Maths Connect for Jamaica Grade 3 Pupil Book

J A M A I C A Rosalyn Kelly Benita Byer-Bowen Delia D. Samuel Series editor: Karen Morrison Heinemann Educational P...

Author: Heinemann Educational Publishers

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J A M A I C A

Rosalyn Kelly Benita Byer-Bowen Delia D. Samuel

Series editor: Karen Morrison

Heinemann Educational Publishers Halley Court, Jordan Hill, Oxford OX2 8EJ Part of Harcourt Education Heinemann is a registered trademark of Harcourt Education Limited © Harcourt Education Ltd, 2006 First published 2006 11 10 09 08 07 06 10 9 8 7 6 5 4 3 2 1

British Library Cataloguing in Publication Data is available from the British Library on request. 10-digit ISBN: 0 435891 50 2 13-digit ISBN: 978 0 435891 50 3 Copyright notice All rights reserved. No part of this publication may be reproduced in any form or by any means (including photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright owner, except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Applications for the copyright owner’s written permission should be addressed to the publisher. Typeset by Tech-Set Ltd, Gateshead, Tyne and Wear Original illustrations © Harcourt Education Limited, 2006 Illustrated by Adrian Barclay and Tech-Set Ltd. Cover design by mccdesign ltd. Printed in the United Kingdom by CPI Antony Rowe, Eastbourne Cover photo: © Getty images Acknowledgements Every effort has been made to contact copyright holders of material reproduced in this book. Any omissions will be rectified in subsequent printings if notice is given to the publishers. The publishers would like to thank the following for permission to use photographs: Mike Van der Wolk pp 24, 42, iStockPhoto.com/John Arder pp 48, Mike Van der Wolk pp 48, PhotoLibrary.com 52, Mike Van der Wolk pp 99, 100, Getty Images/PhotoDisc pp 178, Mike Van der Wolk pp 179, 181. Jamaica Maths Connect has been adapted from the Maths Connect (UK) series by permission of Harcourt Education. The authors and publishers would also like to thank AW Binks, Y Ramsey, I Edwards-Kennedy, S Jackson and R Hollands for permissions to use tests from Fundamental Mathematics 3.

ii

Jamaica Maths Connect 3

Contents

To the teacher How to use this book 1

Number

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Basic numeracy Writing numbers using indices Roots Adding and subtracting with decimals Multiplying with decimals Dividing with decimals Recurring decimals Scientific notation Working with numbers in different bases

2

Sets

2.1 2.2 2.3 2.4 2.5

Review of set theory The union and complement of sets The union and intersection of three sets Venn diagrams Formulae

3

Perimeter and area

3.1 3.2 3.3 3.4

Perimeter and circumference Areas of parallelograms and trapezia Area Surface area

Revise and consolidate 1 4

Rates and proportions

4.1 4.2 4.3

Rate and speed Time, distance and speed Direct and inverse proportion

5

Working with indices

5.1 5.2

The three basic rules of indices Roots, fractional and negative indices

vii viii

1 3 5 8 10 12 14 16 19

24 26 28 30 32

35 38 40 43

49

51 53 56

61 63

Contents

iii

6

Expressions, equations and formulae

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Sequences The general term Quadratic sequences Solving simple equations Simplifying algebraic fractions Finding the subject of a formula Special algebraic notation Factorising algebraic expressions


Working with money

7.1 7.2

Foreign exchange Hire purchase

8

Volume

8.1 8.2

Cubes and cuboids Prisms

9

More about shapes and solids

9.1 9.2

The properties of quadrilaterals Viewing and drawing solids

66 68 70 72 74 76 78 79

82

84 86

89 91

93 97

10 Construction 10.1 10.2 10.3 10.4 10.5 10.6

Constructing triangles More constructing triangles Bisecting angles and lines Perpendiculars Parallel lines Constructing angles

Revise and consolidate 3

101 103 106 108 110 112

116

11 Statistics: graphs and diagrams 11.1 Frequency tables 11.2 Graphs and charts 11.3 Other statistical diagrams

119 121 125

12 Statistics: analysing data

iv

12.1 Processing data

129


133

Math Connect Jamaica Maths Connect33

13 Simultaneous equations and inequalities 13.1 Solving simultaneous equations by elimination 13.2 Solving simultaneous equations by substitution 13.3 Solving inequalities

135 137 139

14 Algebraic graphs 14.1 Graphs of functions 14.2 Straight lines and their gradients 14.3 Solving real-world problems using graphs

144 148 151


155

15 Right-angled triangles 15.1 15.2 15.3 15.4

Pythagoras’ theorem The tangent ratio The sine and cosine ratios Using trigonometry to solve real-life problems

158 161 165 169

16 Transformations 16.1 Enlargement 16.2 Scale factors, ratio and proportion 16.3 Problems involving enlargement

173 175 177


182

17 Probability 17.1 Probability 17.2 Estimating probabilities 17.3 Comparing probabilities

184 186 188

Test your knowledge

191

Answers

202

Index

227

Contents

v

To the teacher Jamaica Maths Connect is a unique course organised to fit the curriculum. It covers key areas of knowledge, understanding and skills which provide a firm foundation to raise competence and confidence in mathematics. The lively approach of the text is designed to make the study of mathematics both interesting and enjoyable. Jamaica Maths Connect, with its wealth of stimulus material, will help students to understand the relevance of mathematics to everyday problems. There are ample Investigations for project work, and illustrated case studies to show how mathematics is used by real people in many different jobs. The modules are designed to be used sequentially. Each ‘Looking back’ relates to the previous section, and ‘Revise and consolidate’ applies to all previous material. The ‘Test your knowledge’ tests at the end relate to topics throughout the book. Students should be able to use calculators where appropriate, but they should not become over-dependent on them.

To the teacher

vii

How to use this book This book is divided up into 17 colour-coded chapters. Number chapters are orange; Algebra chapters are green; Geometry chapters are blue; Measure chapters are purple; and Data Handling chapters are red. Each chapter is divided up into units.

Key words

Square root Cube root Root

Unit 1.3 Roots In this unit you will: find the square roots of positive integers find other roots of positive integers use a calculator to find roots.

Directly under the unit heading are the lesson targets. These tell you what you will learn in the lesson. To help you remember important vocabulary, there is a Key words box here. You already know that 16 is a square number and that 16 4 4. We say that 4 is the square root of 16. Finding the square root of a square number is like working out the length of the side of a square when you know its area.

In the explanation box, you can see a summary of the key ideas that are covered in the lesson. The key words are printed in bold type.

Worked examples 1. Find the square roots of a. 16 1. a. 兹16 苶 4

viii


b. 49 b. 兹49 苶

7

The worked examples show you how to answer the exercise questions. On the blue paper you can see the kind of working you should be writing in your exercise book. The green hint boxes help explain how to calculate the answers. The exercises for each lesson are made up of three kinds of question: ● practice questions, which allow you to practise the basic skills ● problem questions, which ask you to apply the skills you have learned ● investigation questions, which give you practice at solving open-ended problems At the end of each chapter are ‘Looking back’ exercises to help you review what you have learned. There are also six ‘Revise and consolidate’ exercises spread throughout the book for more review and practice work, and finally, five ‘Tests of your knowledge’ at the end.

How to use this book

ix

Key words

Natural Integers Rational Factor Multiple Prime BOMDAS

Chapter

Number You have already developed a wide range of skills in working with numbers. In this chapter we will briefly review some of the number concepts covered in Year 1 and 2 and consolidate and combine the skills you have developed so that you can apply these in new contexts and to real-life situations. The aim is to become a competent mathematician who can solve number-related problems quickly and efficiently.

1

Unit 1.1 Basic numeracy In this unit you will: revise and apply the basic number concepts you will use as you work through this book use the four operations (, , and ) to perform simple computations.

The set of natural numbers consists of the counting numbers N {1, 2, 3 …}. The set of positive integers includes 0: {0, 1, 2, 3 …}. The set of integers includes numbers less than 0, the negative numbers Z {… 3, 2, 1, 0, 1, 2, 3 …}. Rational numbers are numbers which can be written in the form of a fraction where 2 6 the numerator and denominator are both whole numbers. For example: 4, 7 and 1.5 are all rational numbers. n All whole numbers are rational because they can be written in the form . 1 Even numbers can be divided exactly by 2. For example, 4, 8, 100, 1 000 008. Odd numbers cannot be divided exactly by 2. For example, 3, 9, 11, 121, 1 000 121. A factor is a number which divides exactly into a given number. For example, 5 is a factor of 20. A multiple is formed by multiplying a given number by any integer. For example, 10 is a multiple of 1, 2 and 5 (1 10, 2 5 and 5 2). Prime numbers are numbers which have only two factors, the number itself and 1. The first five prime numbers are 2, 3, 5, 7 and 11. When we perform operations on numbers we work in a very specific order. Brackets are always solved and removed first. Remember the fraction line acts like a bracket! Multiplication and division are performed next. Addition and subtraction are performed from left to right. This order is sometimes called BOMDAS – this is a mnemonic to help you remember the order of operations, not a rule in itself. The answer to an addition problem can be called the sum. The answer to a subtraction problem can be called the difference. The answer to a multiplication problem is the product. The answer to a division problem is the quotient.

Number

1

Worked example 12 17 26 21 49 100 100 From the numbers above, write down a. a multiple of 3 b. a prime number c. all the odd numbers d. a negative integer e. the difference between the first and the last number f. the sum of all the numbers. a. b. c. d. e. f.

12 17 17, 21, 49 21 or 100 88 83

Exercise From the numbers 6, 17, 32, 41 and 49, write down a) b) c) d) e)

a multiple of 4 two prime numbers a factor of 36 a number which is the sum of two other numbers in the set the product of the largest and smallest numbers.

What is the sum of the first six odd numbers? Find the product of 100 and 4 023. Write your answer in figures and words. Write down all the prime numbers smaller than 60. Find the difference between 1 000 and 878. Write down all the factors of 19. What does this tell you about this number? Mrs James has $300.35 in her bank account. She writes a cheque for $388.75 and the bank accepts the payment. What would be the new balance in her account?

Apply the correct order of operations to calculate a) 4(12 6) (6) (3) 1 c) 2 of 36 21 72

2


b) 3(16 12) (11)(22 16) d) 12 (7 3) 2 8 2.

Key words

Base Index Indices Power Exponent

Unit 1.2 Writing numbers using indices In this unit you will: write expressions using index form evaluate expressions written in index form use your calculator to find powers.

You have already learnt about square numbers. Look at the following number pattern:

11 1

22 4

33 9

44 16

The numbers 1, 4, 9, 16, 25, 36 and so on, are called square numbers. Square numbers are formed by multiplying a whole number by itself. To check whether a number is a square number, you can ask yourself whether you could represent it as the area of a square shape. For example:

4

4

4

3

16 is a square number. 12 is not a square number.

12 is a rectangular number.

16 4 4 We can write 4 4 as 42. 42 16 We read this as ‘four squared equals 16’. 42 is 16 written in index form. Any number can be written in index form. For example, ab is read as ‘a to the power of b’. ‘a’ is called the base. ‘b’ is called the index, power or exponent. The plural of index is indices. A number written in the index form can be expanded as a product. The index tells us the number of times we multiply the base by itself. ab a a … a a

b times

XY X X … X

Y times

Number

3

Worked examples 1. Write the following numbers out in full as products. a. 34 b. 25 2. Rewrite the following in index form. a. 4 4 b. 6 6 6 3. Evaluate a. 23 b. 54

c. 22 32.

1. a. 34 3 3 3 3 b. 25 2 2 2 2 2 2. a. 4 4 42 b. 6 6 6 63 3. a. 23 2 2 2 8 b. 54 5 5 5 5 625 c. 22 32 2 2 3 3 4 9 36

On your calculator, the key xy To evaluate 64: Press

6

is used to evaluate expressions in index form.

(the base)

Press xy Press

4

(the power)

Press

Exercise Identify the base and index in each of the following expressions. a) 115 f) 48

b) 24 g) 19

c) 31 3 h) (4)6

d) 70 i) (4)2

e) 5–3 1 j) (3)5

d) 37 i) 114

e) 82 j) 62

Rewrite the following in full as products. a) 53 f) 15

b) 44 g) 28

c) 26 h) 103

Rewrite in index form: a) c) e) g) i)

66 77777 5555 8 999999

b) d) f) h) j)

3333 222 44444 2222222 444

Evaluate the following and use your calculator to check your answers. a) 24 e) 25 i) 52 31 4


b) 34 c) 43 f) 53 g) 17 j) (3)2 (2)3

d) 122 h) (4)2 23

Key words

Square root Cube root Root

Unit 1.3 Roots In this unit you will: find the square roots of positive integers find other roots of positive integers use a calculator to find roots.

You already know that 16 is a square number and that 16 4 4. We say that 4 is the square root of 16. Finding the square root of a square number is like working out the length of the side of a square when you know its area. For example: 3 2 1 cm2 1 1

2

3

area 1 cm2 side 1 cm 11



We use a special sign to show square root: For example 81 9. We read this as ‘the square root of 81 is 9’.

Worked examples 1. Find the square roots of a. 16

b. 49.

2. Evaluate the following. a. (4 ) 9 3. 182 324. Use this to find 1. a. 16 4

a. 19 18

c.

36 49

b. 17 18.

b. 49 7

2. a. (4 ) 9 4 9 23 6 c.

b. (144 81)

36

b. (144 81) 144 81 12 9 108

36 49 7 49 6

3. a. 19 18 (18 18) (1 18) 324 18 342 b. 17 18 (18 18) (1 18) 324 18 306

Number

5

Exercise Find the square roots of the following positive integers. a) 4 f) 144

b) 121 g) 9

c) 36 h) 16

d) 49 i) 64

e) 81 j) 10 000

Write the square roots of the products of these numbers. a) 16 25 e) 400 9 i) 49 900

b) 4 81 f) 64 100 j) 36 10 000

c) 100 36 g) 121 4

d) 9 49 h) 81 400

Find the value of the following by factorising into well-known square numbers. a) 900 d) 196

b) 400 e) 4 900

c) 2 500 f) 360 0 00

2 500 25 100

Evaluate a)

64

b)

25

d)

16

e)

121

81 25

4

c)

9 1

100

A square tile has an area of 49 cm2. What is the length of one side? Find the perimeter of a square field which has an area of 625 m2. Use a calculator to determine whether the following are square numbers or not. a) 225

b) 7 921

c) 16 348

d) 55 696

e) 0.9

f) 0.3249

122 144, 162 256 and 402 1 600. Use these to write the value of a) 17 16 d) 15 16

b) 41 40 e) 39 40

c) 13 12 f) 11 12.

Investigation The rules for finding the length of the missing side of this type of triangle are ● find the square of both of the given sides ● add the two answers together ● find the square root of this answer to get the length of the missing side. Follow these steps to find the lengths of the missing sides of these triangles.

7 3

B 7

A

4

5

C 14

You will learn more about this in Chapter 15 when you study Pythagoras’ theorem. 6


Other roots It is possible to find roots of numbers other than square roots. For example, 2 2 2 8 can be written as 23 8. We say ‘2 cubed equals 8’. The cube root of 8 is 2. Again, you can think of this as finding the length of the side of a cube if you know the volume of the cube. To show that we are working with a cube root, we write a small 3 in front of the root sign. 3

8 2

So

This is read as ‘the third root of 8 is 2’ or ‘the cube root of 8 is 2’. Because we can write numbers to any index, it follows that mathematically, we must be able to find any root: 4 5. so 54 5 5 5 5 625 and 625 This is read as “the fourth root of 625 is 5”. x

On your calculator, the key y is used to find roots other than squared or cubed roots. The method you will use on your calculator will depend on the make and model you are using. These are the two most common methods but you should refer to your calculator manual to make sure you know how to do this. 4

: To evaluate 625 Method 1: Press

4

Method 2: (the root)

Press

6

2

5

x

Press y 6

Press

x

2

Press y

5

(the number you are finding the root of)

Press

Press

Press

4

Exercise In the expressions below, there are missing numbers. Complete the expressions. 4

a) 24 16, so 16 □

b) 53 5

c) 10 100 000, so 0 100 00 10 e) 43 64, so 3

3

, so 125 5 □

d) 35 243, so 243 3

4

Use your calculator to evaluate the following. 3

a) 216

4

b) 2 401

8

c) 256

5

d) 3 125

6

e) 729

Number

7

Key words

Unit 1.4 Adding and subtracting with decimals

Estimate Rounding

In this unit you will: add whole numbers and decimal numbers using standard written methods subtract whole numbers and decimal numbers using standard written methods estimate answers by rounding. When adding or subtracting using column methods: a) Write an estimate of the answer by rounding each number. For example, you can find an estimate of 87.19 19.58 by adding 90 to 20, to give an estimate of 110. b) Write the numbers in columns underneath each other, making sure each digit is in its correct column. c) Compare the answer with the estimate as a check. In the example above, the accurate answer is 106.77, quite close to the estimate. When adding: Start adding from the right, carrying into the next column on the left, if necessary. When subtracting: Start subtracting from the right, ‘requesting’ from the next column on the left when necessary. You can check an addition by subtracting, and a subtraction by adding. To check if a b c, does c b a? To check if a b c, does c b a? When adding or subtracting fractions, you can convert the answer to a decimal, or convert each fraction to a decimal first.

