¢ 54 000 6060 ¡ 606 ¢ 5444
reads reads reads reads
‘is ‘is ‘is ‘is
equal to’ approximately equal to’ greater than’ less than’
79 £ 8 ¤ 640 7980 ¥ 20 ¤ 400 50 £ 400 ¤ 20 000 3000 ¥ 300 ¤ 10
Number systems The number system we use is called the Hindu-Arabic system. It uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We call them digits. We use them to make up our natural (whole) numbers. Natural numbers are the numbers that we count with (1, 2, 3, 4, 5, 6, .....). Click on the icon to read about other number systems that have been used around the world.
CIV Egyptian
Roman
Mayan PRINTABLE MATERIAL
Chinese Japanese
These all represent the number 104.
9
10
WHOLE NUMBERS (CHAPTER 1)
Unit 2
Operations with whole numbers
Addition and subtraction When we subtract a smaller number from a larger one we find their difference.
When we add numbers we find their sum. For example:
For example:
32 427 + 3274
3 11 9 10
4200 ¡ 326 3874
1 1
3733
sum
difference
Exercise 2
DEMO
1 Do these additions: a
392 + 415
b
601 + 729
c
1917 + 2078
d
913 24 + 707
e
217 106 + 1274
f
9004 216 23 + 3816
b e h
72 + 35 921 + 1234 32 + 627 + 4296
c f i
421 + 327 6214 + 324 + 27 912 + 6 + 427 + 3274
2 Find: a 42 + 37 d 624 + 72 g 90 + 724 3 Do these subtractions: a 97 ¡ 15 d
DEMO
602 ¡ 149
4 Find: a 47 ¡ 13 d 40 ¡ 18 g 503 ¡ 127
b
63 ¡ 19
c
247 ¡ 138
e
713 ¡ 48
f
6005 ¡ 2349
c f i
33 ¡ 27 623 ¡ 147 5939 ¡ 3959
b e h
62 ¡ 14 214 ¡ 32 5003 ¡ 1236
5 The cards have place values as shown. i Find the sum. ii Find the difference.
3
2
ª
5
3
ª
7
ªª ªªª ªª
9 5
2
§ § § § §§§ § §
8
8
§ § § §
§ § § §
4
4 9
§ § § § § § §
9
U
T
§ §
4
7
4 7
H
§ §
4
A
7
A
9 9
A
4
Th
9
7
b
U
T
7
H
8
Th
8
a
An ace has the value 1.
A
WHOLE NUMBERS (CHAPTER 1)
Multiplication and division We multiply numbers to find their product.
23 £ 100 = 2300
53 £ 16
12¡700 ¥ 100 = 127
When we divide one number by another, the result is the quotient. 4 1 7
1
6
318 530
25
product
848 6 Find: (You could do these mentally.) a 50 £ 10 b d 69 £ 100 e g 123 £ 100 h j 49 £ 10 000 k
50 £ 100 69 £ 1000 246 £ 1000 490 £ 100
c f i l
50 £ 1000 69 £ 10 000 960 £ 100 4900 £ 100
7 Find: (You could do these mentally.) a 24¡000 ¥ 10 b e d 45¡000 ¥ 10 h g 72¡000 ¥ 10 j 6000 ¥ 10 k
24¡000 ¥ 100 45¡000 ¥ 100 72¡000 ¥ 100 6000 ¥ 100
c f i l
24¡000 ¥ 1000 45¡000 ¥ 1000 72¡000 ¥ 1000 6000 ¥ 1000
d h l
53 £ 24 642 £ 36 368 £ 73
8 Find: a 24 £ 5 e 27 £ 15 i 274 £ 21
b f j
9 Do these divisions: a 3 42 d
5
c g k
37 £ 4 56 £ 49 958 £ 47 b e
375
4
216
7
6307
62 £ 8 324 £ 45 117 £ 89
c f
10 Find: a 24 ¥ 4 d 240 ¥ 5
b e
125 ¥ 5 624 ¥ 3
c f
11 Find: a d g j m
b e h k n
63 ¥ 4 143 ¥ 2 661 ¥ 8 1201 ¥ 5 8463 ¥ 4
c f i l o
45 ¥ 2 97 ¥ 8 439 ¥ 5 1033 ¥ 4 7349 ¥ 2
quotient
10 42
312 ¥ 6 7353 ¥ 9 81 ¥ 5 275 ¥ 4 955 ¥ 2 4699 ¥ 8 7999 ¥ 5
8 11
168 6809
Sometimes we need to add zeros to complete the division. For example 8 7: 7 5 4
3 5 31 : 3 0 2 0
So, 351 ¥ 4 = 87:75
12 Solve the following problems: a Find the product of 29 and 12. b Find the quotient of 368 and 23. c Find the difference between the product of 7 and 6 and the quotient of 500 and 50. 13 This sum is not correct. By changing only one of the digits, make it correct:
386 + 125 521
DEMO
DEMO
11
12
WHOLE NUMBERS (CHAPTER 1)
Unit 3
Problem solving with whole numbers
Two examples of problem solving are:
² John bought 5 kg of potatoes, 3 kg of carrots, 7 kg of onions and 25 kg of pumpkin. We can find the total weight of John’s vegetables by adding the weight of each vegetable. Total weight = 5 + 3 + 7 + 25 = 40 kg
Exercise 3
² Jason buys 217 baskets of fresh cherries for a supermarket chain at $38 a basket. What will be the total cost? Total cost = 217 £ $38 = $8246
We need a number sentence to answer the question.
Addition and subtraction 1 Jack bought 4 separate lengths of timber. Their lengths were as follows: 5 m, 1 m, 7 m, and 9 m. If all four lengths of timber were put end to end, what would the total length be? 2 Jenny bought a Play Station for $255. She also purchased another controller for $50, a Play Station game for $95 and a bag to store these in for $32. How much did she pay altogether? 3 Kerry needed to lose some weight to be chosen in a light weight rowing team. He weighed 60 kg but needed to weigh 54 kg. How much weight did he need to lose? 4 Stephen made $72 worth of phone calls in one month. His parents said they would only pay $31 of this. How much did Stephen have to pay? 5 Rima went on an overseas trip that required three plane flights. The first flight was 2142 km long, the next one was 732 km long and the third one was 1049 km long. How long were her flights in total? 6 Bill measured out a straight line that was 6010 cm long on the school grounds. He actually went too far. The line should have been 4832 cm long. How much of the line will he need to rub out?
Multiplication and division 7 Carlo lifted five 18 kg bags of potatoes onto a truck. How many kg of potatoes did he lift altogether? 8 My three brothers and I received a gift of $320. If we shared the money equally amongst ourselves, how much did each person receive? 9 A relay team of nine people took 738 minutes to complete a relay race. If each team member took exactly the same time how long did each team member take? 10 A maths textbook is 245 mm long. If I put 10 books end to end how far would they stretch? 11 24 people each travelled 28 km to play sport. How far in total did they travel? 12 If I write 8 words per minute, how long would it take me to write 648 words?
WHOLE NUMBERS (CHAPTER 1)
13
Two step problems This is an example of a problem solved using two steps. How much change from $50 would you receive after buying three bags of potatoes at $14 a bag? Step 1: Total cost of potatoes = $14 £ 3 = $42 Step 2: Change is $50 ¡ $42 = $8 13 Sara bought a shirt costing $29 and a pair of jeans costing $45. How much change did she get from $100? 14 Glen bought three T-shirts costing $42 each and a pair of shoes costing $75. Find the total cost of his purchases. 15 Miki had 65 minutes of time left on her prepaid mobile phone. She made a 10 minute call to Rupesh, a 7¡¡minute call to her mother and a 26 minute call to her boyfriend Michael. How many minutes did she have left after making these calls? 16 Maria bought five 3 kilogram bags of oranges. The numbers of oranges in the bags were: 10, 11, 12, 12 and 10. Find the average number of oranges in a bag. 17 Lachlan had a herd of 183 goats. He put 75 in his largest paddock and divided the rest equally between two smaller paddocks. How many goats were put in each of the smaller paddocks? 18 George had $436 in his bank and was given $30 cash for his birthday. How much money did he have left if he bought a bicycle costing $455? 19 The cost of placing an advertisement in the local paper is $10 plus $4 for each line of type. If my advertisement takes 5 lines, how much will I pay? 20 How much would June pay for 8 iced buns if 3 buns cost her 54 cents? 21 A football team had kicked 12 goals 13 points. They had another kick for goal as the siren sounded. Their final score was 91 points. Did the last kick score a goal or a point? (1 goal = 6 points) 22 Marcia saved $620 during the year and her sister saved twice that amount. How much money did they save in total? 23 Anna had $463 in her savings account and decided to bank $20 a week for 14 weeks. How much was in the account at the end of that time? 24 Tony’s wages for the week were $496. He was also paid for 3 hours overtime at $18 per hour. How much did he earn in total? 25 Alicia ran 6 km each day from Monday to Saturday and 12 km on Sunday. How far did she run during the week? 26 A plastic crate contains 100 boxes of ball point pens. The boxes of pens each weigh 86 grams. If the total mass of the crate and pens is 9200 g, find the mass of the crate.
14
WHOLE NUMBERS (CHAPTER 1)
Unit 4
Rounding and approximation
We round off if we do not need to know the exact number.
Rules for rounding off: ²
If the digit after the one being rounded off is less than 5 (i.e., 0, 1, 2, 3 or 4) we round down.
²
If the digit after the one being rounded off is 5 or more (i.e., 5, 6, 7, 8, 9) we round up.
For example, to round off:
63
to the nearest 10,
63 + 60
fWe round down, as 3 is less than 5:g
275
to the nearest 100,
275 + 300
fWe round up, as 7 is greater than 5:g
8467
to the nearest 1000,
8467 + 8000
fWe round down, as 4 is less than 5:g
+ means “is approximately equal to”
Exercise 4 1 Round off to the nearest 10: a 75 b 78 e 3994 f 1651 i 49 566 j 30 942
c g k
298 9797 999 571
d h l
2379 61 015 128 674
2 Round off to the nearest 100: a 78 b 468 e 25 449 f 14 765
c g
998 130 009
d h
2954 43 951
3 Round off to the nearest 1000: a 748 b 5500 e 65 438 f 123 456
c g
9990 434 576
d h
43 743 570 846
4 Round off to the accuracy given: a $45 387 (to the nearest $1000) b 328 kg (to the nearest ten kg) c a weekly wage of $485 (to the nearest $100) d a distance of 4753 km (to the nearest 100 km) e the annual amount of water used in a household was 362 498 litres (to the nearest 1000 litres) f the profit of a company was $487 374 (to the nearest $10 000) g the population of a town is 37 495 (to the nearest one thousand) h the population of a city is 637 952 (to the nearest hundred thousand) i the number of times the average heart will beat in one year is 35 765 280 times (to nearest million) j a year’s loss by a large mining company was $1 517 493 826 (to nearest billion).
DEMO
DEMO
DEMO
WHOLE NUMBERS (CHAPTER 1)
One figure approximations Rules:
² ²
Leave single digits as they are. For all other numbers, round the left most digit and put zeros in the other places.
For example: 57 £ 8 + 60 £ 8 + 480
15
Estimating with money For example: Estimate the cost of 19 pens at $1:95 each. 19 £ $1:95 When we estimate + 20 £ $2 with money with cents we round to the + $40 nearest whole dollar.
294 ¥ 48 + 300 ¥ 50 +6
We estimate using approximations to get a good idea of what the answer should be. 5 Estimate the cost of: a 195 exercise books at 98 cents each c 18 show bags at $3:45 each e 4 dozen iceblocks at $1:20 each
b d f
27 sweets packets at $2:15 a packet 12 bottles of drink at $2:95 a bottle 3850 football tickets at $6:50 each.
6 Estimate the following products using 1 figure approximations: a 55 £ 3 b 62 £ 7 c 88 £ 6 e 389 £ 7 f 4971 £ 3 g 57 £ 42 i 85 £ 98 j 3079 £ 29 k 40 989 £ 9 7 Estimate the following quotients using 1 figure approximations: a 397 ¥ 4 b 6849 ¥ 7 d 6000 ¥ 19 e 80 000 ¥ 37 g 549 ¥ 49 h 3038 ¥ 28
c f i
d h l
275 £ 5 73 £ 59 880 £ 750
79 095 ¥ 8 18 700 ¥ 97 5899 ¥ 30
8 Multiply the following. Use estimation to check that your answers are reasonable. a 79 b 445 c 3759 £7 £8 £9 9 Divide the following. Use estimation to check that your answers are reasonable. a b c 6 366 8 1080 4 392
In the following questions, round the given data to one figure to find the approximate value asked for: 10 In her bookcase Lynda has 12 shelves. Estimate the number of books in the bookcase if there are approximately 40 books on each shelf. 11 Miki reads 217 words in a minute. Estimate the number of words she can read in one hour. 12 A bricklayer lays 115 bricks each hour. If he works a 37 hour week, approximately how many bricks will he lay in one month? 13 Joe can type at 52 words per minute. Find an approximate time for him to type a document of 3820 words. 14 In a vineyard there are 189 vines in each row. There are 54 rows. Find the approximate number of vines in the vineyard.
PRINTABLE MATERIAL
16
WHOLE NUMBERS (CHAPTER 1)
Unit 5
One million and beyond
One million is 1 000 000. 1 000 000 = 100 £ 100 £ 100
One million $1 coins placed side by side would stretch 25 km. 1 000 000 £ 25 mm = 25 000 000 mm = (25 000 000 ¥ 1 000 000) km = 25 km
1m
A cube made of one million MA unit blocks would measure 1 m long £ 1 m wide £ 1 m high.
Millions hundreds tens 5
25 mm
units 3
Thousands hundreds tens units 4 7 9
Units hundreds tens 6 8
units 2
The number displayed in the place value chart is 53 million, 479 thousand, 682.
Exercise 5 1 In the number shown on the chart above, the digit 9 has the value 9000 and the digit 3 has the value 3 000 000. Give the value of the: EXTRA a 8 b 5 c 6 d 4 e 7 f 2 ACTIVITIES
2 Write the value of each digit in the following numbers: a 3 648 597 b 34 865 271 3 Read the following stories about large numbers. Write each large number using figures. a A heart beating at a rate of 70 beats per minute would beat about thirty seven million times in a year. b Australia’s largest hamburger chain bought two hundred million bread buns and used seventeen million kilograms of beef in one year. c The Jurassic era was about one hundred and fifty million years ago. d One hundred and eleven million, two hundred and forty thousand, four hundred and sixty three dollars and ten cents was won by two people in a Powerball Lottery in Wisconsin USA in 1993. e A total of twenty one million, two hundred and forty thousand, six hundred and fifty seven Volkswagen ‘Beetles’ had been built to the end of 1995. 4 In the following questions, how many times does the given container need to be filled to hold 1 000 000 units? a fuel tank holding 50 litres b packet containing 250 sugar cubes c school hall seating 400 students d rainwater tank holding 2000 litres e case packed with 100 oranges f carriage for 80 passengers g restaurant feeding 125 diners h computer disk cartridges with 40 disks i crates holding 160 cans j stackers storing 8 CDs
WHOLE NUMBERS (CHAPTER 1)
17
5 Arrange these planets in order of their distance from the Sun, starting with the closest. Venus 108 200 000 km Saturn 1 427 000 000 km Earth 149 600 000 km Uranus 2 870 000 000 km Mercury 57 900 000 km Jupiter 778 300 000 km Pluto 5 900 000 000 km Neptune 4 497 000 000 km Mars 227 900 000 km 6 Use the table to answer these. a Which continent has the greatest area? b Name the continents with an area greater than 20 million square kilometres. c Find out which continents are completely in the Southern Hemisphere.
Continent Africa Antarctica Asia Australia Europe North America South America
Area in square km 30 271 000 13 209 000 44 026 000 7 682 000 10 404 000 24 258 000 17 823 000
7 How long would a car, travelling non-stop at 100 kilometres per hour, take to travel a million kilometres? 8 How long would a motor cyclist travelling non-stop at 50 kilometres per hour take to travel one million kilometres? 9 How many hours would a jumbo jet, flying non-stop at 500 kilometres per hour, take to fly 1 million kilometres?
10 A $5 note is 135 mm long. a How far would one million $5 notes laid out end to end in a straight line stretch? b If you walked from one end of the line to the other at a speed of 5 kmph, how long would it take? 11 How long would a satellite orbiting the earth at 8000 kmph take to fly 1 million kms? 12 One million one dollar coins stacked on top of one another would be 2700 metres high. That is about 8 times higher than Auckland’s Sky Tower, 9 times higher than Sydney’s Centrepoint Tower and over 3 times higher than the DIB-200 in Tokyo.
800m 800m
447m
How many coins are needed to build a stack:
400m
a one metre high (to the nearest 10 coins)
300m
b the height of each of the illustrated buildings (to the nearest 1000 coins)?
Comparative sizes of structures 380m 328m 305m
300m
200m
100m
0 DIB-200, Tokyo
NUMBER SEARCH PROBLEMS
Click here for some Number Search Problems.
Sears Tower, Chicago
Empire State Building, New York
Sky Tower, Auckland
Centrepoint Tower, Sydney
Eiffel Tower, Paris
18
WHOLE NUMBERS (CHAPTER 1)
Unit 6
Number opposites
Negative numbers are the opposite of positive numbers.
Some words showing number opposites are:
Zero is our reference point. negative numbers
positive numbers
-4 -3 -2 -1
0
1
2
3
Negative 5o C below zero (¡5o C) a decrease of 2 kg a loss of $1000 10 km south
Positive 5o C above zero an increase of 2 kg a profit of $1000 10 km north
¡3 is the opposite of +3
4
Negative numbers can be shown as ¡3 or ¡3 Positive numbers can be shown as +3 or +3
Opposites are the same distance from 0.
Exercise 6 1 Copy and complete the table:
Statement a b c d e f g h i j
20 m above sea level 45 km south of the city a loss of 2 kg in weight a clock is 2 min fast she arrives 5 min early a profit of $4000 2 floors above ground level 10o C below zero an increase of $400 winning by 34 points
Directed number +20
Opposite to statement 20 m below sea level
2 Write positive or negative numbers for the position of the lift, the car, the parking attendant and the rubbish skip.
(Use the bottom of each object.)
Directed number ¡20
+3 +2 +1 0 -1 -2 -3 -4 -5
ground level
3 If right is positive and left is negative, write the numbers for the positions of A, B, C, D and E using zero as the reference position. B
A -4
E
D
0
C +4
4 Write these temperatures as positive or negative numbers. Zero degrees is the reference point. a 11o above zero b 6o below zero c 8o below zero d 29o above zero e 14o below zero 5 Write these gains or losses as positive or negative numbers: a $30 loss b $200 gain c $431 loss d $751 loss e $809 gain
WHOLE NUMBERS (CHAPTER 1)
19
6 If north is the positive direction, write these directions as positive or negative numbers: a 7 metres north b 15 metres south c 115 metres south d 362 metres north e 19:6 metres south +6 is the same as 6. Usually we simply say “6”. If a number is negative we must use the minus sign.