Worked example

Work out a. 25.7 12.39 4.28 a. Estimate 26 12 4 10

b. 1.73 6.5 7 0.29

Method 1 25.7 12.39 4.28 25.7 12.39 gives your first result, then subtract the 4.28 from your first result to get the answer. Step 1

61

25 . 70

Insert the zero.

Round to whole numbers. Method 2 25.7 12.39 4.28 Find out how much you want to take away from 25.7 altogether, then take that total away. 1 2 . 39 Step 1 4 . 28

1 2 . 39

1 6 . 67

13.31

11

Step 2

21

13.31 4 . 28 9 . 03

Check estimate. ✓

8


Step 2

61

25 . 70 1 6 . 67 9 . 03

Check estimate. ✓

b. Estimate 2 7 7 0 16 1 . 73

7 7 units, so it goes to the left of the point.

6 . 50 7 . 00 0 . 29

Starting from the right: 3 9 12, put 2 down and carry 1 ten to the next column.

1 5 . 52 1 1

1

Check estimate 15.52 is close to 16.

Exercise Work out the following.

Remember to estimate first.

a) 6.2 4.39 7.8 b) 5.9 0.37 6 2.15 c) 4.8 13 5.72 0.325 d) 23.2 1.87 165 4.9 e) 3.42 5.8 6.09 18 4.6 f) 23.1 5.78 6 9.02 0.005 Now check your answers with a calculator. Remember to estimate first.

Work out:

a) 45.7 23.4 b) 12.9 8.3 d) 42.91 3.25 e) 6.2 5.42 Now check your answers with a calculator.

c) 26.43 2.51 f) 9 6.8

If a 3.42, b 6.7 and c 5.29, find the value of: a) a b

b) c a

c) a b c

d) b a

e) a b c

Four lengths of wood measure 3.2 m, 4.35 m, 2.95 m, 2.725 m. How much is this altogether?

Sam is 152 cm tall, Andy is 124.5 cm tall. How much taller is Sam than Andy? Find the missing values in these additions and subtractions. a)

b)

12.34 ?

15.83

9.35 ? 2.15

c)

8.57 ?

d)

? 6.53

6.79

19.7

e)

? 5.327 9.513

Use the calculation 37.64 2.3 35.34 to write down the answer to these mentally. a) 37.64 12.3 c) 27.64 2.3

b) 37.64 22.3 d) 35.34 2.3

Using the digits 2, 3, 4, 5, 6, and 7 only once, find pairs of numbers that have these totals: a) . . 6 . 0

b) 3

. . 1 2 . 6

9

Using a set of 0–9 digit cards, make two three-digit numbers. Place a decimal point in each number. Toss a coin – heads for addition and tails for subtraction. Now complete the calculation. Do this several times. Number

9

Key words

Unit 1.5 Multiplying with decimals In this unit you will: multiply 3-digit by 2-digit whole numbers or decimal numbers using a column method understand where to position the decimal point estimate the result of a calculation by rounding check the result by comparing it with the estimate.

Estimate Multiplication Standard method

Always start with an estimate. a) Estimate the answer by rounding, then multiplying the rounded numbers. b) Convert the multiplication into one involving whole numbers and division by powers of 10. c) Complete the multiplication. d) Divide by the powers of 10 to locate the position of the decimal point. e) Compare the result with the estimate to check if it is sensible. A standard method for: 37.4 0.93: a) Estimate is 40 (40 1).

➝

374 93 1 000

34.782

34 782 1 000

➝

37.4 0.93

➝

b) Convert the multiplication to 374 93 1 000. c) Complete the whole number multiplication, giving 34 782. d) Divide, i.e. 34 782 1 000 34.782. e) Compare with the estimate.

Worked examples 1. Calculate 6.83 2.4 1. Estimate 7 2 14 6.83 2.4 683 24 1 000 68 3

6.83 100 683 2.4 10 24 Reverse 100 10 by 1 000.

24 1 3 660 2 732 1 6 392

20 4 add together

16 392 1 000 16.392 Check estimate.

10


The multiplication can be done by any method, e.g. 683 683 20 4 13 660 2 732

2. A garden path measures 9.7 m long by 0.85 m wide. What is the area of the path? 2. Estimate 10 1 10

9.7 10 97 0.85 100 85 Reverse 10, 100 by 1 000.

9.7 0.85 97 85 1 000

7

7 2 00

80 7 200 560

5 60

5

90 450

35

450 35 8 245

8 245 1 000 8.245 m2 Check estimate.

Exercise Rewrite these calculations to give whole number multiplications. The first one has been done for you. a) 8.91 3.2 891 32 1 000 d) 32.4 4.3

b) 3.8 4.6 e) 9.82 7.6

c) 7.21 5.2

Complete the calculations in question 1 using a method of your choice. Remember to give an estimate for each one first.

Find the area of a room 5.3 m by 4.9 m. Wood costs $109.75 per metre. How much will 3.8 m cost? 24 people go to the cinema. If a ticket costs $175.50, how much is this in total? I want to put a picture 11.3 cm long by 6.2 cm wide on my T-shirt. If the printers charge 27.5c per square centimetre, how much will this cost?

Use the digits 1, 2, 3, 4, 5 once only in a decimal multiplication to make the answer 71.28. If 3.25 2.8 9.1, find the value of the following without using a full written method. a) 32.5 2.8

b) 3.25 0.28

c) 0.325 0.28

For this question, work in pairs or groups. You want to make the answer 32.5 by multiplying two numbers together. a) Choose two numbers to get as close as you can to 32.5. You may not use 1 or 2. Set a time limit. The person who is closest scores 5 points. b) Now adjust your numbers to get closer to 32.5. The person who is closest scores 5 points. c) Continue to do this two more times. If anyone gets the number exactly, they score 20 bonus points. d) Now the winner chooses an answer to make. Number

11

Key words

Unit 1.6 Dividing with decimals In this unit you will: divide whole numbers and decimal numbers by decimal numbers, using a standard written method understand where to position the decimal point estimate the result of a calculation by rounding check the result by comparing it with the estimate.

Division Divisor Estimate Inverse operation

Division by a number is equivalent to repeated subtractions of that number. We call the number we are dividing by the divisor. Always start with an estimate. a) Estimate the answer by rounding, then dividing the rounded numbers. b) Convert the division into one involving a whole number divisor by multiplying both numbers by a power of 10. c) Complete the division. d) Compare the result with the estimate to check if it is sensible. The answer to a division can be checked using the inverse operation of multiplication, e.g. if a b c, check to see if c b a.

Worked examples 1. Work out 29.67 4.6 2. Jess says that 149.76 4.8 33.2 The key on my calculator is broken. How can I check the answer? 1. Estimate: 30 5 6 29.67 4.6 296.7 46 46 2 9 67 . 2 76.

46 6

20.7 1 8 .4 2 .3 2 .3

46 0.05

0

6.45

46 10 460 46 5 230 46 6 276

46 0.4

Answer is 6.45 Check estimate 2. 149.76 4.8 33.2 33.2 4.8 159.36 Jess’ answer is wrong.

12

To make the divisor a whole number, multiply both numbers by 10.


1 46 0.01 0.46 100 5 46 0.05 2.3 100

46 goes into 296.7, 6.45 times, so 4.6 goes into 29.67, 6.45 times. Check using the inverse operation. Inverse operation of by 4.8 is by 4.8.

3. Work out 0.065 0.0032, correct to 1 d.p. 3. 0.065 0.0032 is equivalent to 650 32 Estimate 600 30 20 2 0.3 1

Multiply both numbers by 10 000 to make the divisor a whole number. Sometimes it’s easier to do this before making the estimate.

3 2)6 5 0 64 0

32 20

10 9.6

32 0.3

0.4 0 . 32

32 0.01

Answer is 20.3, correct to 1 d.p.

Exercise Rewrite these calculations to give a whole number divisor. The first one has been done for you. a) 3.28 0.4 32.8 4

b) 83.4 0.3

c) 36.12 2.1

d) 4 0.32

e) 0.02106 0.065

Remember: the divisor needs to be a whole number – multiply by 100.

Complete the calculations in question 1 using a method of your choice. Remember to give an estimate for each one first.

Use a calculator to check both sides of the equations in question 1. Did multiplying both numbers by the same amount change the answer?

How many 1.6 m pieces of rope can be cut from a piece 37.44 m long? a) How many pens costing $10.35 can I buy with $180? b) How much is left over?

a) Complete the calculation 67.5 5.4 b) Check your answer by working backwards using the inverse operation.

Find the odd one out. a) 220.8 6.9

b) 2208 69

c) 22.08 6.9

d) 22.08 0.69

Check these calculations by working backwards using the inverse operation. a) 43.2 2.4 18

b) 32.86 6.2 5.3

c) 30.55 9.4 3.25

If 85 6.8 12.5, find the value of the following without a full written calculation. a) 850 6.8

b) 85 68

c) 85 0.68

d) 8.5 6.8

Number

13

Key words

Unit 1.7 Recurring decimals

Fraction Decimal Recurring decimal

In this unit you will: recognise recurring decimals convert a fraction into a recurring decimal.

To convert a fraction to a decimal, divide the numerator by the denominator using written division or a calculator. Sometimes the fraction is equivalent to a recurring decimal: . 5 1.6666666, written as 1.6 3 The 6s continue indefinitely, and we show this by writing a dot above the 6. We read the decimal as ‘one point six recurring’. Examples of recurring decimals are: . 1 0.3333333… 0.3 3 . 2 0.6666666… 0.6 3 . 1 0.1666666… 0.16 6 . 1 0.1111111… 0.1 9 .. 19 1.7272727 1.72 This has a pair of recurring digits. 11 . . 6 0.857142857 0.857142 This has a set of 6 recurring digits. 7

Worked examples 1. Is 611 equivalent to a recurring decimal? 0.5454 . 0 1. 116 5 .5

11 0.5

0 .5 0 0 .4 4

11 0.04 You can see the same pattern is going to repeat.

0 .0 6 0 0 .0 5 5

11 0.005

0 .0 0 5 0 0 .0 0 4 4

11 0.0004

6 6 11

14

.. is the recurring decimal 0.54


2. What fractions are equivalent to these recurring decimals? a. 2.3333333… 7 3 4 9 9 9

2. a. b. c.

b. 0.444444…

c. 0.999999…

. 1 . 1 7 0.3 3; 2.3 2 3 3 . 1 . 4 0.1 9; 0.4 9; . 9 0.9 is shown as 9

1

Exercise Without using a calculator, write each fraction as a recurring decimal. a)

5 9

7 11

b)

c)

5 12

7 6

d)

e)

What fractions are equivalent to these recurring decimals? a) 0.333333…

. b) 0.1

. c) 0.6

7 15

. e) 0.7

d) 0.1666666…

Which of the following are recurring decimals? a)

13 27

6 7

b)

c)

9 12

4 11

d)

e)

1 19

Kim has $500 to divide between 3 people. Can they all receive exactly the same amount using the full $500?

Ali and James are trying to decide who has the biggest answer. Ali gets 590 and James 2

gets 11. By changing into decimals, explain who has the biggest answer.

True or false? a) b) c) d) e)

7 divided by 5 will give a recurring decimal. 9 I get a recurring decimal when I divide by 9, except for 9. . 0.3 and 0.3 are the same. You never get a recurring decimal when you divide by 2, 4, or 5. I always get a recurring decimal when I divide by 6.

Investigation a) Copy and complete the table for denominators from 11–20. Write your answers as recurring decimals when possible, otherwise to four decimal places. Denominator 11 12 13 14 15 16 17 18 19 20

1 d

.. 0.09

2 d

.. 0.18

3 d

4 d

5 d

6 d

.. 0.54

7 d

8 d

9 d

0.2

b) Write a sentence or two about each denominator. Can you make any general comments? Number

15

Key words

Unit 1.8 Scientific notation In this unit you will: write and work with numbers in scientific notation (standard form).

Scientific notation Standard form

The range of numbers we work with in mathematics is very large. For example, the population of the world in 2005 was estimated to be 6.4 billion, whilst the diameter of atoms that scientists are working with can be less than one-millionth of a centimetre. When we have to write or type these numbers out using numerals, they take up space, it takes a lot of time and it is easy to make mistakes plus, calculators can only display a certain number of digits for bigger or smaller numbers they have to express them differently. For example, if we enter 6 000 000 12 000 on a calculator the display shows 7.210. This means the answer is 7.2 1010 or 72 000 000 000. 7.2 1010 is a method of writing numbers called scientific notation, or standard form. When a number is written in scientific notation, two parts are multiplied together: ● a number between 1 and 10 (including 1, but not including 10) ● a power of 10. So, we could write the numbers 7 000 and 8 600 in scientific notation like this:

7 000

8 600

7 10 3

8.6 10 3

number from 1–10

power

number from 1–10

power

This means that 7 000 is 7 groups of 103 or 7 groups of 1 000 and 8 600 is 8.6 groups of 103. The power (index) tells you how many times you must multiply the number by ten. “The population of St. Lucia is 1.5 x 105” The population of St. Lucia is written here in standard form or scientific notation. 1.5 105 150 000 Therefore, we can also say that the population of St. Lucia is 150 000. Very small numbers can also be written in scientific notation using negative powers. For example, it takes a laser printer 2.5 105 seconds to print a single letter on a page. In this case, the negative power (index) tells you by what should divide to find the value of the number (work from the decimal point). In this case, 2.5 10 000 0.000 025 seconds.

16


Worked examples 1. Express the following numbers in standard form. a. 1 600 b. 0.56 c. 73 400 2. Express these numbers in ordinary form. a. 7.2 104 b. 4.91 102 1. a. 1 600 1 600 1.6 103 b. 0.56 0.56 5.6 101 c. 73 400 73 400 7.34 104 d. 0.0089 0.0089 8.9 103 2. a. 7.2 104 7.2 10 000 72 000

d. 0.0089

Shift the digits 3 places to the right to obtain 1.6. Shift the digits 1 place to the left to obtain 5.6. Shift the digits 4 places to the right to obtain 7.34. Shift the digits 3 places to left to obtain 8.9. b. 0.0491

Exercise The number 3 600 000 can be written as 3.6 million. Write these numbers in millions. a) 60 000 000

b) 9 900 000

c) 12 890 000

d) 1 342 000

e) 500 000

f) 234 000

a) 1 million

b) 1 hundred

c) 10 thousand

d) 100 000

e) ten.

Write as powers of 10:

Write in scientific notation: a) 49

b) 500

c) 1 800

d) 14.2

e) 165 000

f) 0.7

g) 0.0095

h) 0.04

i) 0.000 832

j) 0.000 076

k) six billion

l) three million.

Express the following numbers in ordinary form. a) 1.2 103

b) 3 102

c) 8 105

d) 2.59 106

e) 9.1 104

f) 5 101

g) 4.73 104

h) 3.6 102

i) 1.8 103

j) 2.07 105

Number

17

The table shows the estimated population of some countries in 2003. Write each population in standard form. Country

Afghanistan

The Bahamas Barbados

Canada

Cuba

Population

29 000 000

297 000

277 000

32 000 000

11 000 000

Country

Dominica

Fiji

Iceland

Japan

New Zealand

Population

69 600

868 500

281 000

127 000 000

3 950 000

The mass of the Earth is approximately 5.98 1024 kg and the mass of the Sun is approximately 1.97 1030 kg. a) Write each mass in full. b) Write your own mass in scientific notation. c) About how many times heavier is the Earth than you?

Give three examples of jobs in which people might use scientific notation regularly. For each one, say how you think they might use it.

Investigation In astronomy, scientists have to deal with vast distances between planets and other bodies in the universe. To talk about these distances, they use a unit called a light year. Find out what a light year is and how it is calculated. Draw a diagram of the Solar System. Name the planets and give their approximate distance from the Sun in light years.

18


Key words

Unit 1.9 Working with numbers in different bases

Denary Base Binary

In this unit you will: review reading and writing numbers in bases other than 10 add, subtract and multiply numbers in any base.

You have already learnt that our number system is a decimal, or denary system. In other words, it uses the number ten as a base for counting. In this system, all numbers can be written using just ten digits and as multiples of powers of ten. You should remember this from your work on place value. 105 (HTh)

104 (TTh)

103 (Th)

102 (H)

101 (T)

100 (U)

8

1

2

3

5

6

Remember that any number to the power of 0 1. If we write this number in expanded form, we get: 8 105 1 104 2 103 3 102 5 101 6 100 8 100 000 1 10 000 2 1 000 3 100 5 10 6 1 800 000 10 000 2 000 300 50 6 812 356 We can use any natural number (except 1) as a base. In Ancient Babylon, the counting system used 60 as a base. For example, if we use 5 as a base, then 3425 means 52

51

50

3

4

2

3 52 4 51 2 50 3 25 4 5 2 1 75 20 2 97 In base 5 we would count: 1, 2, 3, 4, 105, 115, 125, 135, 145 … ● We use the digits up to the one preceeding the base, i.e. up to 4 for base 5, up to 7 for base 8 and so on. ● 105 means 1 five and 0 ones. ● 115 is read as ‘one, one, base five’ and not ‘eleven five’. Two is the smallest base we can use as that leaves us with two digits, 0 and 1! Base two is also known as the binary system and is used extensively in computer programming. The binary number 11012 means: 23

22

21

20

1

1

0

1

1 23 1 22 0 21 1 20 1 8 1 4 0 2 1 1 8401 13 Number

19

Exercise Fill in the missing words. a) The

system uses two digits.

b) The base seven system is based on

digits.

c) In a base 9 system, you would count normally up to numbers differently.

, then express

Write down the digits you would use in the following counting systems. a) base three

b) base six

d) base seven

e) base nine

c) base eight

How would you write the numbers from 1 to 10 in the following base systems? a) base three

b) base five

d) base seven

e) binary

c) base six

Give the value of the underlined digit in the given number system. a) 110001 denary

b) 110001 binary

c) 122 in base 3

d) 3321 in base 4

e) 123 in base 8

f) 123 in base 9

g) 123 in base 12

You should remember that the base is always given when you are working in a different number system: 12 in base 8 12 eight or 128 341 in base 7 341 seven or 3417 To convert a number in the denary system to a different base system: ● Divide the number by the base, writing both the quotient and the remainder. ● If the quotient is not equal to 0, continue dividing until the quotient does equal 0. ● Write the result by writing down the remainders in order, starting with the last remainder.

Worked example Convert a.

a. 124 to base 8

12 4 8 15 r 15 1 r 7 8 1 0 r 1 8

b. 17 to base 2. b.

4 124 1748

17 2 8r1 8 4 r 0 2 4 2 r 0 2

2 2 1 2

20


1r0 0r1

17 100012

Exercise Convert the following base ten numbers to numbers in the respective bases. a) 63 to base 4 e) 1 543 to base 8 i) 450 to base 9

b) 15 to base 2 f) 8 to base 2 j) 60 to base 2

c) 34 to base 5 g) 22 to base 6

d) 110 to base 3 h) 45 to base 7

To convert a number in any base system to a base ten number, use the place value chart for the required base.

Worked example Convert the following numbers to the denary system. a. 2438 b. 14235 a.

64

8

1

2

4

3

b.

125

25

5

1

1

4

2

3

2438 (2 64) (4 8) (3 1) 14235 (1 125) (4 25) (2 5) (3 1) 128 32 3 125 100 10 3 163 238

Exercise Convert the following numbers to base 10. a) 357 g) 5556

b) 101112 h) 768

c) 1234 i) 1012

d) 112113 j) 12113

e) 4356 k) 45689

f) 100002 l) 34658

To add numbers in a base number system other than the denary system: ● Place the numbers in columns, ensuring that the digits corresponding to the same place values are aligned. ● Perform the addition by adding digits with the same place value. If the sum of these digits is not in the base number system, divide the number by the place, putting the remainder as part of the result. Remember to add in the quotient ● Continue adding the digits in the other columns. obtained in the preceding division.

Worked example Find a. 4637 3467 b. 1 1012 1012 11112 a. 4637 3 6 9; 9 7 1 r 2 3467 1 6 4 11; 11 7 1 r 4 1 1427 1 4 3 8; 8 7 1 r 1

Number

21

b.

1 1 1 1 1 1 0 0 0

0 0 1 0

12 12 12 12

1 1 1 3; 3 2 1 r 1 1 1 2; 2 2 1 r 0 1 1 1 1 4; 4 2 2 r 0 2 1 1 4; 4 2 2 r 0; 2 2 1 r 0 2 in binary ⇒ 10

To subtract two numbers in a base number system, do just as you would in the denary number system. However note that you are not subtracting in the base ten system. For example, to find 5426 2546: 4

5 2 2

63 6

4 26 5 46 4 46

To multiply two numbers, perform normal multiplication, but again, if the product is not in the required base, divide it by the base, putting the remainder as part of the result. Continue multiplying digits, ensuring that you add the quotient obtained in the preceding division to the product. For example, to find 1558 278:

1558 278 13738 33208 47138

Exercise Find

a) 13245 + 3325

b) 221023 101123

c) 112 1012 11112

d) 31234 + 12234 111234

e) 455468 55558

f) 88889 77779.

Write the answers to the problems above in the base ten system. Find a) 12516 15426

b) 41057 25267

c) 101102 11112

d) 34215 34425

e) 300004 – 133334

f) 444445 44445.

Convert the answers to the problems above to numbers in the base ten system. Find

22

a) 1112 102

b) 2123 113

c) 2548 68

d) 13214 124

e) 4245 215

f) 5647 457.


Looking back Write down all the factors of 24, 36, 325 and 100. List the prime numbers smaller than 100. Is 441 a square number? Explain why or why not. Calculate a) 122

b) 26

Evaluate

a) 144 9

c) 105

b) 16 9 4

d) 83 c) 9 4 25

e) 124

f) 193.

121

e)

36

d)

81 64

Estimate the answers to the following calculations by rounding them to the nearest whole number. a) 3.71 (4.2 7.5)

b) 17.5 (3.4 1.7)

c) 6.8 (6.3 2.6) (2.5)2

a) Convert the following fractions into decimals. i)

5 9

ii)

1 4 27

iii)

1 8 45

iv)

2 2 7

b) Which are recurring decimals?

Calculate:

a) 63 9 (2) 5 c) 12 (8) 2 12 e) 8 (18) (6 )(4) (4)

b) 48 (2 2) d) 48 (7 (6)) 40 (10)

The table shows the estimated population of some countries in 2003. Write each population in standard form.

The table shows the estimated area of some countries in 2003. Write each area in ordinary form.

Convert: a) 56 to base 5

Country

Population

Antigua and Barbuda 67 900 Australia

19 732 000

Belize

266 000

Brazil

182 033 000

Haiti

7 500 000

Country

Area (square km)

France

5.47 105

Grenada

3.4 102

India

3.288 106

Ireland

7 104

Mauritius

2.04 103

b) 145 to base 3

c) 29 to base 2.

Convert the following to numbers in base 10. a) 869

Evaluate:

b) 1457

a) 10112 112 11012 10012

c) 10234 b) 54326 43556

c) 5418 78.

Number

23

Chapter

2

Key words

Sets In this chapter you will review the vocabulary and concepts related to sets that you have already covered in Year 1 and 2. You will then expand on your understanding of sets to deal with the union and intersection of three sets. You will work with Venn diagrams to represent sets and relationships between them, and learn how to apply a formula to calculate the number of elements in a set.

Subset Intersection Union Complement Disjoint

Unit 2.1 Review of set theory In this unit you will: review the vocabulary used to talk about sets revise basic set theory and solve problems to consolidate your knowledge. The photographs show you three sets. Remember that a set is a well-defined collection of objects. A

B

C

The items in a set are called the elements of a set. Set A has four elements. It is a finite set. A set with no elements is called an empty set {} or . The bangle in set A is not an element of set B. We use the symbol to show that something is not an element of a set. Equivalent sets have the same number of elements, set A set B. Equal sets have exactly the same elements. None of these sets are equal. A subset contains some elements of a set. For example set C set B. The set itself and the empty set are always subsets of a set. The number of subsets of a set with n elements is 2n. The intersection of two sets is the set of elements which belong to both sets. If you look at the sets above, then A B {ring, earring and cross}. The union of two sets is the set of elements which contains all the elements of both sets. If you look at the sets above, then A B {earring, ring, bangle, cross, chain}. The complement of a set X, called X, refers to the elements that are part of the universal set, but which do not fall into set X. So for example, if the universal set was the set of gold and silver jewellery and set X {silver jewellery} then set X {gold jewellery}. Disjoint sets have no common elements. Sets A and C are disjoint. The universal set is the general set to which a set belongs. The universal set to which A, B and C belong could be {jewellery}. The number of elements of a set is written as n(A) 4. 24


Exercise Say whether the following statements are true or false. a) 3 {odd numbers} b) 36 {square numbers} c) {2, 3} {prime numbers} d) U {letters of the alphabet}, A {vowels}, A {consonants} e) {Vowels} {Caribbean capital cities} f) {} {apples, oranges, melons}

Give an example of two equal sets. Without listing the subsets, say how many subsets each of the following sets will have. a) A {a, b, c, d, e} b) B {one, two, seven, nine, eleven, twelve} c) X {Grenada, Dominica, St Lucia, St Kitts} d) Y {the fingers on both your hands} e) M {odd numbers smaller than 15}

Give an example of two disjoint sets. Give a universal set for each of the following. a) {2, 3, 5, 7, 9} b) {33, 44, 55} c) {Castries, Bridgetown, Guadeloupe, St John’s} d) {Peso, Gourde, Franc, Guilder, Dollar} e) {dodecahedron, icosahedron}

If U {whole numbers from 1 to 12}, A {5, 6, 7, 8} and B {8, 9, 10, 11}, write down a) A B

b) A B

c) n(A B)

d) n(A) n(B)

e) A

f) B.

If U {the members of a Caribbean cricket team}, state which of the following are subsets of U. a) wicket keepers

b) linesmen

d) batsmen

e) fielders

c) referees

Sets

25

Key words

Unit 2.2 The union and complement of sets In this unit you will: find the complement of a set find the union of two sets.

Union Complement Universal set

The union of two sets is the set of elements belonging to the first set or the second set. The symbol for union is . To find the union of two sets A and B: ● List the elements of both sets. ● Write down the elements contained in set A or set B. If an element is contained in both sets, only write that element once. A {yellow, red, blue, black, green}, B {red, black, purple, orange, green, pink} A B {yellow, red, blue, black, green, purple, orange, pink}

The complement of a set A, denoted A or Ac, is the set of all elements not contained in A but contained in the universal set. If you look at the family above: M {Males} Find the members of M M {Members of the family who are not males} So, M {Mrs James, Betty, Joyce, Debra, Colleen}

Worked examples 1. U {whole numbers from 1 to 25}

A {factors of 24}

Find A B. 1. A {1, 2, 3, 4, 6, 8, 12, 24}, B {3, 6, 9, 12, 15, 18, 21, 24} A B {1, 2, 3, 4, 6, 8, 9, 12, 15, 18, 21, 24}

26


B {multiples of 3}

2. U {K, E, Y, W, O, R, D} A {W, O, R, D} B {W, O, R, K} Find: a. A b. B d. (A B) e. A B. 3. U {Whole numbers from 1 to 10} A {Odd numbers} Find: a. A b. B d. (A B) e. A B. 2. a. d.

A {K, E, Y} (A B) {K, E, Y, D}

b. e.

c.

AB

B {Prime numbers} c.

AB

B {E, Y, D} c. A B {W, O, R} A B {K, E, Y} {E, Y, D} {E, Y}

3. U {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A {1, 3, 5, 7, 9} B {2, 3, 5, 7} a. A {2, 4, 6, 8, 10} b. B {1, 4, 6, 8, 9, 10} c. A B {1, 2, 3, 5, 7, 9} d. (A B) {4, 6, 8, 10} e. A B {2, 4, 6, 8, 10} {1, 4, 6, 8, 9, 10} {1, 2, 4, 6, 8, 9, 10}

Exercise For each pair of sets, find A B. a) A {chair, table, desk, stool}, B {table, bed, bookcase} b) A {St Lucia, Barbados, Trinidad}, B {Grenada, Dominica, Barbados} c) A {E, Q, U, A, L}, B {E, X, T, R, A} d) A {David, Betty, Jim, Brenda, Cathy}, B {Sharon, Jim, Thomas, Brenda, Cathy} e) A {2, 4, 6, 8, 10, 12},

B {1, 3, 5, 7, 9, 11}

For questions 2–6 find a) A

b) B

c) A B

d) A B

e) (A B)

f) (A B)

g) A B

h) A B

i) A B

j) A B.

U {D, E, P, O, S, I, T}; A {P, O, S, E} B {P, E, S, T} U {F, R, A, C, T, I, O, N, S}; A {R, A, T, I, O, N} B {C, R, A, F, T} U {Days of the week}; A {Days with 6 letters} B {Days beginning with S} U {Months of the year}; A {Months with 31 days} B {Months beginning with J} U {Whole numbers from 1 to 10}; A {Composite numbers} B {Factors of 40} From your answers to questions 2–6, what can you say about the relationship between a) A B and (A B)?

b) A B and (A B)? Sets

27

Key words

Unit 2.3 The union and intersection of three sets In this unit you will: find the union and intersection of three sets.

You already know that: A B is the set of elements belonging to A or B. A B is the set of elements belonging to both A and B. We can expand this to three sets: A B C is the set of elements belonging to A or B or C. A B C is the set of elements belonging to A and B and C.

Worked examples 1. A {O, V, E, N} B {N, O, V, E, R} Find a. A B C b. A B C.

C {V, O, T, E}

2. U {whole numbers from 1 to 10} A {odd numbers} B {factors of 15} C {factors of 21} Find a. A B C b. A B C. 1. a. A B C {O, V, E} b. A B C {O, V, E, N, R, T} 2. A {1, 3, 5, 7, 9} B {1, 3, 5} a. A B C {1, 3} b. A B C {1, 3, 5, 7, 9}

28


C {1, 3, 7}

Union Intersection

Exercise Find A B C and A B C. a) A {chair, table, stool},

B {bed, stool},

C {dresser, bed, chair}

b) A {B, O, T, H, E, R},

B {O, L, D, E, R},

C {Y, O, U, N, G, E, R}

c) A {D, E, L, I, A},

B {S, H, E, I, L, A},

C {O, P, H, E, L, I, A}

d) A {2, 4, 6, 8, 10, 12},

B {1, 2, 3, 4, 5, 6},

C {6, 7, 8, 9, 10}

e) A {Donna, Sarah, Sharon, David}, B {Jonathan, Sandra, Sharon, David}, C {Donna, Sarah, Jonathan, Ricky}

The following table shows the universal set and sets A, B and C. In each case find i) A B

ii) B C

iii) A B C B

iv) A B C.

U

A

C

a)

{letters in the alphabet}

{letters in BIOLOGY} {letters in HISTORY} {letters in CHEMISTRY}

b)


{letters in GRAPE}

{letters in PEAR}

{letters in APPLE}

c)

{whole numbers from 0 to 10}

{even numbers}

{multiples of 3}

{factors of 12}

d)


{multiples of 5}

{factors of 50}

{square numbers}

e)

{integers from 5 to 7}

{natural numbers}

{positive odd numbers}

{prime numbers}

f)

{natural numbers less than 20}

{factors of 35}

{factors of 49}

{factors of 42}

g)

{multiples of 5 between 1 and 51}

{factors of 100}

{multiples of 10}

{factors of 50}

h)

{whole numbers less than 20}

{triangular numbers} {factors of 36}

i)

{whole numbers less than 14}

{prime numbers}

j)

{composite numbers {multiples of 2} between 5 and 15}

{factors of 24}

{prime factors of 60} {odd numbers less than 4} {multiples of 6}

{multiples of 4}

Sets

29

Key words

Unit 2.4 Venn diagrams

Venn diagram Shading

In this unit you will: use Venn diagrams to represent relationships between sets draw Venn diagrams illustrating the intersection of three sets. Before we move onto representing three sets in a Venn diagram make sure you remember how to represent subsets, the intersection and union of sets, and the complement of a set using Venn diagrams. AB

U

B A

AB

U

A

B

U A

U A

B

AB

A (the shaded portion)

To draw a Venn diagram illustrating the intersection of three sets A, B and C: ● List the members of A, B and C. ● List the members of A B C, A B, A C, B C. Draw a circle or rectangle to represent the universal set U, and three intersecting circles to represent A, B and C. Place elements in each portion of the Venn diagram as follows: U

elements in A B C elements in A B but not A B C elements in B C but not in A B C elements in A C but not A B C elements in A only elements in B only elements in C only elements in U but not in A, B or C

30


A

B

C

Worked example Draw a Venn diagram illustrating A B C. U {whole numbers from 1 to 10} A {prime numbers} B {factors of 12} A {2, 3, 5, 7} B {1, 2, 3, 4, 6} C {2, 4, 6, 8, 10} A B C {2} A B {2, 3} A C {2} B C {2, 4, 6}

C {even numbers}

U A

B 7

3

5

2

8

1 4

6

10 C

9

Exercise Draw Venn diagrams to illustrate the following sets. U

A

B

C

a)


{odd numbers}

{prime numbers}

{factors of 14}

b)

{M, A, C, H, I, N, E}

{N, A, M, E}

{M, I, N, E}

{C, H, A, I, N}

c)

{H,U,N,G,R,Y,O,W,E} {G, R, O, W}

{R, O, G, U, E}

{H, O, G}

d)

{days of the week}

{days beginning with T}

{days with 8 letters}

{days between Monday and Friday}

e)


{composite numbers}

{multiples of 3}

{multiples of 4}

f)

{integers from 2 to 10}

{natural numbers}

{square numbers}

{factors of 18}

g)

{odd numbers less than 20}

{multiples of 5}

{factors of 100}

{prime numbers}

For each Venn diagram above, list the members of the following sets. i) A B C ii) (A B C) Sets

31

Key words

Unit 2.5 Formulae

Formula Element

In this unit you will: use formulae to calculate the number of elements in a set.