7 Write the opposite of the following numbers: a +6 b ¡4 c 16 d 0 e ¡2 f ¡40 8 4 + 7 = 11 whereas 4 ¡ 7 = ¡3
DEMO
This can be seen by movement along the number line: -7
+7
0
-5
5
10
4
-3
11
If necessary, use the number line to find: -10
a e i m q
0
-5
b f j n r
5+6 2¡6 ¡2 + 1 4+2 ¡2 ¡ 4
5¡6 2+6 ¡2 + 3 4¡2 7¡3
5
10
c g k o s
2+7 ¡1 + 2 ¡2 ¡ 3 2¡4 3¡7
9 This number line is vertical. As you go up the number line, the numbers increase, and as you go down, the numbers decrease. a Write the directed number for each of the points marked on the number line. b Write i iii v vii 10
True or False for these: B is higher than D ii D is lower than A iv C>E vi B and D are opposites viii °C 40
A 30 D 20 E
10
above freezing
B 0 F C
-10 -20
below freezing
A<E B or 3 647 483 b 5607 ¡ 2489 = 3118 c The difference between 369 and 963 is 396 26 Find the approximate value of 293 £ 39: 27 How many times is the first 5 larger than the second 5 in 28 Find the value of: a 6+9£2¡2 29 In the diagram alongside, what is represented by: a the arrows b PQ Ã ! c RS d T?
b
35 659? c
2 £ (6 ¡ 4) ¥ 2
P
T
R
2£6¡4¥2
S
Q
58
FRACTIONS (CHAPTER 4)
Unit 25
Representation of fractions
The fraction three eighths can be represented in a number of ways.
as a shaded region
Diagram
Number line
CHAPTER 4
Ei_\ means that one whole is divided into 8 equal parts and we are considering 3 of these parts.
three eighths
Words
Symbol
0
Ei_
or
as pieces of a pie
three eighths
1
numerator bar denominator
Exercise 25 1 Copy and complete the following table:
Symbol
Numerator
one half
a
b
3 4
c
2 3
d
Words
Denominator
Meaning
2
One whole divided into two equal parts and one is being considered.
One whole divided into four equal parts and three are being considered.
three quarters
2
two sevenths
g
0 one half
1
0 1 three quarters
0 two thirds
3
1
7
One whole divided into nine equal parts and seven are being considered.
e
f
Number Line
5
8
0
1
FRACTIONS (CHAPTER 4)
2 What fraction of each of these shapes is shaded? a b
d
e
c
f
3 Carefully copy 3 identical sets of each of the following shapes, then answer each question.
a In the first set divide each whole shape into two equal parts. Each part is one half of the whole shape. b In the second set divide each whole shape into three equal parts. Each part is ..... c In the third set divide each whole shape into four equal parts. Each part is ..... d Which shapes were the most difficult to divide equally? 4 Using identical square pieces of paper, make 2 copies of this tangram. Number the pieces on both sheets. Cut one of the sheets into its seven pieces. Use the pieces to help you work out the following: a How many triangles like piece 1 would fit into the largest square?
1 3 4
2
5
b What fraction of the largest square is piece 1? c What fraction of piece 1 is piece 3?
6
d What fraction of the largest square is: i triangle 1 iii square 4
7
ii triangle 3 iv parallelogram 6?
5 Copy the given shape exactly. a If each rectangle is half of the one before it, how much of the shape is unshaded if the whole square is 1? b Check your answer to a by drawing a grid within the large square. Use the boundaries of the shaded square as the dimensions of the smallest squares in your grid.
Qr_
Qw_
c How many of the smallest squares fit into your large square?
Complete the shading of part b to make a chessboard and then answer the following: d What fraction of the whole chessboard is the unshaded area? e What fraction of the total chessboard is the first row of squares? f What fraction of the total chessboard are the unshaded squares in the first row?
59
60
FRACTIONS (CHAPTER 4)
Unit 26
Equivalent fractions and lowest terms Equivalent (equal) fractions represent the same amount.
Multiply to write
1 4
1 4
with denominator 16: 1 4
4 16
and
are equivalent fractions.
=
1£4 4£4
=
4 16
DEMO
9 12
Divide to find equivalent fractions: 9 12
3 4
and
=
9¥3 12¥3
=
3 4
are also equivalent.
Exercise 26 1 Multiply to find equivalent fractions: 5 5 5£2 10 5£2 25 a b = = = = 6 6£2 2 7 7£5 2 2 Divide to find equivalent fractions: 8¥2 2 10 ¥ 5 2 8 10 = = = = a b 10 10 ¥ 2 5 15 15 ¥ 2 3
c
4 4£2 40 = = 5 5£2 2
c
2¥2 2 18 = = 20 20 ¥ 2 10
3 Express with denominator 8: a
1 4
1 2
b
c
3 4
d
1
To obtain equivalent fractions we multiply (or divide) both the numerator and denominator by the same number.
4 Express with denominator 30: a
1 2
b
4 5
c
5 6
d
3 10
b
3 4
c
1
d
0
b
1 4
c
4 5
d
9 10
5 Express in sixteenths: a
1 8
6 Express in hundredths: a
1 2
7 Find 2 if: 2 7 = a 3 21 e
4 16 = 5 2
b
4 2 = 5 15
c
2 9 = 13 39
d
27 2 = 63 7
f
6 3 = 2 4
g
7 35 = 8 2
h
28 14 = 2 25
FRACTIONS (CHAPTER 4)
61
Lowest terms are the simplest form of a fraction. We write a fraction in lowest terms by dividing the numerator and denominator by their highest common factor. Examples: ² = =
12 16 12¥4 16¥4 3 4
²
fas 4 is the HCF of 12 and 16g
= =
80 100 80¥20 100¥20 4 5
fas 20 is the HCF of 80 and 100g
8 Reduce to lowest terms: a e
8 10 24 42
b f
9 36 55 77
c
18 20 39 52
c
45 80 3 51
c
g
21 28 48 84
d
72 96 60 80
d
12 20 24 81
d
h
15 35 6 30
9 Reduce to lowest terms: a e
12 15 49 91
b f
g
h
We divide the numerator and denominator by their highest common factor (HCF).
35 49 15 55
10 Simplify: a e
56 77 250 1000
b f
g
h
15 45 45 180
11 Which of these fractions are in lowest terms? a
15 20
b
1 3
c
13 24
d
64 72
e
6 9
f
21 28
g
22 24
h
5 6
i
75 100
j
14 15
k
9 100
l
39 52
b
175 200
c
32 80
d
875 1000
12 Reduce to lowest terms: 132 144
a
Activity
Representations of equal (equivalent) fractions What to do: 1 Construct 5 identical 4 cm sided squares. Draw lines to create your halves and quarters. Name the equal fractions created in each square. Qw_
2
Wr_
a Write the numbers 1 to 10 on one ice block stick, one centimetre apart. b Write the multiplies of 3, one centimetre apart on another stick.
1
2
3
4
5
6
7
8
9
10
3
6
9
12 15 18 21 24 27
30
c Place one stick directly above the other to show that
1 2 3 4 3 , 6 , 9 , 12 ,
...., ...., ...., ...., ....,
10 30
are equal.
d On a third stick mark off the multiples of 5 from 5 to 50. Place the multiples of 3 stick directly above the multiples of 5 stick and complete these equivalent fractions. 3 6 5 , 10 ,
...., ...., ...., ...., ....,
30 50 .
62
FRACTIONS (CHAPTER 4)
Unit 27
Fractions of quantities
Matthew was given a box of chocolates. 5 had red wrappers, 4 had blue, 4 had gold and 2 had green. There were 15 chocolates in total.
What fraction of 1 metre is 37 cm? 37 cm as a fraction of 1 metre
DEMO
Notice that ² ²
5 had red wrappers so the fraction with red wrappers =
5 15
=
11 did not have gold wrappers so the fraction without gold wrappers =
1 3
=
37 cm 1 metre
=
37 cm 100 cm
=
37 100
11 15
The numerator and denominator must have the same units.
Exercise 27 1 What fraction of each of these different quantities has been circled? a b
c
2 Use a full pack of 52 playing cards for these questions. Calculate which fraction of the full pack are: a
all the red cards, e.g.,
c
all the aces, e.g.,
e
7
3
5
b
all the spades, e.g.,
d
all the picture cards, e.g.,
all the odd numbered cards
f
all the even numbered black cards.
3 What fraction of one hour is: a 30 minutes b 10 minutes
c
45 minutes
d
12 minutes ?
4 What fraction of one day is: a 1 hour b
c
30 minutes
d
1 minute ?
d
1 minute ?
¨
©
A
§
4 hours
5 Use your calculator for this question. What fraction of one week is: a 5 days b 12 hours c 2 hours As the fraction of the week got smaller, what happened to the denominator? 6 In lowest terms, state what fraction of: a 1 metre is 20 cm b 2 metres is 78 cm d 3 kg is 750 g e 1 day is 5 hours g November is two days h a decade is one year
c f i
ª
1 kg is 500 g 1 hour is 23 minutes 2 dollars is 27 cents
7 Jenny scored 27 correct answers in her test of 40 questions. What fraction of her answers were incorrect? 8 James was travelling a journey of 420 km. His car broke down after 280 km. What fraction of his journey did he still have to travel?
J
¨
Q
ª
K
©
1 kg = 1000 g A decade is 10 years.
FRACTIONS (CHAPTER 4)
9 Since
6 ¥ 2 = 3,
1 2
63
of 6 is 3. What number is:
a
1 2
of 10
b
1 2
of 36
c
1 3
of 12
d
1 3
of 45
e
1 4
of 20
f
1 4
of 44
g
1 5
of 30
h
1 5
of 120
i
1 10
j
1 2
of $1:20
k
1 4
of 1 hour (in min)
l
1 12
of 650 g
of 600?
10 Fill in the missing fraction:
is ..... of
a
is ..... of
b
11 Damien only won one third of the games of tennis that he played for his school team. If he played 15 games, how many did he win? 12 One fifth of the students at a school were absent because of colds. If there were 245 students in the school, how many were away? 13 One sixth of the cars from an assembly line were painted white. If 222 cars came from the assembly line, how many were painted white? 14 Draw sketches of the amounts of money which represent Write their numerical value beneath your sketches. a b
d
1 4
of each of the following.
e
15 There are 360o in 1 revolution (one full turn). one quarter turn
a half turn
c
f
a Find the number of degrees in: i one quarter turn ii a half turn iii three quarters of a turn. b What fraction of a revolution is: i 30o ii 60o iii 240o ?
three quarters of a turn
16 One morning two fifths of the passengers on my bus were school children. If there were 45 passengers, how many were school children? 17 Richard spent two thirds of his working day installing computers, and the remainder of the time travelling between jobs. If his working day was 8 hours, how much time did he spend travelling?
Finding the whole amount 2 5
of Freddy’s money was $5260.
So
1 5
was $2630.
)
5 5
was $2630 £ 5 = $13 150
Click on the icon for more problem solving questions.
EXTRA QUESTIONS
64
FRACTIONS (CHAPTER 4)
Unit 28
Fraction sizes and types 3 4
Which fraction is larger,
or 23 ?
The lowest common denominator (LCD) of two or more fractions is the lowest common multiple of their denominators.
If they have the same denominators, we can compare them. The LCM of 4 and 3 is 12. So the LCD of 3 4
Now
3£3 4£3
=
9 12
and
3 4
>
and
=
8 12
2 3
9 12
and 3 4
so
is 12:
>
2 3
=
2£4 3£4
=
> means “is greater than” < means “is less than”
8 12
2 3
Exercise 28 1 Find the LCM of: a 7, 3 e 6, 8, 9
b f
5, 3 10, 5, 6
c g
3, 6 5, 6, 11
d h
12, 18 12, 4, 9
c
5 9
d
4 7
2 Find the lowest common denominator of: 1 4
a
and
5 8
b
2 3
and
3 4
3 4
and
3 Write each set of fractions with the lowest common denominator. Then write the original fractions in ascending order (smallest to largest): a
1 1 2, 4
b
2 3 3, 4
c
1 4 2, 7
d
5 3 8, 4
e
7 5 10 , 6
f
7 3 9, 4
g
5 8 8 , 10
h
9 7 1 25 , 20 , 4
d
1 9 3 2 , 20 , 5
and
5 9
Ascending means going up. Descending means going down.
4 Write each set of fractions with a common denominator. Then arrange them in descending order: 1 2 7 2 , 5 , 10
a
b
1 5 3 2, 8, 4
c
1 7 4 2 , 12 , 6
5 Sanjay scored 16 out of 20 in a test. Robert scored 25 out of 30 in a different test. a Write each of the students’ scores as a fraction. b Write the two fractions with a lowest common denominator. c Which of the two students scored higher as a fraction of the total possible score in their test?
Activity
Fraction strips What to do: 1 Use a sheet of paper 24 cm long and a ruler to copy the fraction chart shown.
2 To compare
left for three 9 10
so
3 4 1 4
and
9 10 ,
1
count across from the
pieces and for nine
1 10
Qw_
pieces. Qt_
Qt_
13 16
Qr_
Qr_
Qt_
Qt_
Qt_
6 ths 8 ths 10 ths 12 ths
> 34 .
2 5
Qe_
Qr_
Use the fraction strips to answer these True or False questions: a
Qe_
Qr_
is further to the right than 34 , 9 10
Qw_
Qe_
c
3 5
12 16
FRACTIONS (CHAPTER 4)
Improper fractions and mixed numbers
Examples: Improper fraction to whole number
2 3
15 5
represents 23 .
is a proper fraction.
= 15 ¥ 5 =3
Its numerator is less than its denominator.
Improper fraction to mixed number 5 4
21 5
is an improper fraction. =
Its numerator is greater than its denominator.
20 5
1 5
+ 1 5
=4+ = 4 15
represents 54 .
Mixed number to improper fraction 5 4
2 45
can be written as a whole number and a fraction. 5 4
1 14
4 5
=2+
= 1 14
is a mixed number.
=
10 5
=
14 5
+
4 5
6 Write as a whole number: a
16 4
b
20 5
c
18 6
d
40 8
e
30 6
f
30 3
g
30 10
h
30 1
i
30 30
j
64 8
k
125 25
l
63 7
7 Write as a mixed number: a
5 4
b
7 6
c
18 4
d
19 6
e
15 2
f
17 3
g
16 7
h
23 8
i
22 7
j
35 9
k
41 4
l
109 12
8 Draw diagrams to show: a
7 6
b 3 12
9 Write as an improper fraction: a
3 12
b
4 23
c
2 34
d
1 23
e
1 12
f
3 34
g
1 45
h
6 12
i
4 59
j
5 78
k
6 67
l
1 11 12
10 Use 2 dice. Use one to roll the numerator and the other to roll the denominator. Find: a the smallest fraction it is possible to roll b the largest proper fraction it is possible to roll c the largest improper fraction (not a whole number) it is possible to roll d the number of different fractions it is possible to roll. e List the different combinations that can be simplified to a whole number.
numerator is the upper face bar denominator is the upper face
65
66
FRACTIONS (CHAPTER 4)
Unit 29
Adding and subtracting fractions
Adding fractions For mixed numbers:
If fractions have the same denominator, add the numerators. + 4 6
1 12 + 2 16
= 1 6
+
5 6
=
If fractions have different denominators: 2 3
+
=
2£4 3£4
=
8 12
=
11 12
1 4
+
+
1£3 4£3
fLCD is 12g
3 12
fadd numeratorsg
=
3 2
+
13 6
=
3£3 2£3
+
=
9 6
13 6
=
22 6
fadd numeratorsg
=
11 3
freduce to lowest termsg
= 3 23
fwrite as mixed numberg
+
fwrite as improper fractionsg 13 6
fLCD is 6g
² If necessary, convert mixed numbers to improper fractions.
Rules
² If necessary, change the fractions to fractions with the lowest common denominator. ² Add the numerators while the denominators stay the same.
Exercise 29 1 Without showing any working, add the following: a
1 4
+
2 4
b
3 10
d
4 7
+
2 7
e
4 9
g
3+
2 3
h
2+
a
1 5
+
3 10
b
3 5
e
3 4
+
1 3
f
7 10
i
3 4
+
1 6
j
5 9
+
5 6
1 4
+
1 3
b
3 5
+
7 10
+
2 3
+
+
3 10
10 9 5 8
+
7 8
c
1 6
+
4 6
f
3 5
+
4 5
i
1+
7 10
+
6 10
2 Find:
+
7 10
c
1 2
+
1 4
d
1 2
+
1 10
1 3
g
2 3
+
1 2
h
5 6
+
5 8
k
3 7
+
3 14
l
4 9
+
2 5
c
5 9
+
5 6
d
3 4
+
7 8
d
2 14 +
h
3 4
+
3 Find: a
+
1 2
+
5 20
+
1 3
+
2 3
4 Find:
5
a
2 + 1 13
b
3 12 + 2 12
c
2 23 + 1 13
e
3 34 +
f
2 23 + 1 12
g
2+
1 3
1 3
+ 1 12
1 8
+ 1 + 1 13
a Find the sum of 1 12 and 2 15 . b Find the average of 3 18 , 2 12 and 3 38 . c Frank has 2 12 m, 3 14 m and 1 13 m of water pipe. He has two pipe joiners. What length of pipe can he make?
FRACTIONS (CHAPTER 4)
Subtracting fractions We use the same rules to subtract fractions as we do to add them, except we subtract the numerators. If fractions have the same denominator: 7 9
5 9
¡
=
7¡5 9
=
2 9
For mixed numbers: 2 ¡ 1 13
fsubtract numeratorsg
If fractions have different denominators: 4 5
3 4
¡
=
4£4 5£4
=
16 20
=
1 20
¡
¡
3£5 4£5
=
2 1
=
2£3 1£3
¡
=
6 3
4 3
=
2 3
¡
¡
4 3
fwrite as improper fractionsg 4 3
fwrite with LCDg
fsubtract numeratorsg
fLCD is 20g
15 20
6 Find without showing any working: a
3 4
1 4
b
7 9
e
1¡
11 13
f
19 20
i
3¡
7 10
j
4¡
a
1 3
¡
1 4
b
5 6
e
3 4
¡
3 8
f
5 6
i
3 8
¡
1 4
j
7 15
1+
2 5
b
1 6
9 Find: a 3 12 ¡ 2 12
b
3 78 ¡ 1 12
f
¡
c
7 8
¡
5 8
d
1¡
5 6
g
5¡
1 2
h
2¡
3 5
6 7
k
1¡
1 7
l
1 ¡ ( 17 + 27 )
¡
1 3
c
3 4
¡
1 5
d
1 2
¡
3 10
¡
1 2
g
2 3
¡
1 6
h
4 5
¡
1 3
k
11 12
¡
3 4
l
7 10
c
5 12
+
5 6
d
3 4
2 23 ¡ 1 13
c
2 35 ¡ 1 25
d
2 23 ¡ 1 12
3 12 ¡ 1 56
g
3¡
h
3 34 ¡ 2 13
¡
4 9 13 20
¡
¡
2 7
7 Find:
1 3
¡
¡
3 15
1 8
¡
8 Find: a
e 10
¡
3 10
+
1 4
¡
1 8
7 10
¡
2 3
+
1 6
a Find the difference between 3 14 and 2 38 . b By how much does 2 12 exceed 78 ? c How much larger than 2 12 is 3 18 ?