Remember that n(A) is the number of elements in the set A. In the Venn diagram below, the letters in the sets denote the number of elements in each set. x number of elements in A but not B y number of elements in both A and B z number of elements in B but not A w number of elements in neither A nor B n(A B) x y z There is also a formula for calculating n(A B)

U A

B x

y

z

w

n(A B) n(A) n(B) n(A B)

Worked examples 1. There are 32 children in Grade 6. 18 like chicken and 16 like fish. How many children like both chicken and fish? 1. There are two methods of solving this problem. Method B (Using the formula) Method A (Venn diagram) n(A B) n(A) n(B) n(A B) First label each set with a capital letter. n(A B) 32, n(A) 18, n(B) 16 A children who like chicken 32 18 16 n(A B) B children who like fish 32 34 n(A B) Then represent what we need to find by a letter. x number of children who like both chicken and fish n(A B) 2 Now draw the Venn diagram and enter the number of elements in each set in the appropriate region. U B

A 18 x

x

16 x

32 (18 x) (16 x) x

Number of children who like chicken but not fish 18 x Number of children who like fish but not chicken 16 x Write an equation using the information that there are 32 children in Grade 6 18 x x 16 x 32 Solve the equation: 34 x 32 x2

32


2. There are 38 children in Grade 5. 20 like Mathematics, 25 like English and 3 like neither. How many children like Mathematics and English? 2. Method A (Venn diagram) A children who like Maths B children who like English x no. of children who like Maths and English n(A B) 38 3 35 20 x x 25 x 35 45 x 35 x 10

U B

A 10

10

15

3

Method B (Formula) n(A B) 35; n(A) 20; n(B) 25 n(A B) n(A) n(B) n(A B) 35 20 25 n(A B) n(A B) 10

Exercise There are 40 students in Grade 4. 23 like netball and 28 like volleyball. How many students like both? Answer this question using both methods.

There are 42 students in Grade 5. 34 students like singing, 29 like acting and 5 like neither. How many students like both? Answer this question using both methods.

There are 120 people registered to take GCE exams in Mathematics and English. 11 people do not show up to take the exam. 80 take Mathematics and 70 take English. How many people take both Mathematics and English?

180 people registered on a voters’ list are asked to vote for two candidates. The voters can vote for either one of the candidates or both. 60 voters voted for candidate A, 135 voted for candidate B, and 8 voted for neither. How many voters voted for both candidates?

There are 45 students in Grade 6. 28 students like beef and 3 like both beef and chicken. How many students like chicken?

There are 37 students in Grade 4. 29 students take Needlework, 6 take both Needlework and Art, and 1 takes neither. How many students take Art?

There are 75 students in Grade 3. 50 students like History, 31 like Geography and 12 like neither. Use a Venn diagram to answer the following questions. a) How many students like both Geography and History? b) How many students like Geography or History? c) How many students like Geography but not History? d) How many students like History but not Geography? Sets

33

Looking back Find A B C and A B C. a) A {T, A, B, L, E},

B {B, E, A, T},

C {L, A, T, E}

b) A {Harry, Dean, Ken},

B= {Ken, Fran, Joe},

C {Harry, Ken, Bob}

c) A {3, 6, 9, 12},

B {2, 4, 6, 8},

C {7, 8, 9}

d) A {apple, pear, orange}, B {grapes, pear, orange}, C {pear, strawberry, orange}

The table shows the universal set and sets A, B and C. In each case, find A B C and A B C. U

A

B

C

a)


{triangle numbers}

{prime numbers}

{odd numbers}

b)


{letters in TELEVISION}

{letters in TELEPHONE}

{letters in TELEPATHY}

c)

{days of the week}

{days beginning with S}

{days between Thursday and Sunday}

{days between Monday and Wednesday}

Draw Venn diagrams illustrating the information below. a) U {whole numbers from 1 to 15} A {multiples of 3} B {factors of 48} C {factors of 45} b) U {whole numbers from 1 to 16} A {square numbers} B {even numbers} C {factors of 32}

There are 78 athletes at a sports meet. 48 athletes are registered to run the 100 metres race and 52 are registered to run the 20 metres race. 10 are registered to run neither. Use an appropriate formula to calculate how many athletes are registered to run both races.

Of the 107 students in Form 2, 73 like Food and Nutrition, 69 like Physical Education and 20 like neither. Use a Venn diagram to answer the following questions. a) How many students like both Food and Nutrition and Physical Education? b) How many students like Food and Nutrition or Physical Education? c) How many students like Food and Nutrition but not Physical Education? d) How many students like Physical Education but not Food and Nutrition?

34


Key words

Perimeter Circumference Diameter Radius

Chapter

Perimeter and area In this chapter you will review what you have already learnt about perimeter and area of shapes, including the circle and composite shapes. You will apply your skills to solve problems related to perimeter and area.

3

Unit 3.1 Perimeter and circumference In this unit you will: determine the perimeter of different polygons find the circumference of circles calculate the perimeter of sectors of circles. Do you remember that the perimeter of a shape is the total distance around the shape? If the shape is a regular polygon, such as an equilateral triangle, a square or a pentagon, the perimeter is calculated by multiplying the length of one side, l, by the number of sides. l l

l

l

l

l

l l

l P 3l

l P 4l

l

l P 5l

The distance around the edge of a circle is called the circumference. We use the irrational number pi when we are working with circles. Pi or is a constant ratio of 22 . The circumference of a circle C is 2 times the radius which is written as C 2r. 7 The diameter is twice the length of any radius, so the formula may also be written as C d. has an approximate value of 3.14. umference circ

di

radius

am

ete

r

Perimeter and area

35

Worked example Calculate the perimeter of the shapes below. a. b. B

6.2 cm 15 cm A

4.7 cm

C

a. The triangle is isosceles, so AB BC 6.2 cm and AC 4.7 cm. The perimeter is: 6.2 6.2 4.7 17.1 cm. b. The circumference is: 2r 2 3.14 15 94.2 cm.

Exercise Calculate the circumference of the following circles, using 3.14. Give your answers correct to one decimal place.

a)

b) 5 cm 28 cm

d)

c) 20 cm

77 cm

36


Work out the perimeter of each shape below. a)

b)

8 cm

4.1 cm 10 cm

17 cm 6.4 cm

c)

25 m

14 m

Work out the circumference of the whole circle then divide it by 5 to get the length of the arc.

Calculate the perimeter of each of the shapes below. a)

b)

c)

10 cm

72°

14 cm

35 m

The diagram shows two lanes of a circular track. Johnathan runs along the outer lane and Jeremy runs along the inner lane. a) What distance did each boy run in one lap? b) If Johnathan runs 10 laps and Jeremy runs 8 laps, who runs the further distance and by how many metres?

25 m 30 m

Aaron is making a birthday cake for his brother. The cake is 20 cm in diameter. When the cake is iced he will put a ribbon around the edge. The ribbon needs to have 1.5 cm overlap. What length of ribbon will Aaron need?

Perimeter and area

37

Key words

Unit 3.2 Areas of parallelograms and trapezia

Parallelogram Trapezium

In this unit you will: use the formula for finding the area of a parallelogram use the formula for finding the area of a trapezium combine formulae to determine the areas of compound shapes. A parallelogram is a quadrilateral with two pairs of parallel and equal sides. The parallelogram is made up of two triangles with equal areas. The area of each 1 triangle is 2(base perpendicular height). Hence the area of the parallelogram is 1 1 (base height) (base height), or 2 2 simply (base height).

base

perpendicular height

perpendicular height

base

a

A trapezium is a quadrilateral with only one pair of parallel sides. The parallel sides have lengths a and b and the perpendicular height is h.

h b

If we put two identical trapezia together, we make a parallelogram. The total length of the parallel base is now (a b) and the height is h. The area of the parallelogram is (base height), which is (a b) height. 1 Hence the area of one trapezium is 2(a b) h.

a

b h

b

a

This formula can be used to find the area of any trapezium where a and b are the lengths of the parallel sides and h is the perpendicular height between them.

Worked examples 1. Calculate the area of the parallelogram.

2.

Calculate the area of the trapezium. 3 cm

7 cm

5 cm

1. Area of parallelogram base height 5 cm 7 cm 35 cm2 1 2. Area of trapezium 2(a b)h 1 2(5cm 3 cm) 4 cm 1 2(8cm) 4 cm 4 cm 4 cm 16 cm2

38


4 cm

5 cm

Exercise Calculate the area of each shape. 8 cm

a)

7.5 cm

b) 4 cm 12 cm

c)

12 cm

d) 5 cm 18 cm

26 cm

5.4 cm

6 cm

Work out the area of each of these shapes. Copy the shape and show how you divided it up and any dimensions that were calculated. 4 cm 8 cm a) b) c) 5 cm 6 cm

10 cm 9 cm

3 cm 4 cm

4 cm

6 cm

10 cm

12 cm

The base of a parallelogram is 5.4 cm and its height is 6 cm. What is the area of the parallelogram?

A trapezium has parallel sides of 7.3 cm and 4.5 cm. If the perpendicular height of the trapezium is 6 cm, what is the area?

The area of a parallelogram is 14.26 cm2.

40

The base of the parallelogram is 3.1 cm. What is the height of the parallelogram?

cm

Yvonne wants to build the kite shown.

20 2

cm

How much paper will she need? Give your answer in m . 60

cm

Perimeter and area

39

Key words

Unit 3.3 Area

Area Parallelogram Trapezium Kite

In this unit you will: use formulae for calculating the area of polygons convert between different units in the metric system.

The formulae for calculating the area of some well-known polygons are given below in case you have forgotten them. Triangle l

Rectangle

h

b

h

h

b

b bh A 12 bh 2

A bl

Circle

h Parallelogram

b a h Trapezium

r

b

b A bh

A πr2

A

h(a b) 12 (a b)h 2

The metric units of area are square metres (m2), square centimetres (cm2) and square millimetres (mm2). 1 m2 10 000 cm2

1 cm2 100 mm2

If you convert linear cm to mm you multiply by 10. However, if you convert square cm to square mm, you are in fact multiplying by 10 10 or 100. So, when you convert between square units, you have to double the decimal places you would normally multiply or divide by on the metric scale. Consider a rectangle with a length of 3 cm and a width of 2 cm. The length may also be expressed as 30 mm and the width as 20 mm. Hence the area 20 mm is either 6 square centimetres or 600 square millimetres.

3 cm

2 cm

30 mm

Worked examples 1. Copy and complete the following. a. 8 cm2 mm2 b. 4 m2 cm2 c. 340 mm2 cm2 d. 72 500 cm2 m2

40


8 100 800 a. b. c. d.

8 cm 800 mm 4 m2 40 000 cm2 340 mm2 3.4 cm2 72 500 cm2 7.25 m2 2

2

4 10 000 40 000 340 3.4 100 72 500 7.25 10 000

2. Determine the area of this shape in a. square metres b. square centimetres.

4m 3m 6m

1

Area of the trapezium 2(a b)h 1 2(4 6) 3 15 m2 15 10 000 cm2

1

Or 2(400 cm 600 cm) 300 cm 150 000 cm2

Exercise Convert these square centimetres to square metres. a) 50 000 cm2

b) 250 000 cm2

c) 35 700 cm2

d) 8 250 cm2

e) 575 cm2

d) 8 254 mm2

e) 375 mm2

Convert these square millimetres to square centimetres. a) 500 mm2

b) 2 500 mm2

c) 72 500 mm2

Convert these square centimetres to square millimetres. a) 5 cm2 d) 6.7 cm2

b) 65 cm2 e) 0.22 cm2

c) 2.75 cm2 f) 0.3 cm2

Convert these square metres to square centimetres. a) 7 m2

b) 75 m2

c) 3.5 m2

d) 0.5 m2

e) 0.3475 m2

A postage stamp measures 2.5 cm by 1.8 cm. a) Convert the lengths to millimetres. b) Work out the area of the stamp in i) square centimetres ii) square millimetres.

Calculate the areas of the following shapes. a)

8 cm

6.5 cm

b)

6 cm

6 cm

10 cm

c) 4.5 cm

7 cm

5 cm

Perimeter and area

41

Calculate the area of these shapes. Show all your working. a)

b)

3 cm

3 cm 5 cm 4 cm

9 cm

6 cm 15 cm

c)

d) 7 cm 10 cm

14 cm

28 cm

The piece of fabric below is used to make bandanas for a carnival band. If each bandana must be 50 cm by 40 cm, how many bandanas can be made from the fabric?

Scale: 1 cm = 80 cm

Investigation Derive a formula for calculating the area of a kite where x and y are the diagonals of the kite.

y

x

42


Key words

Surface area Prism Pyramid Cross section

Unit 3.4 Surface area In this unit you will: calculate the surface area of three-dimensional shapes, including prisms and pyramids.

The surface area of a three dimensional shape is the total area of all the faces of the shape. The formula for the surface area of a cube is 6l2, where l is the length of one of the sides. The surface area of a cuboid is 2lw 2lh 2wh, where w is the width, l is the length and h is the height. l

w l

l

h

l l

w

l

l

Cube Surface area 6 l2

Cuboid Surface area 2(lw) 2(wh) 2(lh)

Worked examples 5 cm

1. Find the surface area of the prism.

5 cm

4 cm

10 cm 8 cm

2. Find the area of the square-based pyramid shown. 6 cm

5 cm 5 cm

1. The cross section of the prism is a trapezium, so there are two faces with this shape. The other four faces are rectangular. 1 1 Area of the trapezium 2(a b)h 2(5 8) 4 26 cm2 The area of the rectangular faces is: (5 10 5 10 4 10 8 10) cm2 220 cm2 The total surface area (220 26 26) cm2 272 cm2 2. The area of the base is 5 5 cm2 25 cm2 1 The triangular faces are identical: 4 2 (6 5) 60 cm2 Total area is 25 cm2 60 cm2 85 cm2

Perimeter and area

43

Exercise Calculate the surface area of each of the following shapes. a)

10 cm

20 cm 8 cm

b) 4 cm 4 cm 4 cm

c)

5 cm 4 cm 7 cm

6 cm

Work out the surface area of each of the shapes below. a)

13 cm

15 cm

12 cm 18 cm

22 cm

b)

4 cm 5 cm

4 cm

5 cm 7 cm

10 cm

c) 7 cm 6 cm

15 cm

7 cm

The diagrams shown below are the nets for solids. Calculate the surface area of each. a)

b)

6 cm

5.5 cm

10 cm

10 cm

12 cm

10 cm

12 cm

10 cm

44


c)

12.7 cm

Looking back The area of a circle is 38.5 cm2, what is the circumference of the circle? A rectangle has a length of 10.2 cm and a width of 6.8 cm. What is the area of a square with the same perimeter?

Determine the area of the triangle shown. 13 cm

5 cm

12 cm

Trudi trains for a marathon by running 10 kilometres around a square track each day. If the square has a length of 120 metres, how many times should she run around the track to cover a total distance of 10 kilometres?

A rectangle has a length of x cm and the width is 3 cm more than the length. Determine the length of the rectangle if the perimeter is 50 cm.

Determine the area of the shaded region.

25 cm

15 cm

The diagram shows a sports ground.

200 m

Calculate the perimeter and area of the ground giving your answer to the nearest whole number.

For each of the diagrams, calculate a) the perimeter b) the area of the shaded region. Remember, you can find the length of the arcs by dividing the circumference of the whole circle.

i)

50 m

ii)

C 6 cm B 8 cm

60°

20 cm 30 cm

A

The floor of an auditorium is in the shape of the trapezium

16 m

shown. How many square tiles each of length 30 cm are needed to cover the floor completely? 20 m

24 m

The surface area of a cuboid is 210 square centimetres. If the base of the cuboid is a square of area 25 cm2, what is the height of the cuboid? Perimeter and area

45

Project This is a scale drawing of a pre-primary classrom on a scale of 1 cm to 1 m. blackboard

door

1 cm : 1 m

a) A storage cupboard is to be built along the back wall. The dimensions of the cupboard are length 2 300 mm, width 800 mm and height 950 mm. What size should the cupboard be drawn on the plan? b) Calculate the floor area of the classroom after the cupboard is in place. c) The school supplies tables which are 80 cm long and 45 cm wide. The teacher places four together to make groups. How much floor space will 10 groups take up? d) The following items also need to go into the classroom: ● a carpet 1.8 m 1.2 m ● 2 bookshelves 30 cm wide and 1 m long ● a display table 1.2 m long and 50 cm wide. e) Redraw the plan of the classroom. Show where you would place the items in it to make best use of the available space. Remember that the door has to open and that groups should not obstruct access to the bookshelves or storage cupboard.

46


Pre-primary teachers have a lot of equipment and toys which need to be safely stored when they are not being used. This is a list of the equipment that one teacher has collected for use with her class: ● Beads (different shapes, colours and materials) ● Beans (different sizes and colours) ● Buttons (different sizes, colours and shapes, and different number and size of holes) ● Bottle tops to use as counters ● Cardboard boxes, sheets and containers ● Clay ● Cotton reels ● Crayons ● Egg boxes and trays ● Elastic bands ● Envelopes (used, in different sizes) ● Flowers ● Leaves (different sizes and shapes) ● Matchboxes ● Pencils ● Plastic bags, bottles and lids ● Sand ● Sticks (different lengths and thicknesses) ● Stones ● Straws ● String and wool ● Styrofoam trays and cups ● Tins (different sizes and shapes) ● Toys (fruit, vegetables, transport, animals and so on) ● Cut out pictures from magazines, newspapers and calendars Your task is to design a simple, easy-to-use, inexpensive storage system for keeping all this material in the classroom storage cupboard. Start by drawing a scale model of the storage cupboard. You have three shelves which you can place where you choose. Decide how you will organise the material, what you will use to store it in and how you will make it easy for anyone to use. Draw your design and write short notes to explain how your system will work. Calculate how much it will cost to implement your system.

Perimeter and area

47

Case study – Pre-primary school principal and teacher St Joseph’s Educare Centre Ms Loretta Bowen-Williams Qualifications/Experience: Diploma in Pre-Primary Education (Miami College of Education) Montessori Teaching Diploma (London) Ten years teaching experience

I run a small private pre-primary school in a large town. We have seventy children in all and five staff members. I currently rent the premises from a local businessman – they are very lovely and spacious with large grounds for the children to play in. My work has two elements. As the principal I am responsible for the administration and day-to-day running of the school. I spend about three hours a day doing this part of my job. I have to make sure that the teachers are doing their jobs and that they are paid. I also report to the board of trustees and spend time fund-raising for the school. I have to make sure that the financial records are kept up-to-date and that fee payments are entered and receipts issued. We charge fees on a monthly basis and I have to send out accounts to parents. We charge extra for outings, food and additional equipment. I also have to pay the school accounts – telephone, electricity, rent and daily expenses such as food and equipment that we buy. Once I have dealt with the financial aspect of the job, I have to keep records for the Ministry of Education. This involves attendance records for both pupils and teachers – when a teacher takes a leave of absence, we have to complete a form and record the details in a register. We keep an overall attendance record for the pupils too. I also deal with the ordering of equipment and books and make appointments to see parents if necessary. I keep a diary and find that it helps to manage my time efficiently. In addition to my work as principal, I also teach art and drama for two hours each day. I really love this part of my job as it allows me to have contact with the children. I plan lessons in advance and I try to collect all the materials that I will need for the lessons before I get to school. For example, this week we are making templates and painting on fabric and I have to work out what I will need for each class. For each group of four children I need a set of four plastic paint containers and four brushes as well as four pairs of scissors, rulers and thin plastic for making the templates. I also need at least a metre of plain fabric for each group. I have three classes with twelve to sixteen children in each, so it takes a little organisation to get it all sorted out. I make up a basket or box with the equipment for each class and keep it all together. Some lessons require much more planning than others. For example, when we built models out of drinking straws, I had to draw money from school funds to buy straws and plasticine before the lessons. However, it is all worth it when the children are creative and they make beautiful artworks.

48


Revise and consolidate

R1

Calculate: a) 25 (5 8)

b) (3 9) (24 3)

c) 4 8 2 3

d) 64 4 8

e) 60 5 2 3

f) 2 8 2 4

Evaluate the following. a) 25

c) 24

b) 43

e) 91

d) 125

f) 50

Say whether the following statements are true or false. a) 23 8

b) 32 6

c) 53 125

d) 21 12

e) 102 210

f) 22 2

g) (3)2 9

h) (2)2 22

i) 34 43

j) 23 1

k) (2)3 1

l) 72 301

Find the roots of the following. 3

a) 9

b) 125

c)

81 64

d)

25 16

Which of the following are prime numbers? 1 29 73

2 33 87

3 37 99

11 39 101

15 47 111

17 51

19 59

21 60

23 61

27 72

Write down the first five multiples of the first three prime numbers. Write out all the factors of the following numbers. a) 4

b) 12

c) 48

What is the HCF of 36 and 72? If bird food costs $125.75 per kg, how much will 4.2 kg cost to the nearest cent?

How many pieces of string, each 1.4 m long, can I cut from a ball that is 38.5 m long? If a 4.583, b 16.79, c 143.8, d 25.37 and e 9.648, calculate: a) a b

b) c d

c) e a

e) c e

f) 2(a b)

g) b d

d) b c d

What must be added to or subtracted from: a) 3.268 to make 3.28?

b) 4.156 to make 4.14?

Convert the following fractions to decimals and say whether each is recurring or not. a)

1 8

b)

2 9

c)

3 7

d) 113

e) 2272


Write the following numbers in standard form. a) 2 million

b) 67 580 000

c) 8 100 000 000

d) 0.004

e) 0.000 000 98

f) 0.000 000 234

Rewrite the following as ordinary numbers. a) 1.2 103

b) 3.456 106

c) 8 101

d) 2.2 103

e) 9.034 104

f) 7 105

Convert the following base ten numbers to numbers in the respective bases. a) 12 to base 4

b) 16 to base 2

c) 110 to base 8

Convert the following numbers to the denary system. a) 2658

b) 101102

c) 10112

d) 1332104

Add: a) 12436 13546

b) 14057 45567

Subtract: a) 101102 11112

b) 34125 43325

Multiply: a) 2123 213

b) 4546 156

a) Draw a Venn diagram to show the following information: U {whole numbers from 1 to 25} A {prime numbers} B {triangular numbers} C {odd numbers} b) What is A B C? c) What is A B C? The area of a circle is 61 cm. What is its circumference? A rectangle has an area of 54 cm2 and a length of 6 cm. What is its breadth? Calculate the shaded area below. The side of the square is 12 m.

50


Key words

Rate Per Speed

Chapter

Rates and proportions In real life we often have to compare two quantities of different things, for example, km per hour or dollars per kilogram. In this chapter you will work with different rates and proportions to solve problems and become more aware of how mathematical rates can help you compare prices and otherwise become a more critical consumer.

4

Unit 4.1 Rate and speed In this unit you will: carry out practical activities to record time in minutes and seconds calculate speed by dividing distance by time. A rate is a comparison of two different quantities measured in different units. We normally express rates using the word ‘per’, for example, kilometres per hour. Per means ‘for each’ and it can be shortened to ‘p’ or indicated using a ‘/’ as in km.p.h. or km/h. To calculate a rate, you divide one quantity by another. For example, to find a speed in kilometres per hour, we would divide the kilometres covered by the number of hours. Most rates are given ‘per’ one unit. For example, ‘per kilometre’ means for every one kilometre.

Worked examples 1. Angela ran the 100 m race in 20 seconds. What was her speed in metres per second? 2. Andrew ran the 3 500 m race in 11 minutes 40 seconds. What was his speed in metres per second? 1. In 1 second she ran 100 m 20 5 metres. Angela’s speed was 5 m/s. 2. 11 minutes and 40 seconds is 700 seconds. In 1 second, Andrew ran 3 500 m 700 5 metres. Andrew’s speed was 5 m/s.

You need to convert the units before you start.

Exercise Work in pairs. a) Count how many times you can sign your name in one minute. b) Complete this sentence: My signing rate is

times per minute. Rates and proportions

51

The picture shows how to measure your partner’s pulse (heart rate). a) Measure each other’s pulse for 30 seconds. b) Calculate your own pulse in beats per minute.

Three students recorded how often they blinked in a given time. These are their results: Jason

Benita

Delia

5 blinks in 20 seconds



a) Calculate the blink rate per minute for each student. b) Find and record your own blink rate per minute. c) Rank the four blink rates (including your own) in order, from fastest to slowest.

A hummingbird beats its wings about 9 600 times in 2 minutes. a) What is the bird’s wingbeat rate in beats per second? b) How many times would the bird beat its wings in an hour at the same rate?

When Jessica applied for a data capturing job, she had to complete a typing test. She managed to type 300 words in five minutes. The company requires a typing rate of at least 40 words per minute. Do you think they would employ Jessica?

Work in groups of three. You need a watch which can record seconds and a measuring tape. a) How many metres can each of you walk, at normal speed, in 10 seconds? b) Calculate your walking speed in metres per second. c) Repeat this experiment for running.

Different athletes competed in a

Athlete

Distance

Time

i) Victor

200 m

24 s

a) Calculate each one’s speed in m/s to the nearest whole number.

ii) Sandra

400 m

65 s

iii) Indira

100 m

14 s

b) Three of the athletes were runners, one was a cyclist, one was a swimmer and one was a rower. Which was which?

iv) Ravi

500 m

30 s

v) Sarah

100 m

58 s

vi) Francis

2 000 m

8 min 25 s

school competition. Their results are given in the table.

52


Key words

Unit 4.2 Time, distance and speed

Time Distance Speed Formulae Rate

In this unit you will: develop and apply the relationship between time, distance and speed by investigation, experiment, observation and using a formula solve a variety of problems involving time, distance and speed.

In order to work with time, distance and speed, you need to know the following formulae: Distance Speed Time

Distance Speed Time

Distance Time Speed

Total distance Average speed Total time

We can represent this relationship and remember the formulae using this triangle:

D S

T

Rates can also be represented graphically.

Distance (metres)

A distance–time graph has distance on the y-axis and time on the x-axis. 250 200

moving back to starting point

150 100 travelling away

50 0

● Speed is shown by the slope, or gradient on the graph.

stopped

speed increase

0

60

120 180 Time (seconds)

240

● If the graph slopes up it means you are moving away from the starting point, if it slopes down, you are moving back towards the starting point. ● A horizontal line indicates no movement. ● Time and distance at different points can be read from the axes.

A speed–time graph has speed on the y-axis and time on the x-axis. ● Distance is the area between the line of the graph and the x-axis.

Speed (m/s)

5

● An upwards slope shows an increase in speed; a downwards slope shows a decrease in speed.

constant speed

4 3

distance 2

● A horizontal line means the speed is constant but that you are moving.

increasing speed

1 0

0

1

2 3 Time (seconds)

4

● Speed and time at different points can be read from the axes.


53

Worked examples 1. Lisa travels for 25 minutes at 60 km/h. How far does she travel? 1 2. Joseph drives 180 km in 12 hours. What is his speed in km/h? 3. The graph shows Mr Khodra’s walk to work each morning.

The farthest he gets from home is 1 500 m.

1 600 1 400

Distance (metres)

1 200 1 000 800 600 400 200 0

a. b. c. d.

0

2

4

6

8 10 12 14 Time (minutes)

16

18

20

How far does Mr Khodra live from work? How long does it take him to walk to work in the morning? What happens 6 minutes after he leaves home? During which period was Mr Khodra travelling the fastest?

1. D S T

Before you use the formula you need to check the units. The speed is in km/h and the time is in minutes so you need to convert the speed to minutes.

S 60 km/h 60 km 60 minutes 1 km/min So D 1 km/min 25 25 km Lisa travels 25 km. 2. S D T 1 180 km 12 hours 120 km/h Joseph’s speed is 120 km/h. 3. a. He lives 1 500 m or 1.5 km from work. b. It takes him 20 minutes to walk to work. c. Between 6 and 8 minutes the line is horizontal: he doesn’t travel any distance. He could have been waiting to cross a road or stopped to buy a cup of coffee. d. Between 16 and 20 minutes. The slope of the graph is steepest between these times.

54


Exercise Find each speed in kilometres per hour: a) 4 000 km in 7 hours b) 6 hours to travel 701 km c) 842 km in 9 hours d) 72 km in 3 hours e) 1 236 km in 6 hours f) 387 km in 9 hours

Copy and complete the table using the correct units in each case. Speed

Distance

100 m/s

200 m 12 km

2 m/s

Time

3 hours 100 s

15 km/h

400 km

5 cm/day

49 cm 200 km

1 2

hour

5 m/s

25 mins

12 km/min

12 hours

1

A large shark swims at a speed of 15 km/h. How far will it swim in 40 minutes? A bird flies 12 km in 15 minutes. How far will it fly in an hour at the same speed? This graph shows Mr King’s journey from 90

a) How far was Mr King from Hewanorra after 1 hour?

80

b) How far is it from Hewanorra to his cousin’s house? c) How far was Mr King from his cousin’s house after driving for 1 hour and 30 minutes? d) How long did it take Mr King to travel the first 20 miles?

Distance from Hewanorra (miles)

Hewanorra to his cousin’s house.

70 60 50 40 30 20 10 0 0

1 2

e) What did Mr King do 1 hour into his journey?

1 1 12 Time (hrs)

2

f) During which period was Mr King travelling the fastest?

Mr James is taking part in a sponsored walk. He leaves the starting line at 3.30 p.m. and walks 1 km in the first 10 minutes. Mr James then stops for 2 minutes to have a rest. After that, he walks 2 km in 30 minutes. Draw a graph to show this information. Rates and proportions

55

Key words

Unit 4.3 Direct and inverse proportion

Direct proportion Inverse proportion Indirect proportion Varies directly Varies inversely Constant

In this unit you will: distinguish between direct and inverse proportion solve problems involving proportion.

Number of mangoes

1

2

3

4

5

Mass (kg)

0.2

0.4

0.6

0.8

1.0

The table shows the mass of mangoes. As the number of mangoes increases, the mass increases by a constant amount. We say that the number of mangoes is directly proportional to the mass or varies directly as the mass varies. If a quantity Y increases by a constant amount as X increases, then X is directly Y proportional to Y. Y constant. A constant is a fixed number X for all ordered pairs (X, Y). When two quantities X and Y are in direct proportion, the graph of Y against X is a straight line.

0

X

Worked example X In the table, X number of mangoes and Y mass in kilograms. Y a. What is the mass of 6 mangoes? b. How many mangoes have a mass of 15.6 kg? c. Draw a graph to show this relationship. 0.2 1 0.4 2 0.6 3 0.8 4 15

0.2 0.2 0.2 0.2 0.2

Y In each case, 0.2 X

1

2

3

4

5

0.2

0.4

0.6

0.8

1

a. 6 0.2 1.2 kg 6 mangoes weigh 1.2 kg. b. 15.6 0.2 78 mangoes 78 mangoes weigh 15.6 kg. c. Y 1 0.8 0.6

So, Y is directly proportional to X.

0.4 0.2 0 0

56


1

2

3

4

5

X

Number of people

20

10

1

Number of days

5

10

100

The table shows the number of days taken by different numbers of people to build benches. This is an example of inverse or indirect proportion. If a quantity Y decreases as a quantity X increases and XY constant, then X is inversely proportional to or varies inversely or indirectly with Y. 1 We can also say that X is directly proportional to Y Y 1 or Y is directly proportional to . X When two quantities X and Y are inversely proportional, 1 the graph of Y against is a straight line. X 0

1 X

Worked examples 1. In the table, let X number of days X and Y number of people. Y a. How many days will 5 people take to do the same job? b. How many will do the same job in 25 days? 2. Draw a graph to show this relationship:

5

10

100

20

10

1

1 X

0.2

0.1

0.01

Y

20

10

1

1. 20 5 100 10 10 100 1 100 100 XY 100 in each case So, X is inversely proportional to Y. a. 100 5 20 days 5 people will take 20 days to do this job. b. 100 25 4 people 4 people will take 25 days to do this job. 2.

Y 25 20 15 10 5 0 0.01

0.1

0.2

1 X


57

Exercise Determine whether each table illustrates direct or inverse proportion, or none. a)

b)

c)

d)

Number of pens

2

4

6

Cost

$2.40

$4.80

$7.20

Time (hours)

1

3

5

Distance (km)

40

120

180

Time (hours)

2

3

4

Speed (km/h)

45

30

22.5

Number of adults

1

2

3

Number of children

30

12

10

The table shows the cost of different numbers of oranges. Number of oranges

15

30

45

60

90

Cost ($)

1

2

3

4

6

a) Determine whether the information in the table illustrates direct or inverse proportion. b) What is the cost of i) 75 oranges

ii) 150 oranges?

c) Let X number of oranges and Y cost. Draw a graph of Y against X.