11 Complete the magic squares where each row, column and diagonal must have the same sum.
3 Qw_
1 We_
1 Er_ 3 2 Er_
3
0
1 Qw_
6 Qw_
2 Qe_
Qe_ 5
Test your friends by making up two of your own.
67
68
FRACTIONS (CHAPTER 4)
Unit 30
Multiplying fractions
During the basketball season, a player drinks of a litre of milk five days a week.
3 4
For multiplying fractions:
How much milk does the player drink each week?
=
+
3 4
+
3 4
+
3 4
=
3 4
5 1
=
£
3 4
=
3+3+3+3+3 4
=
15 4
15 4
=
litres
litres
3 5
£
10 2 1
f cancel common factors g f multiply numerators, multiply denominators g
5 6
=
3 2
£
5 6
f write as fractions g
=
13
£
5 62
f cancel common factors g
=
5 4
2
= 1 14
c a£c a £ = b d b£d
The rule is:
f write as fractions g
1 12 £
DEMO
5£3 1£4
10 1
=6
or by multiplication: 5£
£
6 1
We can get the answer by addition: 3 4
3 5 1
=
=
+
£ 10
=
We can show this using diagrams:
3 4
3 5
f write as a mixed number g
Exercise 30 1 Find the missing number: a
5£
¤ 2 = 3 3
b
6£
¤ 3 = 5 5
c
3£
¤ 7 = 8 8
2 Write as a mixed number: a
3£
3 5
b
6£
4 7
c
5£
2 3
d
9£
3 4
e
6£
1 4
f
3£
7 8
g
8£
1 3
h
4£
5 6
i
9£
1 2
j
7£
4 10
k
2£
11 12
l
4£
4 5
m
3£
5 7
n
5£
4 9
o
9£
4 5
p
10 £
3 Simplify the following by cancelling common factors then multiplying: a
2 3
£
3 2
b
3 10
1 3
c
3 4
£
4 5
d
7 6
£
2 5
e
3 5
£
25 6
f
8 3
£
15 4
g
1 2
£
2 3
h
1 2
£
2 3
i
2 3
£
6 5
a
2 3
£
4 5
b
3 8
£
4 5
c
3 4
£
5 9
d
4 7
£
7 9
e
2 23 £
f
2 23 £
g
1 1 14 £ 1 15
h
4 3
i
2 25 £ 2 12
j
6 34 £
k
1 14 £
l
3 7
£
3 1
£
£
3 4
£
1 3
Cancelling common factors keeps the numbers smaller and easier to handle.
15 2
4 Find:
8 9
£
1 7
6 7 9 10
£
2 3
6 7
£ 3 12 £ 1 13
Change a mixed number to an improper fraction before you multiply.
FRACTIONS (CHAPTER 4)
69
5 Find: a
2 3
of 12
b
3 5
d
2 7
of 21
e
3 10
g
4 5
of 60
h
j
3 4
of
3 4
k
To find
2 3
=
c
3 4
of 4
f
3 8
of 16
1 13 of 9
i
3 7
of 49
1 4
l
1 2
of 17 12
of 10 of 20
of 6
“of ” means that we multiply. Qw_\ of 5 = Qw_ £ 5
of an hour, we write
2 3
of 60 minutes
2 3
£ 60
f1 hour = 60 minutesg
= 40 minutes 6 Find: a
3 4
of a metre
b
2 3
of one day
c
3 5
d
5 6
of an hour
e
7 10
of a litre
f
3 20
of a century of a kilogram
7 The whole value of each of the following groups of shapes appears beneath them. What is the value of the coloured shapes in each group? a b c d 90
30
70
200
8 Lisa had $117. She spent one third of her money on new jeans. How much did the jeans cost? 9 While Evan was on holidays, one eighth of the tomato plants in his greenhouse died. If he had 96 plants alive when he went away, how many were still alive when he came home? 10 A business hired a truck to transport boxes of equipment. The total weight of the equipment was 3 tonnes, but the truck could only carry 58 of the boxes in one load. a What weight did the truck carry in the first load? Remember 1 tonne = 1000 kg. b If there were 80 boxes, how many did the truck carry in the first load?
Multiplying fractions using diagrams 11
Use a rectangle to find
1 2
of 35 .
Shade
3 5
of a rectangle.
12
Divide the rectangle into halves and shade 12 of the 35 . 3 10
is shaded.
Checking our answer:
1 2
£
3 5
=
1£3 2£5
=
3 10
13
1 3
of 23 .
a
Use a rectangle diagram to find
b
Check your answer using the rule for multiplying.
a
Use a rectangle diagram to find
b
Check your answer using the rule for multiplying.
1 4
of 13 .
Write down the fraction multiplication and answer for the following shaded rectangles: a b
70
FRACTIONS (CHAPTER 4)
Unit 31
Problem solving with fractions
Donna trains three times a week. On Monday she ran 2 12 km, on Wednesday she ran 1 78 km and on Friday she ran 1 38 km. How many km did she run altogether?
After a party, three eighths of the birthday cake was left over. Usman ate half of this. What fraction of the cake did he eat?
2 12 + 1 78 + 1 38
Donna ran =
5 2
+
15 8
+
11 8
=
5£4 2£4
+
15 8
+
=
20 8
=
46 8
=
23 4
+
15 8
+
1 2
of
1 2
of
3 8
=
1 2
£
=
3 16
Usman ate and
11 8
fLCD is 8g
11 8
3 8
3 8
i.e., Usman ate
3 16
of the cake.
flowest termsg
= 5 34 i.e., Donna ran 5 34 km.
Exercise 31 1
a
Find the sum of 2 13 and 45 .
c
Find the product of
5 8
and
2 Balance the following scales: a
c
e
3 4
+
5 8
1 4
+
5 6
7 10
+
8
12
7 25
100
+
5 8
+ 12
+ 100
4 10 .
b
d
f
b
Find the difference between 2 13 and 45 .
d
How much larger than
2 9
+
5 6
4 18
+ 18
2 3
+
3 8
16
+
4 7
+
15 42
42
15 + 42
3 In a class of 28, four students were late handing in their projects. What fraction of the class were late? 4 Tom paid $2800 deposit on a car. He borrowed a further $8400 to pay for the car. What fraction of the car’s total cost was Tom’s deposit? 5 When Susan drove her car out of the yard the fuel tank was 12 full.
She used 13 of a tank to take her friends for a drive. How much fuel remained in the tank?
9
3 10
is 37 ?
FRACTIONS (CHAPTER 4)
71
6 Alice has 42 birds in an aviary; 26 are canaries and the rest are budgerigars. a What fraction of the birds are budgerigars? b If half the budgerigars are female, what fraction of all the birds are male budgerigars? 7 Tony plays his computer games for an hour and a quarter each week night. On Saturday he plays for three and a half hours and he plays for four and three quarter hours on Sunday. At this rate how much time does Tony spend playing computer games during one year? 8 A swimmer swims 37 of the way in the first hour and the second hour. What fraction has the swimmer left to swim?
2 5
in
9 To make a 20 kg blend of 5 different nuts, a wholesaler mixes 6 kg of peanuts, 4 kg of almonds, 3 kg of walnuts and 2 kg of cashews. The rest are macadamias. What fraction of the blend are the: a macadamias b peanuts c walnuts? 10 Wi filled one aquarium 34 full of water. He filled an identical aquarium 11 16 full of water. If the volume of one aquarium was 48 litres, how much water did he use altogether? 11 Zoe’s development company plans to subdivide 60 hectares of land into a housing development. One tenth of the land must be used for parks and gardens and 14 will be required for roads and walkways. How many blocks with an area of 15 hectare will she be able to create? 12 Which is the better score in a mathematics test: a 17 out of 20 b 21 out of 25? 13 An orchardist picked
1 4
of his orange crop in July and
2 3
of his crop in August.
a How much of his crop remained to be picked in September? b If he picked 600 cases in September, how many cases did he pick that season? 14 A snail crawls 3 35 metres in the first 14 hour, 2 23 m in the second 14 hour and 1 12 metres in the third How much further did it crawl if at the end of one hour it had reached 10 metres? 15
1 4
hour.
Joe’s Burger Shop makes 16 meat patties with every kilogram of minced beef. In his Double Pattie Delight, Joe uses 2 meat patties. His other varieties use only one pattie. If Joe sells 600 burgers in one week and 13 of them are his Double Pattie Delights, how much beef mince does Joe use in one week?
16 What fraction would 4 different pizzas need to be cut into if: a 12 people were to have one piece of each of the pizzas b each person was to have 2 pieces from each pizza? 17 A road crew repainting lines completed 3 23 km on day 1, 2 78 km on day 2 and 3 34 km on day 3. How many kilometres did they complete in total? 18 It takes 7 23 hours to fly from Adelaide to Singapore. The plane flies over Darwin 3 12 hours after leaving Adelaide. How long will it be before the plane lands in Singapore?
72
FRACTIONS (CHAPTER 4)
Unit 32
Review of chapter 4
Review set 4A 1 What fraction is represented by the shading in these diagrams? a b c
2 Find the fractions represented by the points on the number lines: a b 2
3
1
2
3 In a class of 24, three students were late handing up their projects. What fraction of the class was this? 4 What number must
1 2
be multiplied by to get an answer of 4?
5 Write the fractions in lowest terms: 21 28
a
b
6 Express 56 ,
2 3
and
7 9
15 24
c
120 300
with lowest common denominator.
Then write the original fractions in order of size, beginning with the smallest.
7 Write T for true or F for false. 3 9
a
=
15 40
b
3 47 =
24 7
c
76 8
= 9 12
d
375 1000
=
3 8
8 Find: a
2 12 + 3 45
b
6 14 ¡ 3 23
c
2 3
£ 2 12
3 4
b
5 6
c
5 8
of 1 kg
9 Find: a
of $28
of 1 hour (in minutes)
10 In lowest terms, state what fraction of: a
one week is 3 days
b
one metre is 35 cm
11 Solve the following problems: a There were 2728 paying spectators at a match. If three quarters supported the home team, how many supported the visiting team? b Three fifths of the students in a school order their lunch from the canteen. 142 do not. How many students are there in the school? c
3 7
of the students of a school attended a film night. If there were 840 students in the school, how many attended the film night?
FRACTIONS (CHAPTER 4)
Review set 4B 1 Divide and shade each of these shapes to show the fractions written underneath: a b c
7 12
3 8
5 6
2 Find the fractions represented by the points on the number lines: a b 1
2
4
5
3 Write these fractions in lowest terms: 24 27
a 4
b
a Convert
39 8
20 32
c
120 260
c
5 67 =
to a mixed number.
b What fraction of $9:00 is $1:80? c What fraction of 1 km is 800 m? 5 Express 25 , 34 and 13 20 with lowest common denominator. Then write the original fractions in order of size, the largest being first. 6
a If
3 4
of a number is 21, find the number.
b Find the values of 2 and 4 given that
3 4
=
2 20
=
27 4.
7 Write T for true or F for false: 3 7
a 8 Find:
=
6 14
a
=
15 35
2 19 ¡
b 5 6
675 1000
=
5 8
b 3 12 + 2 25
9 Solve the following problems: a A man who weighed 90 kg went on a diet and lost 10 kg. What fraction of his original weight did he lose? b
2 5
of a flock of sheep numbered 240. Find the size of the whole flock.
c Melissa works 2 nights a week after school. On the first night she works 2 23 hours and on the second 3 12 hours. What is her total time worked for the week? 10 Anne can type a 1 hour
2 3
b
of a page in 1 12
hours?
1 4
of an hour. How many pages can she type in
41 6
73
74
DECIMALS
Representing decimals
Unit 33
Australia’s currency (money) is called decimal, because it uses base 10.
CHAPTER 5
one cent
The decimal point separates whole numbers from fractionals.
is a tenth of
i.e.,
1 10
or 0:1 of 10 cents = 1 cent
is a tenth of
i.e.,
1 10
or 0:1 of $1 or 100 cents = 10 cents
is a tenth of
i.e.,
1 10
or 0:1 of $10:00 = $1:00
is a tenth of
i.e.,
1 10
or 0:1 of $100:00 = $10:00
Exercise 33 1 Using the example, change these currency values to decimals of one dollar: a
is $61.10 b
c
2 If seven dollars 45 cents is $7:45, what is: a 4 dollars 47 cents b 15 dollars 97 cents d 36 dollars e 150 dollars g 85 dollars 5 cents h 30 dollars 3 cents?
c f
seven dollars fifty five cents thirty two dollars eighty cents
3 Change these amounts to decimals using the dollar as the unit: a 35 cents b 5 cents d 3000 cents e 487 cents h 638 475 cents g 3875 cents
c f
405 cents 295 cents
4 Express these amounts as a fraction of a dollar. a e i
$1:75 $31:13 105 cents
b f j
$3:25 $243:08 $0:07
79 (e.g., $6:79 is 6 100 dollars)
c g k
$52:40 649 cents 3755 cents
d h l
$0:87 428 cents 100 010 cents
DECIMALS (CHAPTER 5)
Multi Attribute (MA) blocks
75
Decimal Grids
Tenths Hundredths Thousandths
Units
1 • 3 4 7 10 lots of 0.1
represents 1.347
100 lots of 0.01
represents 4¡´¡0.1¡=¡0.4
represents 27¡´¡0.01¡=¡0.27
5 Write the decimal value represented by the MA blocks if the largest block represents one: a b
c
d
6 In question 5, if
represents one cent and
represents one dollar:
a what is the decimal currency value of each example b what is the total value? 7 Write the decimal that represents the shaded area: a b
8
i ii
c
How many rectangles are shaded in these diagrams? What decimals are represented? a b
d
76
DECIMALS (CHAPTER 5)
Place value
Unit 34
Number 7 hundredths
Ten s Un it s De c. Ten Poin t t Hu hs ndr Th edt ous hs and t hs
The decimal point separates the whole numbers from the decimal part.
0 "
23\+\qF_p_\+\q_pL_p_p_ 2 3 "
If a number is less than 1, we write a zero in front of the decimal point.
0 7
Written numeral 0"07
Word form zero point zero seven
4 0 9
23"409
twenty three point four zero nine
the 4 stands for qF_p_
Exercise 34 1 Express the following in 2 different written forms: a 0:6 b 0:45 c 0:908
d
e
8:3
56:864
3 Draw up a place value table in your exercise book using the headings: Number
Place the following into the table: a 8 tenths b 3 thousandths c 7 tens and 8 tenths e 2 hundreds, 9 units and 4 hundredths g 5 thousands, 20 units and 3 tenths
Th ou Hu sand nd s Ten reds s Un it s De c. Ten Poin t t Hu hs ndr Th edt ous hs and t hs
2 Write as decimal numbers: a seventeen and four hundred and sixty five thousandths b two point nine eight three c thirty two point seven five two d twelve and ninety six thousandths e three and six hundred and ninety four thousandths f four and twenty two hundredths
Written numeral
"
d f h
9 thousands and 2 thousandths 8 thousands, 4 tenths and 2 thousandths 36 units and 42 hundredths
4 State the value of the digit 3 in the following: a 4325:9 b 6:374 e 43:4444 f 82:7384
c g
32:098 24:8403
d h
150:953 3874:941
5 State the value of the digit 5 in the following: a 18:945 b 596:08 e 75 948:264 f 275:183
c g
4:5972 358 946:843
d h
94:8573 0:0005
6 A drawing pin has been placed to show the decimal place on these abacuses. i
ii
.
iii
.
a What is the value represented in i, ii and iii? b What is the sum of all 3 amounts? c What is the difference in value between i and ii + iii?
.
DECIMALS (CHAPTER 5)
²
² Write
Write 5:706 in expanded fraction form (as a whole number and fractions). 5:706 7 + = 5 + 10 7 10
=5+
0 100
39 1000
6 1000
+
in decimal form. 39 1000
6 1000
+
77
=
30 1000
=
3 100
+
9 1000
9 1000
+
= 0:03 + 0:009 = 0:039 7 Express in expanded fraction form (as a whole number and fractions): a 5:4 b 14:9 d 32:86 e 2:264 g 3:002 h 0:952 j 2:973 k 20:816 m 9:008 n 154:451
c f i l o
2:03 1:308 4:024 7:777 808:808
8 Write in decimal form: a
6 10
d
8 10
+
9 1000
g
5 10
+
6 100
j
1 10
+
1 1000
+
8 1000
b
19 100
c
4 10
e
52 1000
f
5 100
+
2 1000
h
2 1000
+
3 10 000
i
9 100
+
4 1000
k
4+
3 10
+
l
3 100
+
8 10 000
8 100
+
7 1000
3 100
+
9 State the value of the digit 2: 324 4 1000
b
47 62 100
c
42 946 100
d
695 24 1000
e
3652 1 10 000
f
8 254 10
g
2 57 10
h
652 5 1000
i
1027 59 10 000
7 10 000
§
in expanded fractional form.
Units
What do these hands represent in: i oral and decimal form
5
.
5
§ § § § §
5
5
6
8
ª ª ª ª
b ª ª ª ª
6
Thousandths
.
ªª ªª ªª
4
5
9
§ § § § §§§ § §
d A
2
5 A
2
3
3
ª ª ª
3
.
ªª ª ªª
5
7
.
§ § § § § § §
3
8
8 8
§
7
A
a Which of the hands in question 10 has the highest value in the: i thousandths place ii tenths place iii ten thousandths place iv hundredths place? b Order the hands from highest to lowest value.
§ § § §
§ § § §
8
A
11
Hundredths
4
§ § § §
4
c
6
A
8
A
Tenths
7
expanded fractional form?
A
a
ii
Dec. Point
6
+
4
4 1000
4
5
+
ª ª ª
9
3 100
3
2
+
9
•
9
9 10
§
3
2+
2
4
10 In the decimal place value card game, this hand represents the number 2:9347 or
7
a
Ten Thousandths
78
DECIMALS (CHAPTER 5)
Rounding decimal numbers
Unit 35
Sometimes completely accurate answers are not required and so we round off to the required accuracy. (to 3 decimal places) (to 2 decimal places) (to 1 decimal place)
g
g
+ 6000
(to the nearest 10) 1 zero (to the nearest 100) 2 zeros (to the nearest 1000) 3 zeros g
g
+ 5700
0:5864 + 0:586 + 0:59 + 0:6
g
g
5716 + 5720
Rules for rounding off decimal numbers If, for example, an answer correct to 3 decimal places is required, we look at the fourth decimal place. ²
If the number in the fourth decimal place is 0, 1, 2, 3 or 4, leave the first 3 digits after the decimal point unchanged.
²
If the number in the fourth decimal place is 5, 6, 7, 8 or 9, increase the third digit after the decimal point by one.
Exercise 35 1 Write these numbers correct to 1 decimal place: a 2:43 b 3:57 c 4:92
d
6:38
e
4:275
2 Write these numbers correct to 2 decimal places: a 4:236 b 2:731 c 5:625
d
4:377
e
6:5237
3 Write 0:486 correct to: a 1 decimal place
b
2 decimal places
4 Write 3:789 correct to: a 1 decimal place
b
2 decimal places
b
2 decimal places
5 Write 0:183 75 correct to: a 1 decimal place d 4 decimal places 6 Find a c e
decimal approximations for: 3:87 to the nearest tenth 6:09 to one decimal place 2:946 to 2 decimal places
b d f
c
DEMO
3 decimal places
4:3 to the nearest whole number 0:4617 to 3 decimal places 0:175 61 to 3 decimal places
7 Evaluate correct to the number of decimal places shown in the square brackets: a
17 4
[1]
b
73 8
[2]
c
4:3 £ 2:6
[1]
d
0:12 £ 0:4
[1]
e
8 11
[2]
f
0:08 £ 0:31
[3]
g
(0:7)2
[1]
h
37 6
[2]
i
17 7
[3]
To find 27 correct to 3 decimal places we first write 27 as a decimal to 4 decimal places, then round. 7
0:2 8 5 7 2 : 0 60 40 50
)
2 7
+ 0:286
DECIMALS (CHAPTER 5)
79
We often shorten very large numbers using letters and decimals to represent them. Salaries, real estate prices and profits or losses of large companies use this form. K represents thousands
mill represents millions
bn represents billions
$27:5 K = $27 500 $19 829 = $19:829 K + $19:83 K rounded to 2 dec. places
$2 378 425
37 425 679 420 37 425 679 420 bn 1 000 000 000 = 37:425 679 420 bn + 37:43 bn
2 378 425 mill 1 000 000 = $2:378 425 mill + $2:38 mill =$
=
rounded to 2 dec. places
8 State the value in whole numbers of the following: a $38:7 K b $43:2 K
c
rounded to 2 dec. places
$98:9 K
9 Round these figures to 1 decimal place of a thousand dollars: a $56 345 b $32 475 c $23 159 10 Convert these salary ranges to 1 decimal place of a thousand dollars: a $70 839 - $73 195 b $158 650 - $165 749
c
$327 890 - $348 359
11 Round these figures to 2 decimal places of a million: a 3 179 486 b 91 734 598 d 1 489 701 e 30 081 896
c f
23 456 654 9 475 962
12 Expand these to whole numbers: a 21:65 mill b 1:93 mill
c
16:03 mill
13 Round these figures to 2 decimals of a billion: a 3 867 900 000 b 2 713 964 784 d 2 019 438 421 e 4 209 473 864 000 14 Expand the following to whole numbers: a 3:86 bn b 375:09 bn d 4:13 bn
d
212:45 mill
c f
97 055 843 899 549 000 000 000
c
21:95 bn
e
0:97 mill
Remember one million = 1¡000¡000 one billion = 1¡000¡000¡000.
15 Leo resigned from his job where his salary was $58:5 K. He accepted a new position with a salary of $82:7 K. Write these salaries in whole numbers and find Leo’s increase in salary. 16 A real estate agent sold properties valued at $170:2 K, $295:8 K and $672:1 K. Write the values in whole numbers and find their total. 17 The value of the grape harvest in the Barossa Valley was $67:4 mill in 2003. In the same year the value of the grape harvest in McLaren Vale was $69:9 mill. a Write as whole numbers: i the Barossa value b
i ii
ii
the McLaren Vale value
Which area had the better value? By how much was it better?
18 McGyver and Sons Engineering made a record profit of $1:2 bn in 2003. In 2002 the profit was $86:7 mill. i $1:2 bn ii $86:7 mill a Write as whole numbers: b Find the increase in profit from 2002 to 2003.
80
DECIMALS (CHAPTER 5)
Ordering decimals
Unit 36 We can mark decimal numbers on a number line. 1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
You can write zeros at the end of decimal numbers as this does not affect the place value of the other digits. e.g., 1.6 = 1.60
1.60
1:58 < 1:60 because 1:58 is to the left of 1:60: 1:56 > 1:51 because 1:56 is to the right of 1:51. Arrange 0:1, 0:12 and 0:102 in order from smallest to largest. 0:1 = 0:100 =
100 1000
²
0:12 = 0:120 =
120 1000
²
0:102 = 0:102 =
102 1000
Write all numbers with the same number of decimal places by adding zeros. Compare the numbers.
) the numbers are 0:1, 0:102, 0:12 from smallest to largest.
Exercise 36 1 Write down the value of the number at A on these number lines: a b A 6
7
c
48
e
f
3.7
0.09
A
0.1
2.41
2.42
2.43
2.44
2.45
2.46
0.2
A
l
0.1
2.40
4.20
j 0.41
A
2.39
A
4.19
A
2.38
2.1
h 1.96
0.40
2
2.0
A
1.95
k
152
A
3.8
i
A
151
A
g
14
d
A
47
A
13
0.04
2.47
2.48
0.05
2.49
2.50
2.51
2.52
2.53
2.54
Use the number line to compare the pairs of numbers. Write which is the greater: a 2:42, 2:5 b 2:48, 2:38 c 2:45, 2:54 3 Use a number line to show these numbers and then write them in order from smallest to largest. a 1:73, 1:70, 1:69 b 0:79, 0:77, 0:76 c 5:431, 5:427, 5:425 4 Insert the correct signs f>, < or =g to make the statements true: a 0:7 2 0:8 b 0:06 2 0:05 d 4:01 2 4:1 e 0:81 2 0:803 g 0:304 2 0:34 h 0:03 2 0:2 j 0:29 2 0:290 k 5:01 2 5:016 m 21:021 2 21:210 n 8:09 2 8:090
c f i l o
0:2 2 0:19 2:5 2 2:50 6:05 2 60:50 1:15 2 1:035 0:904 2 0:94
DECIMALS (CHAPTER 5)
5 A
81
Maha went to the greengrocer and bought some apples, bananas, grapes and pears.
B
a Which fruit was the most expensive per kg? Price/kg: $4.15 weight: 2.68kg Cost: $11.12
C
Price/kg: $3.65 weight: 2.80kg Cost: $10.22
D
b Which fruit did she spend the least amount of money on? c Which fruit did she buy the largest amount of (in kg)? d Which scales show the highest total price? e Which scales show the lowest weight (in kg) of fruit?
Price/kg: $4.35 weight: 2.58kg Cost: $11.22
Price/kg: $3.95 weight: 2.56kg Cost: $10.11
6 Arrange in ascending order (lowest to highest): a 0:8, 0:4, 0:6 b 0:4, 0:1, 0:9 c 0:14, 0:09, 0:06 d 0:46, 0:5, 0:51 e 1:06, 1:59, 1:61 f 2:6, 2:06, 0:206 g 0:095, 0:905, 0:0905 h 15:5, 15:05, 15:55
7 Arrange in descending order (highest to a 0:9, 0:4, 0:3, 0:8 b c 0:6, 0:596, 0:61, 0:609 d e 6:27, 6:271, 6:027, 6:277 f g 8:088, 8:008, 8:080, 8:880 h
lowest): 0:51, 0:49, 0:5, 0:47 0:02, 0:04, 0:42, 0:24 0:31, 0:031, 0:301, 0:311 7:61, 7:061, 7:01, 7:06
8 At the athletics meet, 5 competitors recorded these times for an event: Matthew - 10:05 seconds, Sam - 10:015 seconds, Jason - 10:5 seconds, Saxby - 10:55 seconds Eli - 10:055 seconds. a Show their times on a number line. b Write their times in order with the winning time first. c Write their names in order from first to last. 9 Continue the number patterns by writing the next three terms: a 0:1, 0:2, 0:3, .... b 0:9, 0:8, 0:7, .... c 0:2, 0:4, 0:6, .... d 0:05, 0:07, 0:09, .... e 0:7, 0:65, 0:6, .... f 2:17, 2:13, 2:09, .... g 7:2, 6:4, 5:6, .... h 3:456, 3:567, 3:678, ....
Ascending means lowest to highest.
Descending means highest to lowest.
82
DECIMALS (CHAPTER 5)
Adding and subtracting decimals
Unit 37 Examples: Addition
Subtraction
Find 3:84 + 0:372
Find 6:7 ¡ 0:637
3:8 4 0 + 0:3 7 2 1
1
6 9 10
4:2 1 2
6:0 6 3
Notice that the decimal points are vertically underneath each other.
Exercise 37
add zeros so that the place values line up.
6:7 0 0 ¡ 0:6 3 7
add a zero so that the place values line up.
DEMO
1 Find: a 0:4 + 0:5 d 0:17 + 0:96 g 0:4 + 0:8 + 4 j 30 + 0:007 + 2:948
b e h k
0:6 + 2:7 23:04 + 4:78 0:009 + 0:435 0:0036 + 0:697
c f i l
0:9 + 0:23 15:79 + 2:64 0:95 + 1:23 + 8:74 0:071 + 0:677 + 4
2 Find: a 1:7 ¡ 0:9 d 2 ¡ 0:6 g 4:5 ¡ 1:83 j 5:6 ¡ 0:007
b e h k
2:3 ¡ 0:8 4 ¡ 1:7 1 ¡ 0:99 1 ¡ 0:999
c f i l
4:2 ¡ 3:8 3 ¡ 0:74 10 ¡ 0:98 0:18 + 0:072 ¡ 0:251
3
DEMO
a Add 2:094 to the following: i
36:918
ii
0:04
iii
0:982
iv
5:906
iii
13:06
iv
24
b Subtract 1:306 from the following: i
2:407
ii
1:405
4 Add: a 31:704, 8:097, 24:2 and 0:891 c 1:001, 0:101, 0:011, 10:101 and 1 e 4, 4:004, 0:044 and 400:44
b d f
3:56, 4:575, 18:109 and 1:249 3:0975, 1:904, 0:003 and 16:2874 0:76, 10:4, 198:4352 and 0:149
5 Use two step operations to find: a 0:18 + 0:072 ¡ 0:251 d 5 + 0:444 ¡ 3:222
b e
4:234 ¡ 3:26 + 1:4 5:26 ¡ 3:111 + 6
c f
2:11 + 0:621 ¡ 0:01 15 ¡ 3:29 + 10:2
6 Subtract: a 29:712 from 35:693 d 3:7 from 171:048 g 3:333 from 22:2 j $109:75 from $115:05
b e h k
6:089 from 7:1 9:674 from 68:3 38:018 + 17:2 from 63 $24:13 from $30:10
c f i l
19 from 23:481 8:0096 from 11:11 (47:64 ¡ 18:79) from 33:108 $38:45 and $16:95 from $60
7
a What length is 1:6 cm less than 4:22 cm? b What distance is 4:2 km more than 3:55 km?
8
a Add three point seven nine four two, eleven point zero five zero nine, thirty six point eight five nine four and three point four one three eight.
DECIMALS (CHAPTER 5)
83
b Find the sum of seventeen and four hundred and twenty five thousandths, twelve and eighty five hundredths, three and nine hundred and seven thousandths and eight and eighty four thousandths. c Add thirteen hundredths and twenty thousandths and one and four hundredths. d Find the sum of fourteen dollars seventy eight, three dollars forty, six dollars eighty seven and ninety three dollars and five cents. 9
a By how much is forty three point nine five four greater than twenty eight point zero eight seven? b How much less than five and thirty eight hundredths is two and six hundred and forty nine thousandths? c What is the difference between nine and seventy two hundredths and nine and thirty nine thousandths? d How much have I got remaining from my sixty four dollars seventy five if I spend fifty seven dollars ninety?
10 Helena is 1:75 m tall and Fred is 1:38 m tall. How much taller is Helena than Fred? 11 On the first day of school the morning minimum temperature was 18:6o C and the maximum afternoon temperature was 35:9o C. What was the range of temperatures on this day? 12 John gets $5:40 pocket money, Pat gets $3:85 and Jill $7:85. How much pocket money do they get altogether? 13 What is the total length of these three pieces of timber: 2:755 m, 3:084 m and 7:240 m? 14 Our class went trout fishing and caught five fish weighing the following amounts: 10:6 kg, 3:45 kg, 6:23 kg, 1:83 kg and 5:84 kg. What was the total weight of all five fish? 15 In a fish shop, four large fish weigh 4:72, 3:96, 3:09 and 4:85 kg. What must the minimum mass of a fifth fish be if the customer wants a minimum of 20 kg of fish? 16 How much change from $100 is left after I buy items for $10:85, $37:65, $19:05 and $24:35? 17 Shin needed to save $62:50 for a computer game. He had $16:40 in his bank to start with and earned the following amounts doing odd jobs: $2:45, $6:35, $19:50, $14:35. Does he have enough money? If he does not, how much more does he need to earn? 18 At a golf tournament two players hit the same ball, one after the other. First Jeff hit the ball 132:6 m. Janet then hit the ball a further 204:8 m. How far did the ball travel altogether? 19 Out of interest I weighed myself weekly. In the first week I put on 1:2 kg while in the second week I lost 1:6 kg. Unfortunately I put on another 1:4 kg in the third week. If at the beginning I weighed 68:4 kg, how much did I weigh after the three weeks?
84
DECIMALS (CHAPTER 5)
Unit 38
Multiplying and dividing by powers of 10
Multiplication Remember 101 = 10 102 = 100 103 = 1000 104 = 10 000 etc.
Examples: 8:3 £ 10 = 8:3 £ 101 = 83 0:0932 £ 100 = 0:0932 £ 102 = 9:32 4:32 £ 1000 = 4:3200 £ 103 = 4320
When multiplying by 10 = 101 , shift the decimal point 1 place right.
The index or power number indicates the number of zeros.
When multiplying by 100 = 102 , shift the decimal point 2 places right. When multiplying by 1000 = 103 , shift the decimal point 3 places right.
Exercise 38 1 Multiply the numbers to complete the table:
Number
2 Find: a e i m q
43 £ 10 4:6 £ 10 0:8 £ 100 0:24 £ 1000 0:0094 £ 101
a
0:0943
b
4:0837
c
0:0008
d
24:6801
e
$57:85
b f j n r
£10
£100
8 £ 1000 0:58 £ 100 3:24 £ 100 2:085 £ 102 0:718 £ 100 000
£1000
c g k o
£104
£106
5 £ 106 3:09 £ 100 0:9 £ 1000 8:94 £ 103
3 Write the multiplier to complete the equation: a 5:3 £ 2 = 530 b 0:89 £ 2 = 890 d 38:094 £ 2 = 3809:4 e 70:4 £ 2 = 704 2 g 65:871 £ 2 = 6587:1 h 0:0006 £ 2 = 600
d h l p
c f i
0:6 £ 10 2:5 £ 100 0:845 £ 1000 0:053 £ 1000
0:89 £ 2 = 8900 38:69 £ 2 = 386:9 0:003 934 £ 23 = 3:934
4 A cinema ticket costs $13:50. If ten friends went to see a film together, what would be the total cost? 5 1 km = 1000 m. So 4:75 km = 4:750 £ 1000 m = 4750 m. Convert to the smaller units by multiplying by a power of 10: a $4:75 to cents b 12:56 kL to litres d 13:86 tonnes to kg e 9:847 m to mm
c f
3:86 cm to mm 2:08 kg to g
DECIMALS (CHAPTER 5)
Division
When dividing by 10n shift the decimal point n places to the left.
Examples: 0:6 ¥ 10 = 0:6 ¥ 101 = 0:06
85
fWhen dividing by 10 = 101 , shift the decimal point 1 place to the left.g
0:37 ¥ 1000 = 000:37 ¥ 103 = 0:000 37
6 Divide the numbers to complete the table:
fWhen dividing by 1000 = 103 , shift the decimal point 3 places to the left.g
a
647:352
b
93 082:6
c
42 870
d
10:94
7 Find: a 2:3 ¥ 10 d 3 ¥ 10 g 394 ¥ 10 j m p
¥10
Number
8:007 ¥ 10 579 ¥ 100 0:03 ¥ 10
¥100
¥1000
DEMO
¥105
b e h
3:6 ¥ 100 58 ¥ 10 7 ¥ 100
c f i
42:6 ¥ 100 58 ¥ 100 45:8 ¥ 100
k n q
24:05 ¥ 1000 579 ¥ 1000 0:03 ¥ 100
l o r
632 ¥ 10 000 579 ¥ 10 000 0:046 ¥ 1000
8 Write the divisor to complete the equation: a 9:6 ¥ 2 = 0:96 b 38:96 ¥ 2 = 0:3896 d 5:8 ¥ 2 = 0:0058 e 15:95 ¥ 2 = 1:595 g 3016:4 ¥ 2 = 30:164 h 874:86 ¥ 2 = 0:874 86
c f
6:3 ¥ 2 = 0:063 386 ¥ 2 = 0:0386
9 When a group of 100 employees won second prize of $13 352 in a lottery, they divided the money equally between them. How much did each person receive? 10 1 L = 1000 mL. So, 987 mL = 987: 0 ¥ 1000 L = 0:987 L Convert to the units given by dividing by a power of 10: a 4975 m to km b 5685 g to kg c 3095 mm to cm d 75 400 cents to dollars e 47 850 litres to kL f 2348 kg to tonnes g 26 cm to m h 5655 mm to m i 500 m to km
1000 m 100 cm 10 mm 1000 mm 1000 kg 1000 g 1000 L
= = = = = = =
1 1 1 1 1 1 1
km m cm m t kg kL
11 How many cents are there in $96:55? 12 Jess was 1:65 m tall and Tom measured 149:5 cm. How much taller was Jess? 13 Emma needed 1:5 kg of sugar, but discovered that she was 300 g short of that amount. How much sugar did Emma have?
86
DECIMALS (CHAPTER 5)
Multiplying decimal numbers
Unit 39 Examples: ²
Step 1: Remove the decimal point, i.e., £ by 101 .
3 £ 0:6 = 18: = 1:8
²
0:4 £ 0:03 = 0012: = 0:012
Step 2: Find 3 £ 6 = 18 Step 3: Replace decimal point,
i.e., ¥ by 101 .
Step 1: Remove decimal points,
i.e., £ by 103 .
Step 2: Find 4 £ 3 = 12 Step 3: Replace decimal point,
i.e., ¥ by 103 .