Put X on the horizontal axis and Y on the vertical axis.

d) From your graph, find the cost of 75 oranges. e) What would you have to do with the line graph to find the cost of 105 oranges? f) Find the cost of 105 oranges i) from your graph

ii) by calculation.

The table shows the times taken by a cyclist to cover a given distance at different speeds. Time (hours)

1

2

10

Speed (km/h)

40

20

4

a) Determine whether the information in the table illustrates direct or inverse proportion. b) If the cyclist takes 4 hours to cover the distance, what is his speed? c) If the cyclist’s speed is 8 km/h, how much time would he take to cover the same distance? 1

d) Let X time and Y speed. Find X and complete the table below. 1 X

Y

40

20

4 1

e) Draw a graph of Y against 58


1 . X

Put X on the horizontal axis and Y on the vertical axis.

Worked examples 1. 12 erasers cost $1.68. If the cost varies directly with the number of erasers, find: a. the cost of 6 erasers b. how many erasers can be bought with $1.96. 2. 2 people take 12 days to build a small hut. If the number of days varies inversely with the number of people: a. how many people will build the hut in 4 days? b. how many days will it take 5 people to the same job? 1. a. $1.68 12 14c 6 14c 84c 6 erasers cost 84c. b. $1.96 14c 14 erasers $1.96 will buy 14 erasers.

2. a. 2 12 24 24 4 6 people 6 people will build the hut in 4 days. 4 b. 24 5 45 days 4 It will take 45 days for 5 people to do the same job.

Exercise 15 notebooks cost $39.00. If the cost varies directly with the number of notebooks a) find the cost of i) 7 notebooks

ii) 18 notebooks.

b) how many notebooks can be bought with i) $10.40 ii) $62.40?

Tom gets paid $41.00 for 5 hours of work. If the payment varies directly with the number of hours of work a) how much does Tom get paid for i) 2 hours of work ii) 14 hours of work? b) how many hours does Tom work if he is paid i) $123.00 ii) $328.00?

8 people take 25 days to build a storeroom. If the number of days varies inversely with the number of people a) how many people will build the storeroom in i) 50 days ii) 20 days? b) how many days will it take: i) 5 people ii) 60 people to do the same job?

A man travelling at a constant speed of 80 km/h takes 4.5 hours to complete a journey. If the speed is inversely proportional to the time taken a) find the time taken to complete the journey with a constant speed of i) 90 km/h ii) 72 km/h. b) find the speed if the journey took i) 8 hours ii) 7.2 hours to complete. Rates and proportions

59

Looking back Scientists estimate that your blood travels 19 302 km in a day (24 hours). At what speed is it moving through your body?

A car travels at a speed of 73 km/hr. How far will it travel in 1

b) 22 hours?

a) 4 hours

How long will it take to travel a) 63 km at 9 km/h

b) 20 km at 80 km/h?

Aisha swims 30 lengths of a pool in 30 minutes. a) If the pool is 25 m long, how far does she swim? b) If she swims at a steady rate of 1 length per minute, draw a graph to illustrate her swim. c) Deborah also swims 30 lengths in 30 minutes. However, she swims 10 lengths in 8 minutes and then takes a 2 minute break. Add Deborah’s swim to your graph.

1.5 metres of cloth cost $48.00. If the cost varies directly with the length of cloth a) find the cost of i) 4.6 metres

ii) 2.5 metres of cloth.

b) what length of cloth costs i) $208.00 ii) $409.60?

The table shows the mass of pumpkins. Number of pumpkins Mass (kg)

3

8

10

24

64

80

a) Determine whether the information in the table illustrates direct or indirect proportion. b) What is the mass of i) 12 pumpkins

ii) 7 pumpkins?

c) How many pumpkins have a mass of i) 72 kg ii) 112 kg? d) Let X number of pumpkins and Y mass. Illustrate the information in the table on a graph. e) Use your graph to find i) the mass of 4 pumpkins

60


ii) how many pumpkins have a mass of 120 kg.

Key words

Index/indices Powers Base

Chapter

Working with indices You have already worked with powers and indices in Chapter 1. In this chapter you will learn more about the rules for operating with indices. You will review and learn how to work with indices when multiplying, dividing and raising an index by another. You will also learn to use fractional and negative indices.

5

Unit 5.1 The three basic rules of indices In this unit you will: determine some basic laws of indices apply the laws of indices in simplifying expressions. You already know how to use powers or indices to write numbers in mathematics. For example 63 is read as ‘6 cubed’ or ‘6 to the third power’, and vm is read as ‘v to the m power’. 3 and m are indices.

Singular – index, plural – indices

When we write expressions out in full, we are able to find quick methods of working with indices and this has allowed mathematicians to develop rules for operating with indices. 45 4 4 4 4 4 and a3 a a a 5 2 54 56 This leads us to the first rule.

(5 5) (5 5 5 5) 5 5 5 5 5 5 56

Rule 1 p a p b p a b

With the same base, add indices when multiplying. p is the base. 7777777777 7 7 7 7 7 7 7777

710 74 76 This leads us to the second rule. Rule 2 ma mb ma b

With the same base, subtract indices when dividing. m is the base.

(42)3 46 This leads us to the third rule.

(4 4) (4 4) (4 4) 46

Rule 3 (ga)b gab

Multiply indices when one index is raised to another power.

Worked example Simplify a. 2n6 m3n2

b. (h8g4)2

c. x4 x3y2

a. 2n6 m3n2 2 n6 n2 m3 2n8m3 b. (h8g4)2 (h8g4)(h8g4) (h8)2 (g4)2 h16g8 c. x4 x3y2 x4 3y2 xy2

Multiplication is commutative.


61

Exercise Simplify the following algebraic expressions. a) x2 3x2

b) 2m3 m4

c) 4z3 5z3

d) 5n3 2n n

e) q q 8q

f) 2a2 2a2 2a2

Simplify the following algebraic expressions. 4a5 b) a 14y2 e) 7y5

6x4 c) w3 21z3 f) 3z3

a) t3 t4 t5

b) k4 k6 k7

p5g2 c) p3

(y5)2z3 d) 7 yz

e) 4ax 2ay

16q9r6 f) 32 (8q r)

x3 a) 2 x 4n3 d) 2 2n

Simplify the following.

If a 3, b 2 and c 4, find the value of the following. a) abc

b) (cb)a

c) (ac)b bca

d) ab ac

(ab)c e) ab

f) 4ab 6ac 12ac

Write the answers to the following in index form. a) 54 54

b) 4210 4210

c) a6 a6

d) ym ym

Check by expanding.

Give a numerical value for each problem in question 5. Tell a friend what you observe about anything ‘to the zero power’ e.g. 100

?

Simplify the following. 1 2

a) 9 9

1 2

1

1

1

b) 8 3 8 3 8 3

1 4

1 4

1 4

c) h h h h

1 4

What do you observe about the questions in 8 and the results?

Investigation Challenge a classmate to state the squares of {1, 2, 3, 4, … 20}. Challenge a classmate to state the cubes of {1, 2, 3, 4, … 10}. Explain how to use your calculator to find the value of numbers raised to any power.

62


Key words

Unit 5.2 Roots, fractional and negative indices

Roots Square root Fractional indices Negatve indices

In this unit you will: differentiate between powers and roots use a calculator to find powers and roots of numbers perform calculations that include fractional and negative indices.

Remember that 25 is read as ‘the square root of 25’. The square root is that number which when multiplied by itself gives 25. Therefore, 25 5. Because 25 32.

5

32 2 is read as ‘the fifth root of 32 2’. n

m is read ‘the nth root of m’. In question 8 of the previous unit, you would have seen that: 1 2

1 2

1

a) 9 9 91 9

1

1

b) 8 3 8 3 8 3 8

1 4

1 4

1 4

1 4

c) h h h h h We add indices when multiplying same base.

9 b) 8

1 2

3

1 3

h

1 4

a)

c)

4

1

1 2

9 9 9 1

1

1 4

1 4

1 2

2

For square roots, it is not necessary to put the 2 in . 1

3

8 3 8 3 8 8 3 1 4

4

h h h h h

1 4

n

So mn m

The denominator of the fraction tells us the root.

We also know: 77777 1 1. 75 77 2 7777777 7 75 7 72 1 Therefore mn n m

Expand and simplify. Subtract indices when dividing. Divide and use a positive index.

62 2. 62 62 62 2 60 but 62 62 is also 2 1, therefore 60 1. 6 So p0 1.

Anything to the zero power 1.


63

Worked examples 1. What are the two possible values of 64 ? 3 2. Find the value of 27 . 3. Use your calculator to find a. 36 b. 3.60

8 8 64 and (8) ( 8) 64

1. 8 and – 8 3 (3) (3) (3) 27 3 3 9 and 2. 27 3 9 3 27 3 Check by using the calculator to do (3) . 3. a. 36 6 Remember 6 6 is also 36. b. 3.60 1.897 The calculator may not give the negative result. Before you go to the calculator, take a guess. 12 1 and 22 4, 1 3.60 4 so the result is between 1 and 3. Learn to use your calculator to find the powers and roots of numbers.

The rules we learnt before are also used with fractional and negative indices. 1 2

1 4

3

For example: ● 5 5 5 4 2

3 6

● (a

12 3

) a

Using a positive index. 1 a4 4 a

Invert to get a positive index.

h8k2 h5 h58 h3 1 ● 5 8 2 2 2 3 h hk k k h k2 Subtract indices when dividing.

Write as positive indices.

Exercise Find the value of the following. a) 16

b) 1600

c) 4

d) 400

e) 169

Remember there are two answers for each of these.

Work out the value of the following. 3

a) 8

3

b) 125

3

c) 64

d) 216

3

e) 512

3

d) 1.69

e) 16.9

3

e) 100

Use your calculator to find the value of a) 40

b) 160

c) 6.4

Rewrite the following using fractional indices. 5

a) 243

10

b) b

c) g

Rewrite the following using positive indices. 1 a) a5

b) p3 q3 p5 c) (c6)

6

d) h 2

1

d) 49

2

e)

b3 c4

2 1 Remember 4 2 4 1 64


Simplify the following, giving your answers with positive indices. 1 2

a) (d6) d2 p3q3 d) p3q6

b) f 7 f 6 f 9 r8s4 e) r3s5

2 3

c) (y )12 8t9u4 f) 42t9u3

Investigation a) Find two ways of simplifying the following. 81 i) 144

6 ii) 3 4

iii) 25 4 6

iv) 16 64

b) Write a general statement about what you observed. c) Try other examples to test if your statement works all the time.

Looking back Simplify: a) x4 x7

b) y2 y3

c) 4x2 x3

d) a3 2a3

e) 2d4 3d2 5

f) 3s2 3s4 5s

g) 3x2 x3y4

h) 3a2b3 3a2 4b2

i) xy xyz2

j) 2x2y 3x2z3 4z2

k) 3m2 7n2

l) (x2)4

m) (3a2)3

n) (4x4)3

o) 2(m3)3.

x6 a) 4 x

8a4 b) 3 4a

15x2y3 c) 3xy

a8 d) a

12x7 e) 3x9

Simplify:

Find the value of m. a) am 1

b) xm x

1 c) xm 3 x

1 b) x2 4 x

c) (mn )3 m

d) 3m 27

Simplify: a) (4x0)2 1

d) (9x2)2

1

e) (8b 3) 3 .


65

Chapter

6

Expressions, equations and formulae Algebra is used primarily to make generalisations. In this chapter you will explore patterns and sequences to arrive at general statements and use binary operations and function notations as you find the value of expressions. You will also rearrange literal equations and factorise expressions.

Key words

Sequence Terms Consecutive Ascending Term-to-term rule Descending

Unit 6.1 Sequences In this unit you will: find the terms of a sequence given the first term and the term-to-term rule. Individual numbers in a sequence are called terms. Terms that follow each other are called consecutive terms. The first term of a sequence is denoted T(1), the second T(2), the third T(3) and so on. The sequence: 2, 5, 8, 11, 14, 17 … ● is an ascending sequence ● is an infinite sequence

The terms are getting larger. The three dots show that it continues to infinity.

● can be described as T(1) 2 and the term-to-term rule is ‘add 3’. The sequence: 10, 5, 0, 5, 10, …, 95, 100 ● is a descending sequence ● is a finite sequence

The terms are getting smaller. The last term, 100, is given.

● can be described as T(1) 10, the term-to-tem rule is ‘subtract 5’ and the last term is 100.

Worked example a. Describe this sequence by giving the first term and the term-to-term rule: 20, 17, 14, 11, 8, 5 … . b. Find the 10th term of the sequence. a. T(1) 20. The term-to-term rule is ‘subtract 3’. b. T(10) 5 3 3 3 3 We know that T(6) 5. To find T(10) we must subtract 3 four more times. 5 (3 4) 7

Exercise Describe each of these sequences by writing down the first term, the term-to-term rule and, if appropriate, the last term. a) 4, 11, 18, 25 … c) 8, 28, 48, 68 … 66


b) 9, 7, 5, …, 3, 5 d) 0, 4, 8, 12, 16, 20

Find the tenth term of the following sequences. a) b) c) d)

T(2) 20 and the term-to-term rule is ‘subtract 6’. T(3) 10 and the term-to-term rule is ‘add 10’. T(13) 100 and the term-to-term rule is ‘subtract 5’. T(15) 9 and the term-to-term rule is ‘add 3’.

Find the 20th term of each of the sequences in question 1 a) and c) without finding the terms in between. You may check your answers on a calculator.

Find a) the 12th term and b) the 30th term of each of the following sequences. i) ii) iii) iv)

T(1) 20, the term-to-term rule is ‘add 15’. T(1) 56, the term-to-term rule is ‘subtract 4’. T(1) 100, the term-to-term rule is ‘subtract 1’. T(1) 5, the term-to-term rule is ‘subtract 5’.

A decorator uses this table to calculate the number of bags of wallpaper paste she needs for a given number of rolls of wallpaper. Number of rolls of wallpaper Number of bags of paste

1

2

3

4

5

1.5

2

2.5

3

3.5

a) Describe the sequence generated by the number of bags of paste by giving the first term and the term-to-term rule. b) How many bags of paste will the decorator need for i) 10 rolls ii) 15 rolls iii) 25 rolls of wallpaper? c) Describe in words how the decorator calculates the number of bags of paste she will need.

At a party each table can seat a maximum of eight people as shown on the right. Table

The tables are put together to seat more people as shown below.

Table

Table

Work out the sequence generated by the number of people that can be seated per table. Describe in words how you could calculate the number of seats available for the number of tables. How many people sit at each table?

The first term of a sequence is 5 and the term-to-term rule is ‘add number to go into the box so that a) every other number is an integer c) every fifth number is an integer

’. Choose a

b) every fourth number is an integer d) every tenth number is an integer. Expressions, equations and formulae

67

Key words

Unit 6.2 The general term

General term Multiples

In this unit you will: generate a sequence given the general term find and justify the general term of a sequence. We can find the general term of a sequence by looking at the difference between consecutive terms. Consider the sequence 2, 5, 8, 11, 14 … . The difference between the consecutive terms is always 3, so we compare the sequence to the multiples of 3. A general term of 3n gives 3, 6, 9, 12, 15 … Our sequence is one less than the sequence of the multiples of 3, so the general term is 3n 1. We can express the general term of a sequence as T(n). If T(n) 2n 1, then to find the first few terms of the sequence, we replace n with the term number: T(1) 2 1 1 1 T(2) 2 2 1 3 T(3) 2 3 1 5 … By expressing the general term in this way, it is easy to find any term in the sequence. For example, to find the 112th term: T(112) 2 112 1 223

Worked example The general term of a sequence is T(n) 5 3n. a. What is the difference between consecutive terms? b. Is the sequence ascending or descending? c. Find the first four terms of the sequence. a. 3 b. Descending c. T(1) 5 3 1 2 T(2) 5 3 2 1 T(3) 5 3 3 4 T(4) 5 3 4 7

Since the general term includes 3n, the sequence can be compared with the sequence of multiples of 3. The difference between consecutive terms will therefore be 3. Since the difference between the terms is 3, the terms must be getting smaller.

Exercise Copy and complete this table. Sequence

a) T(n) n 4 b) T(n) 7 n c) T(n) 9 3n d) T(n) 4n 12 e) T(n) 84 2n f) T(n) 8 5n 68


Difference between consecutive terms

Ascending or descending?

Find the first five terms of each of the sequences in question 1. Find T(10), T(12) and T(99) of each of the following sequences. a) T(n) n d) T(n) 1 n

b) T(n) n 5 e) T(n) n 100

c) T(n) 5 3n f) T(n) 1 0.5n

Find the general term of each of the following sequences and write it in the form of T(n) … a) 3, 0, 3, 6, 9 …

b) 12, 8, 4, 0, 4 …

c) 7, 6, 5, 4 …

d) 2, 8, 14, 20 …

Find the general term for the number of shapes in each of the following sequences and explain your answer.

a)

b) c) d)

A taxi driver at Miami Airport uses this table to calculate the cost of metered journeys. Distance (km) Cost in US$

1

2

3

4

5

6

7

8

9

10

2.10

2.20

2.30

2.40

2.50

2.60

2.70

2.80

2.90

3.00

a) Find the general term of the sequence of costs and explain your answer. b) Use the general term to calculate how far you would have travelled if the cost was $4.50.