The number of decimal places in the question equals the number of decimal places in the answer. INVESTIGATION
DEMO
Exercise 39 1 State the number of decimal places in the following products. (Do not calculate the answer.) a 8 £ 5:7 b 12:98 £ 7:6 c 1:2 £ 5:3 d 11:296 £ 11:34 e 1:076 £ 5:2 f 0:0006 £ 0:13 2 Find a d g j m p
the value of: 0:8 £ 7 2:4 £ 3 0:3 £ 0:02 1:2 £ 0:12 30 £ 0:003 700 £ 1:2
3 Given that 34 £ 28 = 952, a 34 £ 2:8 d 0:34 £ 2:8 g 0:034 £ 2:8
b e h k n q
c f i l o r
9 £ 0:04 6:5 £ 4 0:04 £ 0:004 0:12 £ 11 (0:6)2 (0:09)2
find the value of the following: b 3:4 £ 2:8 e 0:034 £ 28 h 0:034 £ 0:028
0:4 £ 0:6 2:7 £ 5 7 £ 0:005 5:05 £ 0:09 0:08 £ 80 0:4 £ 0:3 £ 0:2
c f i
34 £ 0:028 0:34 £ 0:28 340 £ 0:0028
4 Given that 57 £ 235 = 13 395, find the value of the following: a 5:7 £ 235 b 5:7 £ 23:5 d 5:7 £ 0:235 e 57 £ 0:235 g 0:57 £ 0:235 h 5:7 £ 0:000 235
c f i
5:7 £ 2:35 0:57 £ 2:35 570 £ 0:235
5 Find a d g j m
c f i l o
0:5 £ 5:0 3:8 £ 4 0:04 £ 0:04 (0:03)2 2:5 £ 0:004
the value of: 0:4 £ 6 0:03 £ 9 0:9 £ 0:8 0:16 £ 0:5 1:2 £ 0:06
b e h k n
0:11 £ 8 0:03 £ 90 0:007 £ 0:9 (0:2)2 (1:1)2
6 Find the perimeter of these regular polygons: a b 4.09 m
c 30.75 cm
6.045 km
DECIMALS (CHAPTER 5)
d
e
87
f
36.5 mm
2.56 m 3.75 cm
g
h
i 3.68 m
1.25 mm
8.51 cm
7 A stone weighed 5:6 kg. If Duncan was able to lift 8 stones of this weight, how much weight could he lift? 8 I need 4:5 m of hose to water my garden. If hose costs me $3:40 per metre, how much will it cost me to buy my hose? 9 Fred needed at least 25 metres of timber. He found 6 pieces of timber in a shed, each 3:9 m long. Did he have enough altogether? How much timber did he have over or did he still need to find? 10
a Find the cost of 45 litres of petrol at 87:8 cents per litre. b Find the cost of 9:6 metres of pipe at $3:85 a metre. c Find the capacity of 6 dozen 1:25 litre bottles.
11 A caterer orders 5700 pies and 3600 pasties to sell at a football match. The pies and pasties each have a mass of 0:16 kg. What is the total mass of the: a pies b pasties c pies and pasties? 1 d How many heated vans ( 2 tonne capacity) are needed to deliver the pies and pasties? e If the caterer has a profit margin of 29:7 cents on each pie or pasty, what is her total profit if she sells the lot? 12
HAZEL’S PIZZA SHOP MENU Pizza Supreme Mexican Hawaiian Pasta Bolognese Napoletana Chips Drinks Cola Juice
Small $13:50 $11:80 $9:90
Large $15:50 $13:60 $11:70
$7:50 $6:50 $2:50
$13:80 $12:00 $4:10
$2:50 $3:00
$3:50 $3:80
Family $19:80 $17:50 $15:80
Find the cost of: a 4 large Hawaiian pizzas and 4 small chips b 1 family Mexican pizza, 3 large chips and 4 large juices c 5 large colas and 5 small chips d 6 large Napoletanas and 6 large juices e 2 small Supreme pizzas, 2 small Bolognese and 4 large colas. f 3 small Hawaiian pizzas, 2 small Napoletanas, 4 small chips, 1 large chips and 5 small colas.
88
DECIMALS (CHAPTER 5)
Dividing decimals by whole numbers
Unit 40 Examples: ²
4 )
²
4
)
1:1 6 4 : 6 24
Put a decimal point directly above the decimal point of the number to be divided (the dividend).
4:64 ¥ 4 = 1:16
So 4 divides into 4:64 exactly 1:16 times.
0:8 7 5 3 : 5 30 20
Add zeros if necessary to complete the division.
3:5 ¥ 4 = 0:875
Exercise 40 1 Find: a e i m
3:2 ¥ 4 24:16 ¥ 8 5:004 ¥ 9 0:354 ¥ 6
2 Find: a $0:84 ¥ 4 e $5:20 ¥ 8 i $114:75 ¥ 9
b f j n
7:5 ¥ 5 2:46 ¥ 6 52:5 ¥ 5 3:44 ¥ 8
c g k o
b f j
$0:57 ¥ 3 $50:65 ¥ 5 $787:50 ¥ 7
c g k
1:26 ¥ 3 0:72 ¥ 9 8:004 ¥ 6 0:045 ¥ 3 $2:68 ¥ 4 $82:56 ¥ 3 $1040:00 ¥ 8
d h l p d h l
3:57 ¥ 7 81:6 ¥ 4 0:042 ¥ 6 4:25 ¥ 5 $3:90 ¥ 5 $5:22 ¥ 9 $189:96 ¥ 6
3 Solve these problems: a If 5 pens cost $7:75, find the cost of 1 pen. b How much money would each person get if $76:50 is divided equally among 9 people? c One 3:5 m length of timber is cut into five equal pieces. How long is each piece? d How many 7 kg bags of potatoes can be filled from a bag of potatoes weighing 88:2 kg? e If $96:48 is divided equally among six people, how much does each person get? f The football club spent $189:20 on 8 trophies for their best players. How much did each trophy cost? g Paul worked at the local supermarket for 9 hours and was paid $69:30. How much did he earn per hour? 4 The perimeter of each of the following regular polygons is given. Find the length of one side to the nearest 2 decimal places. a b c d
P = 48:88 metres
P = 30:72 km
P = 138:72 cm
P = 34:2 millimetres
DECIMALS (CHAPTER 5)
Calculator practice with decimals 5 Choose the correct answer and then check using your calculator: a 4:387 £ 6 i 263:22 ii 26:322 iii 2:6322 b 59:48 £ 9 i 5:3532 ii 5353:2 iii 535:32 c 18:71 £ 19 i 355:49 ii 35:549 iii 35 549 d 0:028 £ 11 i 3:080 ii 0:0308 iii 0:308 6 Estimate the following using 1 figure approximations: a 8:6 £ 5:1 b 9:8 £ 13:2 d 1:96 £ 3:09 e 15:39 £ 8:109 g 0:976 £ 92:8 h 109:4 £ 21:84
iv iv iv iv c f i
2632:2 53:532 3554:9 30:800 12:2 £ 11:9 39:04 £ 2:08 1446 £ 49:2
Find the actual answers using your calculator. Solve these problems using your calculator. 7 How many $3:60 hamburgers can be bought for $104:40? 8 Thirteen people share a $47 446:75 lottery jackpot. How much do they each collect? 9 21 DVDs cost $389:55. How much does one DVD cost? 10 A square has a perimeter of 12:66 metres. Find the length of each side of the square. 11 How many 2:4 metre lengths of piping are needed to make a drain 360 metres long? 12 The heights of the girls in the Primary School Basketball team were measured in metres and the results were:
1:56, 1:43, 1:51, 1:36, 1:32, 1:45, 1:39, 1:38 Find the mean height. 13 Janine’s weekly earnings for 6 weeks were: $272:25, $301:50, $260:40, $278:85, $284:70 and $288:30. Find the average amount Janine earned per week. 14 A piece of wood is 6:4 m long and must be cut into short lengths of 0:36 m. a How many full lengths can be cut? b What length is left over? 15 When Emily’s family travelled from Adelaide to Eston they used 1:5 tanks of petrol. The tank held 62 litres of petrol.
a How many litres of petrol did they use travelling to Eston? b If petrol cost $0:90 per litre, what was the cost of fuel to travel from Adelaide to Eston? c If the car used 10 litres of petrol per 100 km, how far is it from Adelaide to Eston and back? d If they travelled a total of 2040 km while they were away, how many kilometres did they travel while in Eston?
89
90
DECIMALS (CHAPTER 5)
Fractions and decimals conversion
Unit 41 To convert fractions to decimals we can:
Use Division
Use Multiplication Examples: 4 5
²
=
4£2 5£2
=
8 10
9 25
²
= 0:8
²
=
9£4 25£4
=
36 100
=
7£125 8£125
=
875 1000
0: 4 2: 0
2 5
= 0:4
²
0: 4 4 4 4 4: 0 4 0 4 0 4 0
9
4 9
= 0:4444 :::: = 0:4
= 0:36 Fractions can be written as terminating decimals or recurring decimals.
7 8
²
5
Terminating decimals end. 0:4 is a terminating decimal.
DEMO
Recurring decimals repeat themselves without end. 0:4 is a recurring decimal. The bar above the 4 indicates this.
= 0:875
0:37 is a recurring decimal also. 0:37 = 0:373737:::: without end.
Exercise 41 1 Write as decimals using multiplication: a
7 10
b
1 2
c
2 5
d
3 10
e
4 5
f
1 4
g
4 25
h
3 4
i
1 8
j
5 8
k
7 20
l
6 25
m
13 20
n
11 125
o
4 14
p
2 15
q
5 35
r
9 2 20
s
7 1 25
t
358 500
c
3 8
d
9 8
g
4 78
h
5 38
2 Use division to write as a decimal: a 35 b 95 e
2 34
5 45
f
3 Convert the following fractions to decimals. Use a bar to show the repeating pattern of digits. a
1 3
b
2 3
c
1 6
d
1 7
e
2 7
f
1 12
g
2 9
h
5 6
i
3 11
j
7 12
4 Copy and complete the following pattern:
Fraction:
1 9
2 9
Decimal:
0:1
0:2
3 9
4 9
5 9
6 9
7 9
8 9
9 9
5 Write as decimals: a
23 32
b
11 16
c
17 80
d
11 25
e
3 1 16
f
3 14
g
2 15
h
9 11
i
7 2 30
j
97 50
k
6 13
l
49 160
m
5 3 12
n
31 123
o
23 45
DECIMALS (CHAPTER 5)
91
To convert decimals to fractions we write the decimal with a power of 10 in the denominator then simplify if possible. Examples:
0:6 =
6 10
6:44 44 = 6 + 100
=
3 5
= 6 11 25
²
²
0:625
²
=
625 1000
=
5 8
6 Write as fractions in simplest form: a 0:1 b 0:7 e 0:9 f 0:6 i 0:18 j 0:65 m 0:75 n 0:025
c g k o
0:5 0:19 0:05 0:04
7 Write these as fractions in simplest form: a 0:8 b 0:88 e 0:49 f 0:06 i 0:085 j 0:702
c g k
0:888 0:064 0:3
d h l
0:551 0:096 0:6
8 Write as mixed numbers in simplest form: a 2:8 b 4:5 e 22:32 f 46:19 i 3:260 j 4:014
c g k
3:6 28:42 13:025
d h l
7:2 5:002 12:001
d h l p
0:2 0:25 0:07 0:375
For you to remember This table contains commonly used fractions. Copy and complete the table by calculating the decimal form. 1 2
=
1 3
=
1 4
=
1 5
=
1 6
=
1 8
=
1 9
2 2
=
2 3
=
2 4
=
2 5
=
2 6
=
2 8
=
1 11
=
3 3
=
3 4
=
3 5
=
3 6
=
3 8
=
1 20
=
4 4
=
4 5
=
4 6
=
4 8
=
1 25
=
5 5
=
5 6
=
5 8
=
1 40
=
6 6
=
6 8
=
1 99
=
7 8
=
8 8
=
=
Challenge Convert each of the following mixed numbers into a decimal number and then fit the decimals into the grid alongside. One of the numbers has been inserted to get you started. 315 15
5
.
7
7 27 25
9 23 40
2 78
54 11 20
7 2 10
7 5 10
9 392 10
Notice that the decimal point occupies one square on the grid.
92
DECIMALS (CHAPTER 5)
Review of chapter 5
Unit 42 Review set 5A
1 Given that the boundary of the square represents one unit, what decimals are represented in the following grids? a
2
b
a Express 2:049 in expanded rational form (whole number and fractions). b State the value of the digit 2 in 51:932
3 Round off correct to 1 decimal place: a 0:465 b
$35 650 to $K
4 Given that 58 £ 47 = 2726, evaluate: a 5:8 £ 47 b 5:8 £ 0:47
c
c
5:8 £ 4:7
c
5:6 ¥ 10
8 094 387 to mill
5 Find: a
6:2 £ 1000
b
2:158 £ 100
d
4:2 ¥ 100
6 Convert a 7
$352:76 to cents
b
8:94 L to mL
a Find the difference between 246 and 239:84 b Find 0:03 £ 0:5 c A square has sides of length 3:7 m. What is its perimeter? d How much would each person get if $82:40 was divided equally between four people?
8 Write the following decimal numbers in ascending order:
0:216, 0:621, 0:062, 0:206, 0:026 9 In 3 seasons a vineyard produces the following tonnage of grapes: 638:17, 582:35 and 717:36. a What was the total tonnage for the 3 years? b Find the average tonnage for the 3 years. 10 A marathon runner stops for a drink 13 of the way at the 14:1 km mark. How far has he still to run? 11
a Write in decimal form: i 12 ii
3 8
iii
2 3
b Convert these decimals to fractions in lowest terms: i 0:6 ii 0:85 iii 0:2
DECIMALS (CHAPTER 5)
Review set 5B 1 If
represents one thousandth, what are the decimal values of the following?
a
b
c What is the sum of a and b? 2
a Convert 8 +
7 10
+
9 1000
to decimal form.
b State the value of the digit 6 in 9:016 3 Round off correct to 2 decimal places: a
b
9:4357
4 Given that 26 £ 53 = 1378, a
2:6 £ 5:3
b
$29 762 to $K
c
3 472 613 250 to bn
c
2:17 ¥ 100
evaluate: 2:6 £ 0:053
5 Find: a
1:89 ¥ 10
b
1:114 £ 1000
b
97:82 kg to mg
6 Convert to decimal form: a 7
7408 cm to metres
a Find the product of 4:2 and 1:2 b Evaluate 3:018 + 20:9 + 4:836 c Find the difference between 423:54 and 276:49 d Determine the total cost of 14 show bags costing $7:85 each.
8 Write these decimal numbers in descending order:
0:444, 4:04, 4:44, 4:044, 4:404 9 The first horse in a 1000 metre sprint finished in 56:98 seconds. The second and third horses were 0:07 seconds and 0:23 seconds behind the winner. What were the times of the: a second horse b third horse?
10
a Write as fractions in lowest terms: i 0:46 ii 0:375
iii
0:05
b Write as decimals: i 34 ii
iii
7 9
4 25
93
94
PERCENTAGES (CHAPTER 6)
Percentages and fractions
Unit 43
Percent means ‘out of one hundred’. 100 100
1 100
or
or 100%
1%
5 out of 100 =
5 100
= 5% = 5 percent
50 out of 100 =
50 100
= 50% = 50 percent
Most common fractions can be changed into percentages by first converting into fractions with a denominator of 100. For example: ²
²
²
=
CHAPTER 6
1 5
=
20 100
=
1 4
= 20%
=
25 100
=
7 25
= 25%
=
7£4 25£4
=
28 100
= 28%
Exercise 43 1 In each of these patterns there are 100 tiles.
a
b
Write the number of coloured tiles as a fraction of 100.
2 In this circle there are 100 symbols. Count each type then write the number of each type of symbol as a fraction of 100. a M= b C= c L= d X= e V=
3
i ii a
X M V M X V C X X C L X X C L V C X C X X X V M X V M C X C V X V X V V X M V LM X C M X V X V CX V L X C L C X X M C V X V L CL VV VM X C X XC X X V L V X LV X V X V L M X V C X C XM V X V V
What percentage is shaded in these diagrams? What percentage is unshaded? b c
4 Estimate the percentage shaded: a b
0 10 20 30 40 50 60 70 80 90 100
Check to see that your numerators total 100.
d
c
0 10 20 30 40 50 60 70 80 90 100
d
0 10 20 30 40 50 60 70 80 90 100
PERCENTAGES (CHAPTER 6)
In a class of 25 students, 6 have black hair.
To change a fraction to a percentage, we write the fraction with 100 in the denominator.
The fraction with black hair =
Examples:
= 13 25
²
=
13£4 25£4
=
52 100
²
= =
95
557 1000 557¥10 1000¥10 55:7 100
=
6 25 6£4 25£4 24 100
So the percentage with black hair is 24%.
= 55:7%
= 52%
5 Write the these fractions as percentages: a
19 100
b
3 100
c
37 100
d
54 100
e
79 100
f
50 100
g
100 100
h
85 100
i
6:6 100
j
34:5 100
k
75 1000
l
356 1000
6 Write as fractions with denominator 100, and then convert to percentages: a
7 10
b
1 10
c
9 10
d
1 2
e
1 4
f
3 4
g
2 5
h
4 5
i
7 20
j
11 20
k
7 25
l
19 25
m
23 50
n
47 50
o
1
7 Copy and complete these statements: a Fourteen percent means fourteen out of every ....... b If 53% of the students in a school are girls, 53% means the fraction c 39 out of one hundred is ......%. 8 In a class of 25 students, 13 have blue eyes.
::::::: : :::::::
Remember to write the fraction with 100 in the denominator.
a What percentage of the class have blue eyes? b What percentage of the class do not have blue eyes? 9 There are 35 netballers. 14 of them are boys. What percentage are girls?
10
A pack of 52 playing cards has been shuffled and 25 cards have been dealt as shown. a What percentage of the cards shown are: i hearts ii black iii picture cards iv spades? b If an ace is 1 and picture cards are higher than 10, what percentage of the cards shown are: i 10 or higher ii 5 or lower iii higher than 5 and less than 10? c From a full pack, what percentage are: i red ii picture cards iii diamonds iv spades or clubs? (J, Q and K are picture cards.)
96
PERCENTAGES (CHAPTER 6)
Percentage, decimal and fraction conversions
Unit 44 Percentages ²
85% 85 100 85¥5 100¥5 17 20
= = =
Fractions ²
Percentages ²
2:5% = = = = =
2:5 100 2:5£10 100£10 25 1000 25¥25 1000¥25 1 40
²
= 21 ¥ 100 = 21:0 ¥ 100 = 0:21
fto remove the decimalg fto simplifyg
21%
Decimals
²
First convert to a fraction with denominator 100, then write in simplest form.
12 12 % = 12:5% = 12:5 ¥ 100 = 12:5 ¥ 100 = 0:125
140% = 140 ¥ 100 = 140 ¥ 100 = 1:4
Divide the percentage by 100 to obtain the decimal.
Exercise 44 1 Write as a fraction in lowest terms: a 43% b 37% e 90% f 20% i 75% j 95% m 25% n 60% q 5% r 44%
c g k o s
50% 40% 100% 80% 200%
2 Write as a fraction in lowest terms: a 12:5% b 7:5% e 97:5% f 0:2%
c g
0:5% 0:05%
d h
17:3% 0:02%
3 Write as a decimal: a 50% e 85% i 15%
b f j
30% 5% 100%
c g k
25% 45% 67%
d h l
60% 42% 125%
4 Write as a decimal: a 7:5%
b
18:3%
c
17:2%
d
106:7%
h
6 12 %
l
4 14 %
e
0:15%
f
8:63%
g
37 12 %
i
1 2%
j
1 12 %
k
3 4%
d h l p t
70% 25% 3% 300% 350%
97
PERCENTAGES (CHAPTER 6)
Fractions
(2 methods)
Percentages
Decimals
3 4
0:27 = 0:27 £ 100% = 27%
Write as a decimal. Multiply by 100%.
= 0:75 = 0:75 £ 100% = 75% or 3 4
=
3£25 4£25
=
75 100
Percentages
Multiply the decimal by 100% to obtain the percentage.
Write as a fraction with denominator 100.
= 75% 5 Change to percentages by writing as a decimal first: a
1 10
b
8 10
c
4 10
d
3 5
e
2 5
f
1 2
g
3 20
h
1 4
i
19 20
j
3 50
k
39 50
l
17 25
m
3 8
n
1
o
11 100
p
3 8
q
1 3
r
2 3
6 Copy and complete these patterns: a 1 is 100% b 15 = 20%
c
1 3
is 33 13 %
d
1 4
is ......
1 2
is 50%
2 5
= ::::::
2 3
is ......
2 4
=
1 4
is ......
3 5
= ::::::
3 3
is ......
3 4
= ::::::
1 8
is ......
4 5
= ::::::
4 4
= ::::::
5 5
= ::::::
1 16
is ......
1 2
7 Change the following into percentage form by multiplying by 100%: a 0:37 b 0:89 c 0:15 e 0:73 f 0:05 g 1:02
d h
0:49 1:17
8 Change the following into percentage form by multiplying by 100%: a 0:2 b 0:7 c 0:9 e 0:074 f 0:739 g 0:0067
d h
0:4 0:0018
9 Copy and complete the table below:
Percent a
20%
b
40%
Fraction
0:2 2 5
c d
3 4
Fraction
g
k l
Decimal 0:35
12:5% 5 8
i j
0:85 2 25
Percent
h
0:5
e f
Decimal
100% 3 20
0:375
is .......
98
PERCENTAGES (CHAPTER 6)
Percentages on display and being used
Unit 45 We can convert
1 4,
0:42, 33% to percentages and plot them on a number line.
²
1 4
²
0:42 = 0:42 £ 100% = 42%
²
33%
=
1 4
£ 100%
= 25% = 33%
Using the percentages we can arrange the numbers in order from lowest to highest. Qr_ 0%
10
20
33% 30
0.42 40
50
60
70
80
90
100%
Exercise 45 1 Convert each set of numbers to percentages and plot them on a number line: a
f 35 , 70%, 0:65g
b
f55%,
9 20 ,
0:83g
c
f0:93, 79%,
17 20 g
d
f0:85, 34 , 92%g
e
f 27 50 , 67%, 0:59g
f
f47%, 0:74,
18 30 g
g
f 34 , 0:65, 42%g
h
f0:39, 58%,
i
f 58 , 73%,
7 2 20 , 5 g
13 20 ,
0:47g
2 Write each of the following number line positions in fraction notation with 100 as the denominator, as decimals and using % notation: a 0%
20
40
60
80
100%
0%
20
40
60
80
100%
0%
20
40
60
80
100%
Which is bigger, 24% or Qw_ ?
24%
b
1 2–
c
This is a table of conversions between fractions, decimals and percentages that are frequently used. Try to learn them. Percentage
Fraction
Decimal
Percentage
Fraction
Decimal 0:05
100%
1
1:0
5%
1 20
75%
3 4
0:75
33 31 %
1 3
0:3
50%
1 2
0:5
66 32 %
2 3
0:6
25%
1 4
0:25
12 21 %
1 8
0:125
20%
1 5
0:2
6 41 %
1 16
0:0625
10%
1 10
0:1
1 2%
1 200
0:005
PERCENTAGES (CHAPTER 6)
99
3 Refer to the illustration given and then complete the table which follows:
Students a b c d e f g
4
a
b
c
Number
Fraction
Fraction with denominator of 100
Percentage
wearing shorts with a ball wearing skirts and dresses wearing shorts and with a ball wearing track pants, baseball cap and striped top wearing shorts or track pants every student in the picture
Column A represents the students of room 16 who are driven to school. i What percentage are driven to school? ii What percentage find some other way to get to school? Column B represents the students of room 16 who play a musical instrument. i What percentage play a musical instrument? ii In lowest terms, what fraction play a musical instrument?
Percentage
Room 16 of Greenfields School 100 90 80 70 60 50 40 30
20 Column C represents the students of room 16 who 10 play sport for the school teams. 0 i What percentage play sport for the school A B teams? ii In lowest terms what fraction does not play sport for the school teams?
C
D
E
F
d
Column D represents the students who regularly use the internet or CD-Roms. i What percentage regularly use the internet or CD-Roms? ii If there are 30 students in this class, how many do not use CD-Roms or the internet regularly?
e
Column E represents the students who have been overseas. i What percentage have not been overseas? ii What fraction of the students is still to go overseas? (lowest terms)
f
Column F represents the students who can type more than 20 words a minute. If there are an equal number of boys and girls in this class of 30 and 3 more girls than boys can type more than 20 words a minute, what percentage of the girls can type over 20 words a minute?
100
PERCENTAGES (CHAPTER 6)
Representing percentages
Unit 46 Graphical representation
Passenger 20.4% Motor Cyclist 7.9% Pedestrian 21.7% Cyclist 5.3% Driver 44.7%
We often see percentages marked on pie charts and other statistical graphs. On pie charts the sector angle must accurately show the actual percentage. This pie chart shows the percentages of different road users who were killed in road accidents.
FATALITIES BY ROAD USERS
The sum of the percentages should be 100%. Can you explain why it may not be exactly 100% for a graph like this?
Exercise 46 1 The sectors of this pie chart of percentages represent 3 age groups of people living in Australia in 1996. Match your prediction with the graph and give reasons for your choice. a Under 15 b 15 - 64 c 65 and over.
21%
14%
2 Look at these diagrams. Find the unknown percentages: a b
15%
lemon squash¡/ lemonade
8%
mineral water
17%
other
Steel garbage is 5% of the total. The depth of steel garbage measures 2:5 mm on this graph. i
Use your ruler to find the percentage of plastic garbage.
ii
Then find the percentage of food garbage.
food ?%
paper 21%
?%
65%
cola brands
glass 16% plastic ?% garden 7% steel 5% other 4%
Sales of all carbonated softdrinks
aluminium 1%
Contents of a garbage can
2.97%
3 Name the states and territories whose percentage of Australia’s total area is represented by the figures shown on the graph. You may find it helpful to study a map of Australia to compare the areas with the percentages shown on the graph.
Click on the icon for an activity.
ACTIVITY
22.48%
17.52%
32.88%
0.089% 12.80% 10.43%
0.026%
PERCENTAGES (CHAPTER 6)
101
Geometric representation 20 50
There are 50 squares. 20 are shaded. What percentage is shaded?
= 0:4 = 0:4 £ 100% = 40%
so 40% is shaded and 60% is unshaded f100% ¡ 40%g 4 Copy and complete the following table, filling in the shading where necessary:
Figure
Fraction Percentage Percentage shaded shaded unshaded
a
b
Figure
Fraction Percentage Percentage shaded shaded unshaded
e
3 4
37:5%
f
30%
PRINTABLE WORKSHEET
c
d
2 3
g
1 6
h
30%
Click on the icon for the activity that matches percentages with geometric representations. 5 Construct a square with 10 cm sides. Divide it into 1 cm squares. a How many squares must you shade to leave 65% unshaded?
ACTIVITY
EXTRA PRACTICE
b In lowest terms, what fraction of the overall square is unshaded? 6 Construct a rectangle 10 cm by 5 cm. Divide it into 1 cm squares. Shade 7 squares blue, 9 squares red and 20 squares yellow. What percentage of the rectangle is: a red b blue c not shaded d either red or blue? 7 Use a compass to draw a circle. Colour 50% of your circle red, 25% blue, 10% orange, 5% green, 5% purple, 5% yellow.
[Hint: 100% of a circle = 360o, so 1% of a circle = What fraction of the whole circle is: a blue b d orange or blue e
360o 100
= 3:6o and 20% of a circle =
red red or purple or yellow
c f
£ 20 = 72o :]
orange coloured?
8 Divide a circle into 5 equal sectors of 20%. Colour 15 of the circle red, 40% yellow, If you drew 4 such identical circles: a what percentage of all the circles would be i blue ii red? b What fraction of the 4 circles would be yellow?
360o 100
1 5
blue and 20% green.
102
PERCENTAGES (CHAPTER 6)
Quantities and percentages
Unit 47
One quantity as a percentage of another We can only compare quantities with the same units. 3 apples as a percentage of 10 apples is possible.
3 apples as a percentage of 7 bananas is not possible.
To express 800 m as a percentage of 2 km, 800 m 2 km 800 m = 2000 m =
800¥20 2000¥20
=
40 100
Write the quantities as a fraction. Convert to the same units.
Out of 56 cakes baked, a shop sold 49. We express this as a percentage as 49 cakes 56 cakes =
49¥7 56¥7
=
7 8
= 0:875 = 87:5%
= 40%
Exercise 47 1 Choose a common name (denominator) which could be sensibly used to express one quantity as a percentage of the whole in each case. a coffee, tea b hamburgers, pizza c Virgin Blue, Qantas d fins, wetsuit, goggles e train, bus, tram f e-mail, letters, fax, telephone g saxophone, clarinet, recorder, trumpet h Holden, Ford, Mitsubishi, Toyota 2 Express the first quantity as a percentage of the second: a 10 km, 50 km b $2, $8 o o d 120 , 360 e 60 cents, $2 g 400 mL, 2 L h 6 months, 4 years j 48 kg, 1 tonne k 36 cents, $2 m 25 cm, 0:5 m n 48 min, 2 hours p 90 cents, $45 q 5 mg, 2 g
c f i l o r
3 m, 4 m 90o , 360o 50 g, 1 kg 5 mm, 8 cm 180 cm, 3 m 6 hours, 2 days
3 Express as a percentage: a 13 marks out of a possible 25 b 72 marks out of a possible 80 c 427 books sold out of a total 500 printed d 650 square metres of lawn in a 2000 square metre garden e 27 400 spectators in a 40 000 seat stadium f An archer scores 95 points out of a possible 125 points. 4 What percentage is: a 42 of 60 d 3 minutes of one hour g 420 kg of 1 tonne
b e h
34 of 40 175 g of 1 kg 16 hours of 1 day
c f i
48 seconds of 2 min 440 mL of 2 L 174 cm of 1 m?
PERCENTAGES (CHAPTER 6)
103
Finding percentages of quantities We can find a percentage of a quantity using these steps: 10% of 7 m = 0:1 of 700 cm = 0:1 £ 700 cm = 70 cm
35% of 5000 people = 0:35 of 5000 = 0:35 £ 5000 = 1750 people
Write the % as a decimal. ‘of’ means multiply.
Remember that the word ‘of’ indicates that we multiply.
5 Find: a 20% of 360 hectares
b
25% of 4200 square metres
c
5% of 9 m (in cm)
d
40% of 400 tonnes
e
10% of 3 hours (in min)
f
8% of 80 metres (in cm)
g
30% of 2 tonnes (in kg)
h
4% of 12 m (in mm)
i
15% of 12 hours (in min)
j
75% of 250 litres (in mL)
6 A school with 485 students enrolled takes 20% of them for an excursion to the museum. How many are left at school? 7 An orchardist picks 2400 kg of apricots for drying. If 85% of the weight is lost in the drying process, how many kilograms of dried apricots are produced?
8 A council collects 4500 tonnes of rubbish each year from its ratepayers. If 27% is recycled, how many tonnes is that? 9 A marathon runner improves her best time of 3 hours by 5%. What is her new best time?
10 Damian was 1:5 metres tall at the beginning of the school year. At the end of the year his height had increased by 5:6%. What was his new height? 11 The fruit drink made at a packaging plant consists of water (65%), blended with pure juice. If the plant produced 25:5¡¡kL of fruit drink last season, how many litres of this was pure juice? (Remember: 1 kL = 1000 L) 12 Dan played 30 games of baseball in a summer season. If the team lost 27% of those games, how many games did they win? 13 Which is the larger amount? a 40% of a litre or 13 of a litre c 8% of $100 or 85 cents e 33% of 1000 or 13 of 1000
b d f
20% of one metre or 14 of a metre 5% of a kilolitre or 5000 millilitres 30% of a kg or 315 g
104
PERCENTAGES (CHAPTER 6)
Money and problem solving
Unit 48 Finding a percentage of an amount
20 cents = 100 cents
20% of one dollar could look like
or
We can find
20 100
Finding one amount as a percentage of another To find 25 cents as a percentage of $1,
$16 as a percentage of $80
first write $1 as 100 cents.
=
16 80
Then 25 cents as a percentage of $1
=
16¥8 80¥8
=
2 10
=
25 100
£ 100%
= 25%
20% of $3500 = 0:2 of $3500 = 0:2 £ $3500 = $700
A as a percentage of B A is £ 100% B
£ 100% £ 100%
£ 100%
= 0:2 £ 100% = 20%
Exercise 48 1 Copy and complete:
This fraction represents ...... cents out of every ............ cents
This fraction represents ...... dollars out of every ............ dollars.
2 Find: a d g j m
10% of $47 11% of $20 83% of $720 12% of $2950 17:5% of $4000
3 Express: a $5 as % of $20 d $20 as % of $80 g $1:50 as % of $30 j $40 as % of $60
b e h k n
30% of $180 37% of $700 36% of $4:50 45% of $9700 6:8% of $40
c f i l o
b e h k
$15 as % of $150 $25 as % of $125 35 cents as % of $1:40 $334 as % of $33 400
c f i l
=
:::::: ::::::
=
= ...... %
:::::: 100
= ...... %
70% of $21 27% of $150 8% of $48:50 54% of $2500 10:9% of $50 000 $3 as % of $20 $6 as % of $120 $8 as % of $24 $9:95 as % of $99:50
PERCENTAGES (CHAPTER 6)
105
4 Write these scores as percentages. Arrange them in descending order. a Jan threw 18 goals from 25 shots, Jill 30 from 40, Jessie 38 from 50 and Jenny 20 from 32. b Jeff threw 21 from 30, Jake 40 from 60, Joel 34 from 50 and Juan 50 from 80. 5 In a series of three matches, Kim scored 5 goals from 9 shots, 7 from 11 and 4 from 5, and Kathy scored 15 from 20, 7 from 14 and 9 from 16. a Who was the more accurate scorer overall? b By what percentage was one girl better than the other? 6 Each of the following students saved a percentage of their allowance. Arrange the names of the students and their percentage saved in descending order. a Tom saved $6 from a $10 per week allowance, Tina $35 from $70 per month, Tao $13 from $25 per fortnight and Toni $11 from $20 per week. If each student was promised an extra 10% on the amount they saved over one year, how much more would be received by: b Tao
c Toni?
7 Nicky pitched 9 strikeouts and 4 walks against the 36 batters who faced her. What was Nicky’s percentage of: b walks? a strikeouts 8 A goal kicker had 80 kicks for goal during the football season. He kicked 56 goals. What percentage of his scoring attempts were: a goals b not goals? 9 Out of $1200, Sarah gets paid 30% and Jack gets paid 45%. Peter is paid the remainder. How much does each person receive?
Complete the crossnumber by writing all the clues as percentages. 1
2
3
4
5
6
7 10
8 11
14 18
16 19
21 25
13
15
22 26
20 24 27
2 5
16
4 5 6 7
17
23
15
3
9
12
Across 1 1 4 0:07 6 1 2 1 100
8 10
0:67
12 13 14
1 0:09 0:87
11 20
18 19 20 21 22
1 20 326 100
0:03 1 100 2 25 4 5 4 25
24 25
0:05
26 27
0:21 0:85
37 50
Down 2 0:56 3 0:4 3 4 4 7 8 9 11 12 13 17
1:58 3 5 703 100
0:57 3 20 24 25 14 50
18
3 10
19 21
1:11
23
0:06
24 26 27
21 25 11 20 1 50
0:08
PRINTABLE PUZZLE
106
PERCENTAGES (CHAPTER 6)
Discount and GST
Unit 49
If the marked price of a wetsuit is $200 and 15% discount is offered, the discount is The selling price is found by:
15% of marked price = 15% of $200 = 15% £ $200 = 0:15 £ $200 = $30 discount
Normal price Less 15% discount Selling price
$200 ¡ $30
Discount % is a percentage decrease.
$170
Exercise 49 1 If there is a 10% discount on a pair of shoes, originally priced at $90, find the amount of discount. 2
take
a If the marked price of a DVD player is $320 and 15% discount is offered, find the actual selling price. b A camera’s normal price is $460. By buying it duty free, it is 25% less. What is the selling price duty free? c A supermarket is offering 2% discount on the total of your shopping docket. How much will you pay if your docket is $130? d If the marked price of a computer is $2600 and 12% discount is offered, what is the new selling price?
3 Find the selling price after these discounts have been made: a b c
d
e
f
g
h
i
j
k
l
4 A television set is priced at $456 with 10% discount in store A. In store B a similar television set is priced at $525 with 20% discount. Which television set would cost less?
Remember to round off to the nearest cent.
PERCENTAGES (CHAPTER 6)
Goods and services tax (GST) Many goods and services sold in Australia include a goods and services tax (GST).
107
The rate of GST is 10%. This is a percentage increase.
For example, in order to make a profit a shop must sell an item for $80, and the GST must be added to this price. The GST is
10% of $80 = 0:1 £ $80 = $8
The shop sells the item for $80 + $8 = $88
5 What is the GST which must be added to the following items? a b
Selling price = $20 + GST
Selling price = $800 + GST
6 What is the selling price of these items? a b
Selling price = $2200 + GST
c
Selling price = $56 + GST
c
Selling price = $9.50 + GST
Selling price = $135 + GST
7 What is the GST on an item which would otherwise sell for: a $100 b $10 c $16
d
$320?
8 What is the price, GST included, on a service which would otherwise cost: a $100 b $48 c $2000
d
$640?
9 A shopkeeper needs to sell a pair of shoes for $160 to make the profit she wants. a What is the GST she must add on? b What must she sell the shoes for? 10 A bicycle shop sells bicycles for $250 and GST is to be added to this price. a What is the GST amount? b How much will the customers have to pay for a bicycle? 11 Challenge a Rachael pays her hairdresser $44 for a cut and colour. How much GST was included in the bill? b Martin receives a bill for $528 from the plumber. How much GST was included in the bill?
The bill is: hairdressers charge + 10% GST
108
PERCENTAGES (CHAPTER 6)
Unit 50
Simple interest and other money problems
When a person borrows money from a bank or a finance company, the borrower must repay the loan in full, and pay an additional charge which is called interest.
Simple interest is often called flat rate interest and is not used as often as compound interest.
The total amount to repay on a loan of $5000 for 3 12 years at 8% per annum simple interest can be calculated in this way: The simple interest charge for 1 year = 8% of $5000 = 0:08 £ $5000 = $400 ) simple interest for 3 12 years = $400 £ 3:5 = $1400 The borrower must repay
$5000 + $1400 fprincipal + interestg = $6400
DEMO
Exercise 50 1 Copy and complete the following table, following the example above.