The general term of a sequence is T(n) 3n 1. a) Copy and complete the table below. n

1

2

T(n)

4

7

3

4

5

b) Copy and complete the next row of the table which shows another way of generating the same sequence of numbers. T(n)

c) d) e) f) g)

4

4 (1 3)

4 (2 3)

To find the nth term of the sequence, how many times would we need to add 3? How could we express this algebraically? Write down the general term of the sequence using your answer to part d). Simplify your answer to part e). What do you notice about your answer to part f)?

Use the method given in question 7 to find the general term of each of these sequences. a) 7, 9, 11, 13, 15 …

b) 1, 4, 7, 10 …

c) 1, 5, 9, 13 … Expressions, equations and formulae

69

Key words

Unit 6.3 Quadratic sequences

Quadratic sequence Square

In this unit you will: generate the terms of a quadratic sequence.

A quadratic sequence is one whose general term includes a square. For example, T(n) n2 5 is the general term of a quadratic sequence. To find the terms of a quadratic sequence we substitute the term number into the expression. T1 12 5 6 T2 22 5 9 T3 32 5 14 T4 42 5 21

Worked examples 1. Find the first five terms of the sequence T(n) n2 3. 2. a. Write down the first five square numbers. b. Without substituting in the term numbers, find the first five terms of the sequence T(n) 2n2. 1. T(1) 12 3 2 T(2) 22 3 1 T(3) 32 3 6 T(4) 42 3 13 T(5) 52 3 22

2. a. 1, 4, 9, 16, 25 b. T(1) 2 1 2 T(2) 2 4 8 T(3) 2 9 18 T(4) 2 16 32 T(5) 2 25 50

The general term of the sequence of square numbers is T(n) n2. If we know the first five terms of this sequence, it is easy to find the first five terms of the sequence T(n) 2n2 by multiplying these terms by 2.

Exercise Find the first five terms of each of the following sequences. a) T(n) n2 3 d) T(n) n2 7

b) T(n) n2 10 e) T(n) n2 0.5

c) T(n) n2 5

Copy and complete the table below. Term number Sequence

a) T(n) n2 b) T(n) n2 1 c) T(n) n2 2 d) T(n) n2 1 e) T(n) n2 2 70


1

2

3

4

5

Find the first three terms of each of the following sequences. a) T(n) 2n2 d) T(n) 10n2 5

b) T(n) 2n2 5 e) T(n) 6n2 3

c) T(n) 4n2 1

The general term of a sequence is n . T(n) 2 n 1 The first term of the sequence is 1 1 T(1) 2. 2 1 1 Find the next four terms of the sequence. Leave your answers as fractions.

a) Draw the next four terms in this sequence of diagrams.

b) Copy and complete this table. Pattern number

1

2

3

4

Number of white squares Number of green squares Total number of squares

c) What is the general term for the number of white squares? d) What is the general term for the total number of squares?

Find the general term for the number of squares in the terms of this sequence.

Investigation a) Write down the first six terms of the sequence T(n) n2. b) Find the differences between consecutive terms. c) Describe any patterns you notice. d) Repeat for the following sequences: i) T(n) n2 1 ii) T(n) n2 10 iii) T(n) n2 3

iv) T(n) n2 1

a) Write down the first six terms of the each of the following sequences.

ii) T(n) 2n2 iii) T(n) 3n2 i) T(n) n2 b) Look at the differences between consecutive terms and describe any patterns that you notice. c) Predict the differences between consecutive terms for the sequence T(n) 4n2. d) Check your prediction. Expressions, equations and formulae

71

Key words

Unit 6.4 Solving simple equations In this unit you will: review solving equations using brackets and negative integers.

Equation Solve Unknown

Remember that an equation contains an unknown (variable) and an equals sign. We solve an equation to find the value of the variable. When solving equations: Step 1: Simplify both sides of the equation. For example, to solve the equation 3(2x 5) 4x 1, first we must expand the brackets on the left-hand side to get 6x 15 4x 1. Step 2: Solve the equation by doing the same to both sides. For example: Subtract 4x from both sides. 6x 15 4x 1 2x 15 1 Add 15 to both sides. 2x 14 Divide both sides by 2. x7 As long as we do the same to both sides of the equation, it remains balanced.

Worked examples 1. Find the value of m when 5(2m 3) 4(m 6) 27. 2. Find the value of x when 4(2x 5) 3(5x 1) 7(4x 6). Expand each of the brackets. 1. 10m 15 (4m 24) 27 10m 15 4m 24 27 6m 39 27 6m 12 m 2

Remember . Simplify the equation as much as possible. Subtract 39 from both sides and then divide both sides by 6.

Check: LHS 5(2m 3) 4(m 6) 5(2 2 3) 4(2 6) 5(4 3) 4(8) 5(1) 32 5 32 27 RHS 2. 8x 20 15x 3 28x 42 First expand each of the brackets. 23x 17 28x 42 Next simplify. 17 5x 42 Then solve. 25 5x 5 x

72


Check: LHS 4(2 5 5) 3(5 5 1) 20 78 98 RHS 7(4 5 6) 7 14 98 LHS

Exercise Find the value of x in each of the following equations. a) 3(2x 2) 4(2x 1) c) 7(2 4x) 10(5x 3)

b) 5(3x 1) 10(x 1) d) 4(3 2x) 11(x 8)

Find the value of the unknown in each of the following equations. a) 2(m 4) 3(m 5) 8 c) 7(2p 3) 4(3p 1) 1

b) 5(2v 3) 2(7v 1) 9 d) 3(z 7) 2(3 2z) 20

Find the value of t in each of the following equations. a) 5(5t 60) 2(t 3) t 14 c) 6t 5 (t 10) 2(6 2t)

b) 3(2t 3) 3(t 4) 3(2t 12) d) 5(3t 3) 3(2t 1) 3(2t 2)

Find the value of the unknown in each of the following equations. The answer does not have to be a whole number. a) 4(m 0.5) 2(6m 1) c) 5(2x 0.4) 3(x 0.3) 10(x 0.2)

b) 3(z 1) 4(z 0.3) 20z 0.095 d) 74x 6(2x 0.25) 5(7x 3)

Four chefs are making a stew. Patricia puts in x chicken legs. Sheila puts in five more than Patricia. Anne puts in double the amount that Sheila adds, and Winifred adds three times the number that Sheila adds. a) Write down an algebraic expression for the number of chicken legs added by i) Sheila ii) Anne iii) Winifred. b) Write down the total number of chicken legs added. c) There are 58 chicken legs in the stew. Work out how many chicken legs each chef added.

In a year’s time, Alex will be twice as old as Katie and Matthew will be three years older than Alex. Let p represent Katie’s age now. a) Write down an expression for i) Katie’s age in a year’s time ii) Alex’s age in a year’s time iii) Matthew’s age in a year’s time. In a year’s time, the ages of Katie, Alex and Matthew will sum to 33. b) Work out the ages of Katie, Alex and Matthew.

I think of a number. When I multiply it by 2 and add 5, I get the same answer as when I subtract it from 2. Write down and solve an equation to work out the value of the number I thought of.

Make up a question of your own like the one in 7 above and give it to a partner to solve. Expressions, equations and formulae

73

Key words

Unit 6.5 Simplifying algebraic fractions In this unit you will: add, subtract, multiply and divide simple algebraic fractions.

Cancel Numerator Denominator Multiplicative inverse Least common multiple (LCM)

To cancel we divide the numerator (the top) and the denominator (the bottom) by the same number or term. For example: 35 5 Dividing both the numerator and denominator by 7. 14 2 15a3b 3a2b Dividing both the numerator and denominator by 5a. 5a To divide by a fraction we multiply by the multiplicative inverse. For example: 1 1 4 2

1

2

4 1

1

2 is the multiplicative inverse of 2.

1

2 4abc 2ac 4abc 3 15 3 15 2ac

3 2ac is the multiplicative inverse of 2ac 3

2b 5 To add or subtract fractions we must have a common denominator. For example: ● When the given denominators are the same: 1 4 5 8 8 8 7 6 1 12 12 12

a c ac b b b p r pr q q q We do not know the value of a c or p r so they cannot be simplified. ● When the given denominators are not the same:

74

1 1 2 1 3 2 4 4 4 4

The common denominator is 4; the LCM of 2 and 4 is 4.

6 4 2 2 10 3 5 15 1 5 1 5

The LCM of 3 and 5 is 15; 15 3 5 and 5 2 10.

s r sz ry sz ry y z yz yz yz

The LCM of y and z is yz; yz y z and s z sz.

ab cd abe c2d abe c2d c e ce ce ce

The LCM of c and e is ce; ce c e and e ab abe; ce e c and c cd c2d.


Exercise Multiply the following algebraic fractions: b4 7a a) 3 2a b

12e 5efg b) 25eg 2

g2 22hj5 c) 11h g

2b2c 20e3 d) 2 2 5de 8bc

Remember: b4 b b b b and a3 a a a.

Divide the following algebraic fractions: cde de a) pq p

xy8 x6y3 b) z z2

wr r7 c) 2 2 y x vx

2a 10a d) 5b b5

What is the LCM of the following sets of terms? a) 9, 3, 6

b) 15, 10, 25

c) 2p, p2, 6p

d) abc, a, b

e) a2, b2a, ab

f) e2f , ef2, ef

Remember: each term must divide into the LCM.

Simplify the following expressions: a b a) 2 2

p q b) 9 9

2x 3x c) 8 4

16t 6t d) 6 3

3d e c) y x

6 y d) xy x

x w c) 2 p p

7d d d) ab a

Simplify these expressions: 2 3 a) a b

11a 1 b) c 2

Simplify the following expressions: 5 3 a) d e

3x x b) 4 z

a) The following table shows the age of each member of a family. Write an expression for the sum of their ages. Name Mr Jones Mrs Jones Mary Peter

Age in years a 7a 8 a 2 a 3

b) Simplify the expression for the sum of the ages in part a).

The dimensions of a rectangle are shown on the

2x y

diagram opposite. a) Write an expression for the perimeter. b) Write an expression for the area.

y x

c) Simplify the expressions in parts a) and b). Expressions, equations and formulae

75

Key words

Unit 6.6 Finding the subject of a formula

Transpose Subject Inverse operation Powers Roots

In this unit you will: rearrange simple formulae to obtain a required subject.

When you solved equations you used inverse operations. ‘Addition’ and ‘subtraction’ are inverse operations while ‘multiplication’ and ‘division’ are inverse operations. For example 40 5 45 ⇒ 40 45 5

54 20 34 ⇒ 54 34 20

16 3 48 ⇒ 16 48 3

24 8 3 ⇒ 24 3 8

4

54 5 5 5 5 625 so 625 5 6

a6 a a a a a a so a6 a ‘To raise to a power’ and ‘finding the root’ are inverse operations. For example: if a3 8 then 3 a 8 2

p2 64 p 64 8

A formula is a general equation that links variables. For example, the circumference of a circle is C 2 r. C is the circumference, r is the radius. ‘C’ is the subject of the equation. It is by itself on one side of the equation. To put ‘r’ as the subject, rearrange the formula by using inverse operations. C r 2 We do this by dividing both sides by 2 . Note we are not finding a value or solving an equation when we do this.

Worked examples 1

1. Put ‘h’ as the subject of A 2 (a b)h. 2A h Divide both sides by 12(a b). ab

At least ‘One step’

1

2. Put ‘a’ as the subject of A 2 (a b)h. 2A a b Divide both sides by 12h. h 2A b a Subtract b from both sides. h

At least ‘Two steps’

3. Rearrange the formula A r 2 to make r the subject. A Easier to transpose first. r 2 A r Find the square root of both sides. Two steps

76


Exercise For each of the formulae below, make m the subject. a) m 5 t m d) y 2 g) z 2 m

b) m 2 w

c) 3m r

e) x m 5 m h) p 7

f) y 12m

Make t the subject in the following formulae. a) 2t z

b) 5t w

c) y 6t

Make z the subject for each of the formulae below. z a) t 4

z b) m 8

z c) p 6

For each of the following equations, make the letter in brackets the subject. a) y x m (x) d) qr v (q)

t b) z (t) b e) x g b (g)

c) p – r z (p) f) y zx (x)

For each of the following equations make the letter in brackets the subject. 3 a) 5y (r) r F (H) d) HK

b) t s 4k (k) e) d3 e (d)

gh c) 4g2 (h) 3 f) p2 4q (p)

Investigation Look up the exchange rates for American dollars, British pounds and euros in the newspaper or on the Internet and write them down.

Write down formulae for exchanging your own currency to a) US $

b) GB£

c) u

Now write formulae for changing each of these currencies to your own currency.


77

Key words

Unit 6.7 Special algebraic notation

Operations Binary operations Function notation Substitution

In this unit you will: use symbols that represent given operations substitute numbers to obtain the value of an expression.

There are ways of operating on given numbers that involve symbols other than , , , or . Sometimes symbols such as ‘’ are used with a set of rules to describe an operation. These are usually called binary operations. a b means 2a 3b, find the value of 5 8. 2 5 3 8 10 24 34

Substituting 5 for a and 8 for b.

m n means m2 n , find the value of 6 16. 62 16 36 4 32

Substituting 6 for m and 16 for n.

The function notation is another way of writing equations. y 3x 2 may be written f(x) 3x 2 or g(x) 3x 2 and called the ‘f’ function or the ‘g’ function. Find the value of f(4) if f(a) 16a . f(4) 16 4 64 8

Exercise a b means 5a 3b, find the value of a) 3 2

b) 2 3

c) (6) 3

d) 1 1

e) 0.5 2.5

x y

x y means , calculate 2 a) 25 4

b) 4 4

c) 9 (3)

d) 1 6

e) 36 7

f) 8 (2)

Given the function f(x) 8x 7 calculate a) f(3) d) f(1)

b) f(4) e)

1 f(2)

c) f(0) f) f(0.2)

If h(y) y2 4y 1, calculate a) h(2)

b) h(1)

c) h(0.3)

d) h(1)

e) h(5)

f) h(0)

a) If p q means 2(p q), find 4 1 and 1 4. b) Are the results in a) the same? c) Try other numbers including fractions and explain your results to a friend. 78


Key words

Unit 6.8 Factorising algebraic expressions

Factors HCF Expand Term Expression

In this unit you will: recognise the Highest Common Factors (HCF) in algebraic expressions write algebraic expressions as products of factors.

24 can be written as 6 4 or 8 3. 6, 4, 8, 3 are factors of 24. 8 12 4(2 3) 4 is the HCF of 8 and 12. Remember, the highest common factor 4 and the expression (2 3) are factors of 8 12. is the largest number that divides, without remainder, into all terms. In the same way, some algebraic expressions can be written as a product of factors, that is, they can be factorised. You have learnt to expand or remove brackets, for example 12(a 2b) 12a 24b. Factorising is the opposite of expanding. 2, 3, 4, 6 and 12 are common factors of 12a and 24b but 12 is the HCF. Factorising algebraic expressions gives a product of terms and expressions. 6a 12b 18c 6(a 2b 3c) 6 is the HCF of the three terms. The number 6 and the expression (a 2b 3c) are factors. Check your result by expanding 6(a 2b 3c). Remember when you expand, you move the brackets and multiply each term by the factor outside. b

The area of a trapezium is the sum of the areas of two triangles that have the same height ‘h’. 1

h

1

So the area of the trapezium 2 ah 2 bh.

h

1

‘2 h’ is common to the area of both triangles so the 1 area of the trapezium 2 h(a b).

a

1

The term ‘2 h’ and the expression (a b) are two factors.

Worked examples 1. Factorise 5x4y 15x3y2. 2. Factorise 5pq 10p2q.

Remember the rules for indices and the sign rules. 5x3y is common to both terms, divide each term by 5x3y.

1. 5x4y 15x3y2 5x3y(x 3y) Check by expanding. 2. 5pq is the HCF So 5pq 10p2q 5pq(1 2p) Check by expanding.