Principal
Interest rate (p.a.)
Time (years)
Interest for one year
Total interest
$2000
10%
2
0:1 £ $2000 = $200
2 £ $200 = $400
$1000
15%
1
$5000
8%
4
$20 000
12%
1 12 p.a. is short for per annum.
2 Find the simple interest when: a $2000 is borrowed for 1 year at 15% per annum simple interest b $3500 is borrowed for 2 years at 10% per annum simple interest c $5000 is borrowed for 4 years at 8% per annum simple interest d $20 000 is borrowed for 1 12 years at 12% per annum simple interest e $140 000 is borrowed for
1 2
year at 20% per annum simple interest.
3 Find the total amount to repay on a loan of: a $2000 for 5 years at 8% p.a. simple interest b $6500 for 3 years at 10% p.a. simple interest
DEMO
c $8000 for 4 12 years at 12% p.a. simple interest d $10 000 for 10 years at 10% p.a. simple interest. 4 Alex borrowed $4000 for 2 years at 5% simple interest. How much did Alex repay at the end of 2 years? 5 Margot invested $6000 for 3 years at 8% simple interest. How much interest did she earn in that time?
PERCENTAGES (CHAPTER 6)
109
Who uses percentages? Extension: These are some of the words that are used regularly where money is being considered: discount, interest, rates, commission, taxation, rebates, deposit, profit, loss, increase, decrease, gross, nett, deduction. If a shopkeeper buys a mountain bike for $500 and sells it a few days later for $400 he has made a loss of $100 or 20%. 6 Kirsten bought a house for $267 000. Soon afterwards she had to move interstate and sold the house for $253 000. Find her: ¶ µ loss £ 100% a loss b loss as a percentage of her cost find cost
A car dealer buys a second hand car for $3800, spends a further $400 fixing the engine and putting better tyres on it and then sells it for $5040. He makes a profit of $840 on his costs. This is a 20% profit as a percentage of his costs. 7 A store keeper buys a set of golf clubs for $300 and sells them for $420. Find the: µ a profit b profit as a percentage of the cost find
¶
profit £ 100% cost
To buy a block of land for $60 000 you may be asked to put down a deposit of 20%. By paying the $12 000 deposit you can then arrange a loan to borrow the remaining 80% balance. Most lenders will not provide a loan for the full 100% value of the land. 8 Martin is looking for a loan to buy a car costing $25 500. He is asked to pay a deposit of 20%. a How much is the deposit? b What percentage does Martin have to borrow? c How much does Martin have to borrow?
When you start working you may be on a salary of $500 per week. This is your gross salary and represents 100% of what your employer pays you. However, you will be expected to pay about 17% in taxation and you may choose to make a 5% contribution to superannuation. This would mean that 22% is taken off in deductions leaving you with a nett salary of 78% or $390:00. 9 Maria’s gross salary was $800 a week. a If she paid 25% in taxation, what was her nett salary? b If she also paid 2% of her gross salary in superannuation, what was her nett salary?
When a real estate agent sells your home for $150 000 and charges you 4 12 % commission he receives $6750 and you receive $143 250. You receive less than the house was sold for. 10 A real estate agent sold my home for $300 000 and charged me 2% commission. How much commission did she receive?
110
PERCENTAGES (CHAPTER 6)
Review of chapter 6
Unit 51 Review set 6A 1
7 25
a
Write
with a denominator of 100:
b
Change
c d e f
Convert 0:45 to a percentage. Express 6 minutes as a percentage of one hour. Find 30% of $600. Find 140% of 2 kilometres (in metres).
1 3
to a percentage.
2 What percentage of the diagram is shaded?
3 Convert to percentages and plot on a number line: f 18 , 52%, 0:8g 4 Samantha had a budget of $200 to spend on clothes. She paid $58 for jeans and $82 for shoes. She spent 24% of her budget on a jacket and the balance on a baseball cap. a How much was the baseball cap? b What percentage of her budget did she spend on the baseball cap and shoes? 5 Write the first quantity as a percentage of the second: a 45o , 360o b 2 mm, 5 cm 6 A small country town had 280 households. 45% used a wood fire to warm their homes, 30% used electricity, 15% gas and the rest used oil or kerosene. How many households used gas, oil or kerosene?
7 A variety store is having a “20% off the ticket price” sale. If I bought a $38:90 toaster, a $79:90 sleeping bag, 2 bath towels at $12:90 each and a $5:40 blank video tape, how much would I save? 8 A plumber charges $940 for supplying and installing a new hot water service. a How much GST must be added? b What amount is the customer charged? 9 Joshua bought $690 of goods at the hardware store. He was allowed 5% discount for paying cash. How much did he pay to the hardware store? 10 Find the simple interest when $2400 is borrowed for 2 years at 12% p.a.
PERCENTAGES (CHAPTER 6)
Review set 6B 1
a Write 40% as a fraction in lowest terms. b Convert 0:45 to a percentage. c Write
7 25
as a percentage.
d Find 85% of $1200. e Find 16% of 4 m (in cm). 2 What percentage of the diagram is unshaded?
3 Write f 34 , 0:78, 72%g
as percentages and then plot them on a number line.
4 Express the first quantity as a percentage of the second. a 13 goals from 25 shots b 58 cm from 2 m
c
500 mL from 5 L
5 Anthony lost 6 marks in a test out of 25. What percentage did he score for the test? 6 What percentage is 650 kilometres of a 2000 km journey? 7 One hundred students agree to come to a fund raising school disco. What price should the committee charge each student if the DJ costs $180, balloons and streamers cost a further $20 and they want to make a 50% profit on their costs? 8 A fridge has a selling price of $840 but a discount of 15% is given. a Find the discount. b What is the actual price paid for the fridge? 9 A dentist charges $270 for dental treatment and GST must be added to this amount. a What is the GST amount? b How much will the customer have to pay? 10 Maryanne received 12% p.a. simple interest on her $3500 investment. a How much interest did she earn after 2 years? b What was her new balance? 11 The deposit on a new car was 20%. If the car cost $16 800, how much was the deposit? 12 A house was bought for $145 000 and sold for a 10% profit. How much was it sold for?
111
112
REVIEW OF CHAPTERS 4, 5 AND 6
TEST YOURSELF: Review of chapters 4, 5 and 6 1 What fraction of the diagram is shaded? a
b
2 What percentage of the diagram is shaded? 3 Find the fractions represented by the points on the number lines: a b 2
A
B
4 Draw a rectangle and shade
7 10
3
1
C
2
D
of it.
5 If the dollar represents the unit, what is the decimal value of:
6 Write in decimal form: a 26 045 cents as dollars 7
a Write
3 20
b
4500 metres as kilometres.
with denominator 100.
b Write 0:24 as a percentage. c Write
4 9
as a decimal number.
8 Write 6:095 in expanded fraction form. 9 Copy and complete the pattern in lowest terms: 2, 1 56 , 10
a Write
47 3
......, ......, 1 13 , 1 16 , 1, ......
as a mixed number.
b State the value of the digit 5 in the number 41:452 c Write 0:64 as a fraction in simplest form. 11
a Write 750 metres as a percentage of 1 kilometre. b Find 20% of 300 g.
12 Convert to percentages and plot on a number line: f 25 , 55%, 0:63g 13 What fraction of 2 dollars is 45 cents? Answer in lowest terms. 14
a Find the lowest common multiple of 9 and 12. b Write
7 12
and
c True or false, 15 Write 16 If
3 8
4 25
5 9
with lowest common denominator. 7 12
> 59 ?
as a percentage.
of a number is 21, find the number.
REVIEW OF CHAPTERS 4, 5 AND 6
17 Find a 18 Find
2 9
1 12 +
4 9
b 3 23 ¡ 1 34
c 3£
113
5 8
of $36:
19 Find the value of: a 1:27 + 5:063
b
5:063 ¡ 1:27
c
d
6:3 £ 0:9
6:3 ¥ 0:9
20 An electrician charges $615 for parts and labour. a How much GST must he add to his bill? b What amount does the customer pay? 21 Continue the number pattern by writing the next 3 terms: 4:51,
4:42, 4:33, ......
22 Jan received 10% discount when she bought a coat marked at $210. How much discount did she receive? 23 Find the simple interest payable when $1500 is borrowed for 2 years at 7% p.a. 24 Find the values of ¤ and ¢ if
¤ 25 5 = = : 8 16 ¢
25 Round off correct to 1 decimal place: a 0:947 b
$87 500 to $K
c
8 705 059 to mill
26 Jon had two $50 notes, a $5 note, and three 20 cent coins. Write this amount as a decimal of 1 dollar. 27 How many litres of drink are in 7 bottles which each hold 1:25 L? 28
3 4
of a box of apples were eaten. If 7 apples remained, how many had been eaten?
29 How much change would you receive from $50 if you bought five postage stamps costing $1:45 each and an express post bag costing $8:20? 30 6 students ran equal distances in a 4:8 km relay. How many metres did each student run? 31 When Reiko ordered new carpet she was asked to pay 20% deposit. If the carpet cost $3200, how much deposit did she pay? 32 A piece of steak weighing 4:35 kg is cut into 3 equal slices. What is the weight of each piece? 33 The number of shoppers in the mall on Saturday was 20% more than on Friday. If there were 1850 shoppers on Friday, how many were there on Saturday? 34 Find the total cost of 15 bus tickets costing $2:25 each. 35 How many 1:2 kg books would weigh 2:4 tonnes? 36 In a class of 30 students, two fifths of the students play sport after school. How many students do not play sport after school? 37 Sue cut 15 metres of wire mesh from a 45 metre roll. What percentage of the roll was left? 38 Arkie had $50 to spend on food. He spent $12 on fruit and vegetables, 18% on meat and $23 on groceries. a How much did he spend on meat? b What percentage of his money was left? 39 A variety store is having a “15% off everything” sale. How much in total would you pay for 2 CDs normally costing $28:50 each and a shirt normally priced at $33?
114
MEASUREMENT (LENGTH AND MASS)
Unit 52
Reading scales
In everyday life we measure many things. Some common measuring instruments are shown here. 17
18
19
35
cm
37
38
F
100
1 000 10 000 9 0 1
E
8 7
FUEL
40
41
42
°C
6 5 4
2 3
2 3
1 0 9 4 5 6
9 0 1 8 7
8 7
6 5 4
10 2 3
2 3
1 0 9 4 5 6
KWH
8 7
9 0 1 8 7
KILOWATT HOURS
6 5 4
2 3
The electricity meter shows 26 593 kWh.
There are 8 main divisions from empty to full on this fuel gauge. 5 8
39
Each small division is 0:1o C. The thermometer shows 36:8o C
Each small division is 0:1 cm (1 mm). The ruler shows 17:4 cm.
The fuel gauge shows
36
full.
CHAPTER 7
Exercise 52 1 Read these ruler measurements (in cm): a b 20 30 d
16
e
17
10
20
25
26
c
f
10
11
18
19
2 Read the temperature (in o C) for these thermometers: a b 33
34
35
36
37
38
39
40
°C
35
36
37
38
39
40
41
42
°C
c
3 Read these fuel gauges: a F
35
36
37
38
39
40
41
42
°C
35
36
37
38
39
40
41
42
°C
d
b
c
F
E
F
E FUEL
E FUEL
FUEL
4 Read as accurately as possible the speeds shown on these speedometers: a b c 60 40 20 0
80 100 120 140 160 KM/H 180 200
60 40 20 0
80 100 120 140 160 KM/H 180 200
60 40 20 0
80 100 120 140 160 KM/H 180 200
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
5 Find the weights, in kilograms, shown by these bathroom scales: a b
71
72
45
c
46
6 Find the quantity of electricity used as shown by these meters: a b 100 1 000 10 10 000 9 0 1 8 7
6 5 4
2 3
2 3
1 0 9 4 5 6
9 0 1 8 7
8 7
6 5 4
2 3
2 3
KILOWATT HOURS
1 0 9 4 5 6
8 7 8 7
10 000
9 0 1
9 0 1
6 5 4
8 7
7 Find the mass (in grams) on these scales: a b
6 5 4
2 3
2 3
1 0 9 4 5 6
9 0 1 8 7
8 7
6 5 4
64
10 2 3
2 3
1 0 9 4 5 6
9 0 1 8 7
KILOWATT HOURS
500 g 1 kg
KWH
8 7
6 5 4
2 3
c
500 g 0
100
1 000
KWH 2 3
63
0
500 g 1 kg
8 Find the quantity of fluid (in mL) in these jugs: a b
0
1 kg
c
1000 mL
1000 mL
1000 mL
800
800
800
600
600
600
400
400
400
200 100
200 100
200 100
9 For the following lines: i estimate the length ii using a ruler, measure the length to the nearest mm. iii What was the error in your estimation? a b c
d
e
f
g
h
115
116
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
Unit 53
Units and length conversions
The earliest units of measurement used were lengths related to parts of the body. All of these measurements were inaccurate because people are different sizes.
We use the metric system of measurement, which is more accurate.
cubit span
The units of length are millimetres (mm), centimetres (cm), metres (m) and kilometres (km).
pace
The units of mass are milligrams (mg), grams (g), kilograms (kg) and tonnes (t).
Exercise 53 1 State what units you would use to measure the following: a the mass of a person b the distance between two towns c the length of a sporting field d the mass of a tablet
e f g h
the the the the
length of a bus mass of a car width of this book mass of a truck
Activity
Measuring instruments What do these instruments measure? Match the instrument to its name:
pocket watch builders square
thermometer fuel gauge
electricity metre sphygmomanometer
micrometre sextant
a
b
c
d
e
f
g
h
ACTIVITY
Click on the icon to find the activity on ‘Measures and who uses them’.
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
117
Conversion diagram for length To convert smaller units to larger ones we divide. ¥10 mm
¥100
¥1000
cm £10
Learn these conversions
m
1 cm = 10 mm 1 m = 100 cm 1 m = 1000 mm 1 km = 1000 m
km
£100
£1000
To convert larger units to smaller ones we multiply. Convert
²
640 cm to m smaller unit to larger ) divide
²
3:8 km to m larger unit to smaller ) multiply
²
7560 mm to m smaller unit to larger ) divide
640 cm = 640 ¥ 100 m = 6:4 m
3:8 km = 3:8 £ 1000 m = 3800 m
7560 mm = 7560 ¥ 1000 m = 7:56 m
2 Write the following in metres: a 900 cm b 643 cm e 9000 mm f 13 500 mm i 2 km j 6:8 km
c g k
4753 cm 620 mm 0:5 km
d h l
35 cm 58 mm 0:826 km
3 Write the following in centimetres: a 7m b 13:8 m e 85 mm f 1328 mm i 1 km j 0:5 km
c g k
0:34 m 402 mm 0:02 km
d h l
0:02 m 0:4 mm 0:003 km
4 Write the following in millimetres: a 7m b 3:4 cm
c
78 cm
d
0:46 m
e
0:26 cm
5 Write the following in kilometres: a 4500 m b 17 458 m
c
200 m
d
16 400 cm
e
653 000 cm
6 If the distance from your home to school is 750 metres, how far in kilometres do you travel to and from school in a week? 7 Zoe is a triathlete. She has to swim 200 m, ride her bicycle for 7:5 km and run 2500 m. What is the total distance Zoe has to travel in a metres b kilometres? 8 Convert all lengths to metres and then add: a 3 km + 110 m + 32 cm c 153 m + 217 cm + 48 mm
b d
72 km + 43 m + 47 cm + 16 mm 15 km + 348 m + 63 cm + 97 mm
9 Write these in the same units and then put in order from longest to shortest: a 37 mm, 4 cm b 750 cm, 8 m, 7800 mm c 1250 m, 1:3 km d 0:005 km, 485 cm, 5:2 m e 3500 mm, 347 cm, 3:6 m f 0:134 km, 128 m, 13 000 cm g 4:82 m, 512 cm, 4900 mm h 72 m, 7150 cm, 71 800 mm 10 Calculate your answer and write it in the units given in brackets: a 6 m ¡ 23 cm (cm) b 9 cm ¡ 25 mm (cm) c 3:8 km ¡ 850 m (m) d 17 m ¡ 8 m 49 cm (m)
118
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
Unit 54
Perimeter
The perimeter of a figure is a measurement of the distance around its boundary. For:
Triangle
Square
In figures, sides having the same markings show equal lengths.
Rectangle
b
a
s
w
c
l
the formulae for finding the perimeters of these figures are: P =a+b+c
P =4£s
P = (l + w) £ 2
[P = 4 £ side length]
[P = (length + width) £ 2]
7 cm
²
8 cm
3 cm 9 cm
P
Always give the units of measurement, for example, cm.
²
17 cm
= 3 + 7 + 9 cm = 19 cm
P
DEMO
= (8 + 17) £ 2 cm = 25 £ 2 cm = 50 cm
Exercise 54 1
i ii a
Estimate the perimeter of each figure. Check your estimate with a ruler. b
c
2 Find the perimeter of each of the following triangles: a b
c
19 cm
4.2 km
15 m
13 cm 27 cm 11 m
3 Find the perimeter of: a
b
c 4.5 cm
12 cm
10.2 km
9.8 cm 3.1 km
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
4 Find the perimeter of: a
b
c 6 cm
10 cm
9 km
7 km
5 cm
d
4 km
13 km
5 km
e
f 11 cm
8 cm
10 cm 7 cm
g
h
18 cm 11.3 cm
i
10 cm
7.2 cm 7.2 cm
10 cm
9.6 cm
5 Use a piece of string to find the perimeter of the following: a b c
d
6 Solve the following problems: a A rectangular paddock 120 m by 260 m is to be fenced. Find the length of the fence. b How far will a runner travel if he runs 5 times around a triangular block with sides 320 m, 480 m and 610 m?
Draw a diagram to help solve these problems.
c Find the cost of fencing a square block of land with side length 75 m if the fence costs $14:50 per metre. 7
a What is the perimeter of an equilateral triangle with 35:5 mm sides? b If the perimeter of a regular pentagon is 1:35 metres, what is the length of one side? c One half of the perimeter of a regular hexagon is 57 metres. What is the length of one of its sides? d One third of the sum of the lengths of sides of a regular dodecagon is 39 cm. What is its perimeter? e The perimeter of 2 identical regular octagons joined along one side is 98 cm. What is their combined perimeter when they are separated?
8
a Find the length of the sides of a square with perimeter 56 cm. b Find the length of the sides of a rhombus which has a perimeter of 72 metres.
A dodecahedron has 12 sides.
119
120
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
Unit 55
Scale diagrams A scale diagram is a drawing or plan which has the same proportions as the original object. The scale is the ratio scale length : actual length.
If the scale is 1 : 20, we can find the: a actual length if the scale length is 3:4 cm b scale length if the actual length is 1:2 m. a
actual length b = 20 £ scale length = 20 £ 3:4 cm = 68 cm
km 0
10
20
30
40
On this map scale, 5 cm (the drawn length) represents 50 km (the actual length).
scale length = actual length ¥ 20 = 1:2 m ¥ 20 = 120 cm ¥ 20 = 6 cm
Scale = 5 cm : 50 km = 5 cm : 50 £ 1000 £ 100 cm = 5 : 5 000 000 = 1 : 1 000 000
Exercise 55 1 For the following scales, state if the drawing or the actual object is larger than the original: a 1 : 500 b 3:1 c 2:5 d 1:4 e 1 : 10 000 2 Find the scale if: a an aeroplane has wingspan 50 m and its scale length is 50 cm b a truck is 15 m long and the diagram has its scale length 12 cm
wingspan
c a bacterium has body length 0:005 mm and its scale length is 10 cm.
3 Find the actual length for a scale length of 5 cm if the scale is: a 1 : 50 b 1 : 2000 c 1 : 10 000
d
1 : 5 000 000
4 If the scale is 1 : 5000, find: a the actual length if the scale length is i
4 cm
ii
5:8 cm
iii
2:4 cm
iv
12:6 cm
iii
20 m
iv
108 m
iii
8:2 cm
iv
0:8 cm
iii
5:6 m
iv
12:2 m
b the scale length if the actual length is i
500 m
ii
175 m
5 If the scale is 1 : 200, find: a the actual length if the scale length is i
3 cm
ii
4:5 cm
b the scale length if the actual length is i
200 m
ii
18 m
6 The drawing of a gate alongside has a scale of 1 : 100. Find: a the width of the gate b the height of the gate c the length of the diagonal support.
(Note: The posts are not part of the gate.)
width of the gate
onal
diag
50
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
7 Using the scale on the map of Australia, find the distance from a Sydney to Perth b Adelaide to Darwin c Melbourne to Alice Springs d Canberra to Brisbane.
N
Darwin Cairns
Mt. Isa
Alice Springs
Rockhampton Brisbane
Kalgoorlie
Perth
Adelaide Canberra Sydney Melbourne scale: 1 cm represents 600 km Hobart
8 If the plan of a house wall alongside has been drawn with a scale of 1 : 200, find: a the length of the wall b the height of the wall c the measurements of the door d the measurements of the windows. 9 For the truck alongside, find: a the actual length of the truck b the maximum height of the truck.
(Scale:
10 a
1 : 100)
Measure the length of the body of the dragonfly and find the scale for the diagram. Using the scale in a, find: i the length of the head ii the wingspan iii the greatest width of the rear wing.
length of body
b
121
(Real length = 50 mm)
11 Using the scale shown on the map, find: a the actual distance shown by 1 cm b the map distance required for an actual distance of 200 km c the distance from i A to B ii D to E iii C to F.
E
A
F
D Scale: 1¡:¡500¡000
C
B
122
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
Unit 56
Mass The mass of an object is the amount of matter it contains.
Units of mass are milligrams (mg), grams (g), kilograms (kg) and tonnes (t).
Conversion diagram To convert smaller units to larger we divide. ¥1000 mg
¥1000
¥1000
g
Learn these conversions
kg
£1000
1 g = 1000 mg 1 kg = 1000 g 1 t = 1000 kg
t
£1000
£1000
To convert larger units to smaller units we multiply.
Convert:
²
350 g to kg smaller unit to larger ) divide
²
350 g = 350 ¥ 1000 kg = 0:35 kg
7 500 000 mg to kg smaller unit to larger ) divide
²
7 500 000 mg = 7 500 000 ¥ 1000 ¥ 1000 kg = 7:5 kg
8:5 t to kg larger unit to smaller ) multiply 8:5 t = 8:5 £ 1000 kg = 8500 kg
Exercise 56 1 Give a c e g i k m o q s
the units you would use to measure: a person’s mass the mass of an egg the mass of an orange the mass of a raindrop the mass of your school lunch the mass of a refrigerator the mass of a school ruler the mass of a bulldozer the mass of a calculator the mass of an ant
b d f h j l n p r t
the the the the the the the the the the
mass mass mass mass mass mass mass mass mass mass
of of of of of of of of of of
a a a a a a a a a a
ship book lounge suite boulder cricket bat dinner plate slab of concrete leaf computer horse
2 Which of these devices could be used to measure the items in question 1? A
B
spring balance
C
kitchen scales
3 Convert these grams into milligrams: a 2 b 34
c
350
D
bathroom scale
d
4:5
weigh bridge
e
0:3
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
4 Convert these tonnes into kilograms: a 4 b 25
c
3:6
d
294
e
0:4
5 Convert these kilograms to grams: a 6 b 34
c
2:5
d
256
e
0:6
6 Convert these milligrams to grams: a 3000 b 2500
c
45 000
d
67:5
e
9:5
7 Convert these kilograms to tonnes: a 4000 b 95 000
c
4534
d
45:6
e
0:8
8 Write the following in grams: a 8 kg b 3:2 kg e 4250 mg f 75 420 mg
c g
14:2 kg 6:8 t
d h
380 mg 0:56 t
9 Convert the following to kilograms: a 13 870 g b 3:4 t
c
786 g
d
3496 mg
10 Calculate your answers in kilograms: a 520 g + 2:1 kg + 16 kg c 1:5 kg ¡ 750 g e 4:2 t ¡ 3 t + 300 kg
b d f
700 g + 1600 g + 63 g 2 t ¡ 763 kg 15 kg ¥ 2
11 Solve the following problems: a Find the total mass, in kilograms, of 200 blocks of chocolate, each 120 grams. b If a nail has mass 25 g, find the number of nails in a 5¡¡kg packet. c Find the mass in tonnes of 15 000 bricks if each brick has a mass of 2:2 kg. d A box of 150 tins of dog food weighs 205 kg. If the empty box weighs 25 kg, find the mass of each tin. e A carton with a mass of 350 g holds 12 boxes of cereal. Each box of cereal has a mass of 850 g. Find the total mass of the carton full of boxes of cereal. 12 Write in the same units. Then list in ascending order (smallest to largest). a 2400 mg, 2 g b 6700 g, 7 kg c d 0:004 t, 3:6 kg, 3800 g e 1900 mg, 1:5 g, 0:002 kg
1420 kg, 1:4 t
13 Write both masses as kg. Find the cost per kg for each of them. Which is the better buy? a b
Soap Powder 3 kg $6.60
Soap Powder 2 kg $4.50
Muesli 1 kg
$16.75
Activity Sheet
Click on the icon for the Measurement Message Activity Sheet.
Muesli 500 g $9.10
123
124
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
Unit 57
Problem solving
A rectangular housing block 40 m by 22 m is to be fenced. The fencing costs $75 per metre. The length of fencing needed = (40 + 22) £ 2 m = 62 £ 2 m = 124 m
40 m 22 m
The cost of the fence = 124 £ $75 = $9300
²
Think about what the question is asking and the units you will work in.
²
A labelled diagram is often helpful.
²
Set out your answer in a clear and logical way.
²
You may need to write your final answer in a sentence.
Exercise 57 1 A farmer fences a 250 m by 400 m rectangular paddock with a 3 strand wire fence. a Find the total length of wire needed. b Find the cost of the wire if wire costs $2:40 per metre. 2
a A house owner has a block of land 30 m by 75 m (30 m across the back). If he wishes to fence two sides and the back of the block, what is the total length of the fence needed? b If the fence is to be made of “Good Neighbour” panelling which comes in sheets 2 m wide costing $18:50, what will be the cost of the fence?
3 A carpenter has to make a window frame with the dimensions shown. What is the total length of timber he requires?
5 cm 120 cm
150 cm
4 6m
a Henry edges his garden with railway sleepers. If his garden has two plots as shown, find the total length of sleepers required.
Plot 1 2m
Paths 6m
Plot 2
2m
b If each sleeper is 2 m long and weighs 40 kg, find:
20 m
5 A supermarket buys cartons of canned peaches. Each carton contains 12 cans and each can weighs 825 g. Find the mass in kilograms of a carton of peaches.
i the total number of sleepers needed ii the total mass of sleepers.
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
6
125
a A couple wish to build a brick fence along the 60¡¡m front of their block of land. If they want 12 rows of bricks and each brick is 20 cm long, find the number of bricks required. b If each brick weighs 2:5 kg, find the total mass in tonnes of the bricks needed.
7 3.5 m
7.5 m 10 m
A builder needs to construct a pergola with the dimensions as shown. The support posts cost $15 per metre and the timber for the top costs $4:50 per metre. a Find the total length of timber for the top and hence the cost of this timber. b Find the cost of the posts. c Find the total cost of building the frame for the pergola if nails and other extras cost $27.
8 A grazier has a property with the dimensions illustrated. One of the farmhands is asked to check the fence on his motorbike. If he can travel at 15 km/h, how long will it take him to check the whole fence?
5.5 km 12.5 km
6 km
a Using the scale diagram alongside, find the total length of timber required to make the gate frame shown.
9
Scale 1 : 60
b If the timber costs $4:50 per metre, find the total cost of the timber used.
10 A 30 m picket fence is to be built as shown. There is a 2 m post every 2 m, to which the rails are attached. If the timber for the pickets costs $1:80 per metre, for the rails costs $2:50 per metre and for the posts costs $4:50 per metre, find: a the number of posts and hence the total length of timber required for the posts
rail
1.2 m
picket 10 cm
10 cm
b the total length of rails needed c the number of pickets needed and the length of timber needed to make these pickets d the total cost of the fence.
Click on the icon for a worksheet with more problem solving questions.
PRINTABLE WORKSHEET
126
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
Unit 58
Review of chapter 7
Review set 7A 1 Read the scales: a
b
c
d
F
5L
80 100 120 60 140 40 160 KM/H 180 20
E
0
FUEL
2 1
200
2 Convert: a 356 cm to mm d 83 000 kg to t
b e
3
3200 g to kg 7:63 m to mm
3 Find the perimeter of: a
4
kg 5 0 6
c f
450 m to km 630 cm to m
b 12 cm
3.8 m
17 cm
4 If the scale is 1 : 500 000 find: a the actual length if the scale length is i
3:8 cm
ii
6:4 cm
iii
12:2 cm
iii
130 km
b the scale length if the actual length is i
50 km
ii
22 km
5 Look at the scale diagram. Use your ruler to find the actual dimensions given that the scale is 1¡¡:¡¡2000. Which of the following could it represent? A C
a bathroom a swimming pool
B D
a beach towel a sports field
6 Kym competes in the 200 metre, 400 metre, 800 metre, 1500 metre and 5000 metre running events on sports day. How many kilometres does she run? 7
a Find the total mass in kg of 1500 oranges if the average mass of an orange is 180 g. b If a truck can carry 1400 kg of soil, how many truckloads will be needed to remove 42 tonnes of soil?
8 A rectangular farming block with dimensions as shown is to be fenced with a 3-strand wire fence. a Determine the perimeter of the block. 180 m
b Determine the total length of wire required. c If the wire costs $1:75 per metre, find the total cost of the wire.
320 m
MEASUREMENT (LENGTH AND MASS) (CHAPTER 7)
127
Review set 7B 1 Read the gauge for the amount of electricity used: 10 000 9 0 1 8 7
2 Convert: a 3480 g to kg d 5:4 m to cm
b e
3 Find the perimeter of: a
b
100
1 000
6 5 4
8623 mm to m 13:2 t to kg
2 3
2 3
1 0 9 4 5 6
9 0 1 8 7
8 7
6 5 4
10 2 3
2 3
1 0 9 4 5 6
KILOWATT HOURS
c f
KWH
8 7
9 0 1 8 7
6 5 4
2 3
4:6 g to mg 13:3 km to m
13.2 km 10 cm 6 cm
8.3 km
4 If the scale is 1 : 2 500 000 determine: a the actual length if the scale length is
i
4:8 cm
ii
0:7 cm
b the scale length if the actual length is
i
120 km
ii
98 km
5 This is a scale diagram of a toy cat. Use your ruler and the given scale to find: a the length of each whisker b the distance between the tips of the ears.
Scale: 1 : 10
c If the length of the cat’s tail is 2:5 cm on the diagram, how long is the cat’s actual tail?
6 At the hardware store, Max bought 4 offcuts of timber, measuring 500 mm, 750 mm, 400 mm and 800¡¡mm long. How many metres did he buy in total? 7 How many 25 cm rulers placed end to end are needed to measure to a length of 3:5 m? 8 If a bag of nails contains 50 nails and each nail weighs 45 g, find the total weight of 100 bags of nails. 9 How many 1:8 kg bricks can be carried by a truck which has a maximum allowable carrying mass of 3:6 tonnes? 10 Find the total length of edging required to surround the lawn and two garden beds shown.
5m
8m
8m 5m 20 m
16 m
128
MEASUREMENT (AREA AND VOLUME)
Unit 59
Area (square units)
The area of a figure, no matter what shape, is the number of square units (unit2 or u2 ) it encloses.
1 mm 2
1 square millimetre (mm2 ) is the area enclosed by a square of side length 1 mm.
1 cm 2
1 square centimetre (cm2 ) is the area enclosed by a square of side length 1 cm. 1 square metre (m2 ) is the area enclosed by a square of side length 1 m. 1 hectare (ha) is the area enclosed by a square of side length 100 m. 1 square kilometre
(km2 )
This area is 100 mm 2.
10 mm 10 mm
is the area enclosed by a square of side length 1 km.
1 cmX =100 mmX
Exercise 59 1 Find the area in square units of each of the following shapes: a b c
2
a Check to see that the following shapes all have the same area. b What is the perimeter of each? i ii
CHAPTER 8
d
iii
iv
v vi
vii
vii
c What does this exercise tell you about the area and the perimeter of a shape? 3
a In the given sketch, how many tiles have been used for i the floor
ii the walls?
(Do not forget tiles behind and under the sink cabinet and in the shower.) b These tiles are only sold in square metre lots. There are 25 tiles for each square metre. How many square metres need to be bought? c The tiles cost $36:90 per square metre and the tiler charges $18:00 per square metre to glue them. What is the total cost of tiling?
MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)
129
Conversion diagram ¥100
¥10 000
mm2
cm2 £100
For example, to convert:
¥10 000 m2
£10 000
£10 000
6
b e h k n q t
£100
² 350 000 m2 to ha smaller unit to larger ) divide
0:56 m2 = 0:56 £ 10 000 cm2 = 5600 cm2 4 What units of area would most sensibly of the following? a the floor space in a house b c wheat grown on a farm d e a freckle on your skin f g microchip for a computer h i postage stamp j k sheep station l m fingernail n
km2
ha
² 0:56 m2 to cm2 larger unit to smaller unit ) multiply
5 Convert: a 452 mm2 to cm2 d 3579 cm2 to m2 g 550 000 mm2 to m2 j 4400 mm2 to cm2 m 0:7 cm2 to mm2 p 0:8 m2 to cm2 s 0:5 km2 to ha
¥100
350 000 m2 = 350 000 ¥ 10 000 ha = 35 ha
be used to measure the areas
Remember to change larger units to smaller units we multiply, while to change smaller units to larger units we divide.
a dog’s paw carpet for a doll’s house Tasmania bathroom mirror your school grounds suburban railway station pupil of your eye 7:5 m2 to cm2 6:3 km2 to ha 5:2 cm2 to mm2 0:6 ha to m2 480 ha to km2 8800 mm2 to cm2 550 ha to km2
c f i l o r u
5:8 ha to m2 36:5 m2 to mm2 6800 m2 to ha 200 ha to km2 25 cm2 to mm2 6600 cm2 to m2 10 cm2 to m2
a In the given picture, how many pavers were used for: i the driveway ii the patio? b The pavers in the patio are the same as the pavers in the driveway. If there are 50 pavers for every square metre, how many square metres of paving were laid? c If the cost of the pavers is $16:90 per m2 , and the cost of laying them is $14 per m2 , what is the total cost of the paving? d One paver is 20 cm long and 10 cm wide. How far would all the pavers used in this example stretch if they were placed: i end to end in a straight line ii side by side in a straight line? e What do you notice about the answers to d i and d ii?
10 rows of 28 bricks
30 rows of 18 bricks
130
MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)
Unit 60
Area of a rectangle Since a square is a rectangle with equal length and width.
Area of rectangle = length £ width A=l £ w
A = length £ length A= l£l
)
) A=l2 Examples:
²
² 4.2 m
DEMO
1
16.3 m
6m
Area = length £ width = 16:3 £ 4:2 m2 = 68:46 m2
2
12 m
Area = Area 1 + Area 2 = 6 £ 6 + 12 £ 6 m2 = 36 + 72 m2 = 108 m2
Exercise 60 1 Find the area of the following rectangles: a b
c 12 km
28 mm
18 cm 40 cm 18 mm
4 km
2 Find the area of the following squares: a b
c 200 m 200 m
15 m
8.4 cm
3 Find the shaded areas: a
b
in hectares
c
2m
5 cm
12 cm
8 cm
6 cm 4 cm
20 cm
d
6m
15 m
e
f 3 m 10 m 4m 3m
12 m 2m
5 m 3m
3m 3m
20 cm 10 cm 30 cm
20 cm
MEASUREMENT (AREA AND VOLUME) (CHAPTER 8)
131
A rectangle has area 20 units2 and its sides are a whole number of units. We can find all the possible lengths L, widths W and perimeters P for it, in this way: The factors of 20 are: 1, 2, 4, 5, 10, and 20. Now,
area = length £ width, so the possible rectangles are: 20 £ 1, 10 £ 2, 5 £ 4. 1 unit
20 units
Perimeter = (L + W ) £ 2
2 units 10 units 4 units
P = (20 + 1) £ 2 = 42 units
(or 42 u)
P = (10 + 2) £ 2 = 24 units
(or 24 u)
P = (5 + 4) £ 2 = 18 units
(or 18 u)
5 units
4 Using only whole units, write all the possible lengths, widths and perimeters of the following rectangular areas. For each question, use a scale drawing to represent one of the answers: a 12 m2 b 36 cm2 c 64 km2 d 48 mm2 e 64 u2 f 144 mm2 5 Using only whole numbers for sides, write all possible areas which can be found from rectangles or squares with perimeters of: a 12 m b 20 m c 36 km Illustrate the possible answers for a. 6 A rectangular garden bed 3 m by 5 m is cut out of a lawn 10 m by 8 m. Find the area of lawn remaining. 7 A rectangular wheat field is 450 m by 600 m. a Find the area of the field in hectares. b Find the cost of planting the field if planting costs $180 per hectare. 8 A floor 3:5 m by 5 m is to be covered with floor tiles 25 cm by 25 cm square. a Find the number of tiles required. b Find the total cost of the tiles if each tile costs $3:50.
Investigation
Estimating areas of irregular shapes How can we find the area of shapes that are not regular? At best we can only estimate the answer. One method of doing this is to draw grid lines across the figure. Then we count all the full squares and, as we do so, cross them out. Then count squares which are more than half a square unit as 1 (²), and those less than half a square unit as 0. So our estimate for the total area is 26 square units.
Estimate the areas of the shapes. Is b true or false?
a
b