5x4y 15x3y2 x, 3y 5x3y 5 x3y 5pq Remember 1. 5pq


79

Exercise Copy and complete: a) 3 1.2 3

3

b) 1.6 50 1 50 0.6 50 50 6 c) 2.3 300 2

Use factorisation to complete these multiplications. Write down the factorisation and then calculate the answer mentally. a) 3.2 50 b) 2.3 40 d) 46 15 e) 1.3 12

For example, write down 2.1 5 7 then work out the answer in your head.

c) 2.1 35 f) 2.3 600

Copy and complete: a) 288 4 c) 3 150 10 e) 735 7

so 288 16 so 3 150 30

b) 345 5

so 345 15

d) 126 6

so 126 18

so 735 35

Use factorisation to complete these divisions. Write down the factorisation and then calculate the answer mentally. a) 615 15 b) 666 18 d) 448 16 e) 213 30

c) 14.7 21 f) 490 35

Factorise the following algebraic expressions. a) mx mz d) 21efg 14efh

b) 16mn 24mp e) 3xyz 18xy 6xz

c) 2abc 14abcd f) 6ax 3a

Write the following algebraic expressions as a product of factors. a) a4 a3 a2 d) 2ax2 6bx2 8cx3

b) x3 x2y x4z e) 21p3q 14p2q2 p2q

c) m3n2 m2n3 f) k3m2 k2m2 k2m2

Use partitioning to complete these multiplications. Write down the partitioning and then calculate the answer mentally. a) 1.4 13 b) 230 21 c) 3.6 12 f) 1.8 53 g) 0.03 22 h) 0.04 104

Copy and complete:

a) 260 30 200 30 60 30

b) 930 4 800 4 130 4

c) 810 11 770 11 40 11 d) 17.6 8 16 8 1.6 8 e) 430 13 390 13 40 13

d) 2.1 23 i) 2.21 42

e) 3.12 102 j) 11.2 203

a) Write the total cost of 12 shirts and 12 pairs of shorts as i) one expression ii) a product of factors. b) If a shirt cost $50 and a pair of shorts cost $40, use the two results in a) to calculate the total cost of the shirts and pairs of shorts. Explain your working to a friend.

80


Shirt $50.00

Shorts $40.00

Looking back Find the next two terms, an expression for the general term and the value of the twentieth term in each of these sequences. a) 1, 6, 11, 16, 21 …

b) 8, 12, 16, 20, 24 …

c) 3, 7, 11, 15, 19 …

d) 7, 13, 19, 25, 31 …

e) 4.5, 2.0, 0.5, 3.0 …

The general term of a sequence is expressed as 4 3n. a) What is the difference between the terms? b) Is the sequence ascending or descending? c) What is the value of T(1)?

The general term of a sequence is n2 3. a) What type of sequence is this? b) Find the values of the first five terms.

Simplify 4x 7 x a) 6 2

14t 4t b) 6 3

14d 35d2 c) 7 2d

2x2y 20n3 d) 2 2 5mn 8xy

2x 10x e) 5y y5

mn8 m6n3 f) p2 p

Solve the following equations. a) 4x – 5 8 2x

b) 4x 5 3(2x 10)

c) 5(x 4) 3 5

d) 2(x 4) 3(4x 9) 2(3x 3)

e) 2 (x 7) 4(2x 9) 4

Make a the subject of the formula if a) t amn r

b) v x at

(ax w) c) k . p

The length of a rectangular field is 9 m shorter than twice the width. If the perimeter of the field is 78 m, write an equation and solve it to find the length and width of the field.

Given the function f(x) 4x 3 calculate a) f(3) d) f(1)

b) f(4) e)

1 f(2)

c) f(0) f) f(0.2)

Find the highest common factor of a) 3x and 12

b) 4k and m

d) 5pq and mp

e) 12xyz and 3xy

c) 15d and 18f

Factorise a) 10x 35

b) 14d 7

c) 5xy 2y

d) 24xyz 10y

e) 5ab 3ac 2abc

f) 4a 6ab 14ac Expressions, equations and formulae

81

R2

Revise and consolidate A jet travelled a distance of 4 500 km in 3 hours. What was its speed? A large bird flies at a speed of 12 km/h. How far will it fly in 40 minutes? Tom gets paid US$100 for 8 hours of work. The payment varies directly with the number of hours worked. a) How much does Tom get paid for i) working 3 hours ii) working 12 hours? b) How many hours does Tom work if he is paid i) US$75.00 ii) US$162.50?

10 people take 25 days to complete a building job. If the number of days varies inversely with the number of people a) how many people will complete the job in 50 days? b) how many days will it take 6 people to complete the job?

Simplify a) x4 x5

b) c2 c3

c) 4x2 3x3

d) a3 7a3

e) d4 6d2 6

f) 3s2 5s4 2s

g) 2x2 3x4y4

h) 3a2b3 5a2 2b2

i) xy xyz2

j) 2x2y 4x2z3 4z2

k) 2m2 2n2

l) (y2)4

m) (4a2)3

n) (2x4)3

o) 3(m3)3

Simplify x5 a) 4 x

15a5 b) 3a3

12xy3 c) 3xy

a4 d) a

1 b) y2 ÷ 4 y

c) (xy )4 x

d) (6x3)3

x7 e) 9 3x

Simplify a) (6x0)2

1

Find the next two terms, an expression for the general term and the value of the twentieth term in each of these sequences: a) 1, 3, 5, 7, 9, … b) 0.08, 0.12, 0.16, 0.20, 0.24, … c) 5, 9, 13, 17, 21, … d) 7, 13, 19, 25, 31, … e) 45, 20, 5, 30, …

82


1

e) (12b3)3

The general term of a sequence is expressed as 5 2n. a) What is the difference between the terms? b) Is the sequence ascending or descending? c) What is the value of T(1)?

The general term of a sequence is n2 4. a) What type of sequence is this? b) Find the values of the first five terms.

Solve the following equations: a) 2x 5 8 2x b) 6x 5 5(2x 10) c) (x 4) 3 5 d) 3(x 4) 2(4x 9) 5(3x 3) e) 4 (x 5) 3(2x 8) 3

Make a the subject of the formula if: a) b amn k

ax l c) j m

b) x y at

The length of a rectangular field is 2 m shorter than twice the width. If the perimeter of the field is 112 m, write an equation and solve it to find the length and width of the field.

Given the function f(x) 2x 6, calculate: a) f(3)

b) f(4)

c) f(0)

d) f(1)

e) f(12)

f) f(0.2)

Find the highest common factor of: a) 12x and 4 b) 5k and 20m c) 15c and 18cd d) 5pq and pqr e) 120xyz and 30x

Factorise: a) 12x 36 b) 21p 14 c) 10mn 2n d) 12abc 10c e) 5mn 3mp 2mnp f) 6a 4ab 12ac


83

Chapter

7

Key words

Working with money

Foreign exchange Exchange rate

Working with money is an important life skill – the more we learn, the better equipped we become to deal with the issues that confront us on a daily basis. In this chapter you will deal with foreign currencies and hire purchase. You will apply your mathematical skills to solve money-related real-life problems.

Unit 7.1 Foreign exchange In this unit you will: review how to convert from one currency to another using given exchange rates.

You learnt last year that countries have their own currencies and that when you travel to a foreign country you need to exchange the money from your own country to that of the country to which you are travelling. Money is normally changed at a bank or foreign exchange bureau (cambio). The rate at which one currency is exchanged for another is called the exchange rate. When you change your money into another currency you do so at the current rate of exchange. Banks and foreign exchange bureaux take a commission on the exchange of currency. When they sell you another currency, they charge a slightly higher rate than when they buy foreign currency from you – that way they also make a slight profit on the transaction. The rate of exchange is normally displayed on a board in the bank e.g.:

Selling

Buying

$1 US $2.70 EC

$1 US $2.65 EC

You can also go online to check the $1 BDS $1.40 EC current exchange rates and calculate how much money you will get in exchange for different currencies. One site which is very useful is www.xe.com but your local bank may also have a website that publishes this information.

$1 BDS $1.35 EC

Worked examples 1. Julie is travelling to Barbados and needs to change EC$ to BDS$. If the exchange rate is $1 BDS = $1.40 EC, how much in BDS$ currency does Julie get for $250.00 EC? 2. If US $1 = $2.65 EC, how much in EC currency does Dan receive for US$120?

84

1. $250 ÷ 1.40 = $178.57 BDS

$1 BD is less than $1 EC, so we divide.

2. $120 x 2.65 = $318.00 EC

$1 US is more than $1 EC, so we multiply.


Exercise Ms Jones is changing US dollars to EC dollars. If the exchange rate is $1US $2.65 EC, how much in EC dollars does she receive for a) $185 US b) $199 US c) $650 US?

The table shows you how much of each Caribbean currency you would get for one US dollar (in 2005). Caribbean currency for one US dollar (US$1) Bahamian dollar

B$1.00

Barbadian dollar

BD$1.99

Cayman Islands dollar

KYD0.82

Cuban Peso

CUP26.00

Haitian Gourde

HTG36.24

East Caribbean dollar

EC$2.70

Jamaican dollar

Ja$61.60

Netherlands Antilles Guilder

ANG1.77

Trinidad and Tobago (TT) dollar

T&T$6.20

An American Forex bureau at Miami airport changed US dollars to local currency for tourists leaving the USA. The destination and the amount of US dollars are given. How much did each tourist receive? State the currency in your answers. a) St Vincent US$1 500 b) Cayman Islands US$2 000 c) Bahamas US$500 d) Tobago US$4 099 e) Barbados US$350 f) Dominica US$150

Mr Serieux is converting EC dollars to US dollars at an exchange rate of $1 US $2.70 EC, how much US currency does he receive for a) $550 EC b) $1 000 EC c) $125 EC?

Rita is travelling from the United States of America to Barbados. If the exchange rate is $1 US $2.15 BDS, how much BDS currency does Rita receive for a) $1 000 US b) $1 040 US c) $250 US?

When returning from Barbados to the United States of America, how much US currency does Rita receive for $65 BDS if the exchange rate is $1 US $2 BDS?

Five people are travelling from various parts of the world. The table below shows the amounts that these persons need to convert to EC currency and the exchange rate. a) Calculate the EC equivalent of each amount. b) The agent charges 2% commission on foreign exchange. Deduct this amount from each total. Country of origin

Exchange Rate

Amount

Barbados

$1 BDS $1.35 EC

$200 BDS

USA

$1 US $2.65 EC

$600 US

Canada

$1 CAN $2.10 EC

$1 020 CAN

Trinidad

$1 TT $2 EC

$1 500 TT

England

£1 $5.40 EC

£190

Working with money

85

Key words

Unit 7.2 Hire purchase

Credit Deposit Balance Instalments Hire purchase Interest

In this unit you will: understand what is meant by hire purchase (HP) perform calculations to compare prices, work out interest and find total costs related to hire purchase agreements. Sometimes people want to buy something but they do not have enough money to pay for it immediately. In these cases, the shopper can buy on credit. In other words, the shopper pays part of the price as a deposit and agrees to pay the rest of the amount owed, or the balance, in instalments over a fixed period of time. A common form of buying on credit is called hire purchase. In this system, the shopper pays a deposit and signs an agreement to pay fixed monthly amounts till the goods are paid for. The shopper can take the goods home but they technically belong to the shop until they are paid for in full.

CASH

$299

Deposit $45 Six payments of $50.80

Hire purchase costs more than when you pay cash because the shop charges interest on the outstanding balance. To calculate the hire purchase price, add the deposit to the total of the monthly payments. In the example above the hire purchase price is: $45 (6 $50.80) $45 $304.80 $349.80 The interest can be calculated using a formula: Hire purchase price Cash price Interest paid Using the example above: $349.80 $299.00 $50.80 The customer paid $50.80 interest on hire purchase. This means it would be $50.80 cheaper to pay cash for the bicycle.

Worked examples The cost price of a mantelpiece is $1 500. A shopper makes a deposit of 15% and pays 12 monthly instalments of $180. Find a. the hire purchase price b. the interest earned. a.

15 x $1 500 = $225 100

12 x $180 = $2 160 Hire purchase price = $2 160 + $225 = $2 385 b. Interest = $2 385 $1 500 = $885

86


Exercise The cash price of a table is $795 and the hire purchase price is $900. Find the interest charged on hire purchase.

Tonya buys a television on hire purchase. She pays a deposit of $150 and then pays 24 monthly instalments of $116. Calculate the hire purchase price of the television.

Matthew buys a settee on hire purchase. He pays a deposit of $400 plus 18 monthly instalments of $300. Find

Cash

price

0

$300

a) the hire purchase price b) the interest paid.

The cash price of a bed is $2 195. Luke bought the bed on hire purchase. He paid a deposit of $250 and 24 equal monthly instalments. The hire purchase price was $4 500. Calculate a) how much interest he paid

b) his monthly instalment amount.

The cash price of a stove is $4 500. The stove was bought on hire purchase. There was no deposit and 24 monthly instalments of $250 were paid. Calculate

CASH PRICE

$4500

a) the hire purchase price b) the interest charged c) the percentage interest charged on HP.

The cash price of a fridge is $8 000. Calculate a) the deposit in dollars Pay cash $8 000 or 15% deposit + 36 monthly payments of $250

b) the hire purchase price c) the amount paid in interest.

The cash price of a computer is $3 800. It was sold on hire purchase at an interest rate of 60%. Calculate a) the hire purchase price b) the interest in dollars. c) If no deposit was paid, calculate the monthly instalments if equal payments were made over 16 months.

A dressing table was bought on hire purchase. A 20% deposit was paid plus 20 monthly instalments of $360. The hire purchase price was $8 000. Calculate a) the deposit in dollars

b) the cash price

c) the interest charged on hire purchase as a percentage. Working with money

87

Looking back The cash price of a microwave oven is $2 400. If bought on hire purchase, a deposit of 20% is required, plus 12 monthly instalments of $275. Calculate a) the hire purchase price b) the difference between the cash price and the HP price.

A television costs US$899. Mrs Jones can’t afford to pay cash so she puts down 15% deposit and 12 monthly payments of US$67.50. a) How much does she pay for the television in total? b) How much could she have saved if she’d paid cash upfront?

A video camera can be bought for US$650 cash. You can also buy it on HP for a 10% deposit and 26 payments of US$26.75. a) How much deposit is required to buy the camera on HP? b) What is the total price you would pay if you bought the camera on HP? c) What is the difference between the cash price and the HP price?

The following table shows approximately how many JA$ you would get for 1 unit of various currencies. Jamaican dollars per unit of foreign currency US dollars ($)

65.08

British pounds (£)

113.12

Euros (€)

77.50

=) Japanese yen (Y

0.55

a) Change the following amounts of foreign currencies in Jamaican dollars. = 2 000 i) US$150 ii) £225 iii) €150 iv) Y b) How much of each of the following currencies would you get for JA$2 500? i) yen ii) euros iii) US dollars iv) British pounds c) What would a shirt costing JA$3 750 cost in i) yen ii) euros?

88


Key words

Volume Cubic units

Chapter

Volume In this chapter you will review previous work on volume of cubes and cuboids and units of volume (cubic units). You will also learn how to calculate the volume of prisms so that you can solve a range of problems dealing with volume and capacity.

8

Unit 8.1 Cubes and cuboids In this unit you will: convert between units of volume calculate the volume of cubes and cuboids.

The volume of a 3-D shape is the space contained inside it. The formula for the volume of a cuboid of width w, length l and height h, is V w l h.

h

You can also think of the volume as the area of the cross section multiplied by the length.

l w

Common metric units of volume are the cubic millimetre (mm3), the cubic centimetre (cm3) and the cubic metre (m3). Volume (in cm3)

2 cm 3 cm ↓

↓

10

10

↓

↓

5 cm ↓ 10 ↓

30 cm3

5 cm

↓ 1 000 ↓

3 cm

Volume (in mm3) 20 mm 30 mm 50 mm 30 000 mm3 So we can see that 1 cm3 1 000 mm3.

2 cm

In the same way, 1 m3 100 cm 100 cm 100 cm 1 000 000 cm3

Worked example Copy and complete the following. cm3 a. 5 200 mm3 c. 1.4 cm3 mm3 a. b. c. d.

5 200 mm3 5.2 cm3 3.8 m3 3 800 000 cm3 1.4 cm3 1 400 mm3 125 575 cm3 0.125 575 m3

b. 3.8 m3 cm3 d. 125 575 cm3 m3 5 200 1 000 5.2 3.8 1 000 000 3 800 000 1.4 1 000 1 400 125 575 1 000 000 0.125 575

Volume

89

Exercise Convert these cubic centimetres to cubic metres. a) 5 250 000 cm3

b) 754 000 cm3

c) 95 700 000 cm3

d) 82 500 cm3

e) 7 500 cm3

f) 1 450 cm3

Convert these cubic millimetres to cubic centimetres. a) 5 000 mm3

b) 8 534 mm3

c) 72 500 mm3

d) 825 mm3

e) 75 mm3

f) 12 mm3

Convert these cubic centimetres to cubic millimetres. a) 3 cm3

b) 25 cm3

c) 2.5 cm3

d) 1 425 cm3

e) 0.27 cm3

f) 0.6 cm3

Convert these cubic metres to cubic centimetres. a) 7 m3

b) 3.547 m3

c) 6.5 m3

d) 0.5 m3

e) 0.327 58 m3

f) 99 m3

A small specimen box measures 15 mm by 25 mm by 32 mm. a) Convert the lengths to cm. b) Work out its volume in i) cubic millimetres

ii) cubic centimetres.

This box measures 25 mm by 40 mm by 7.5 cm. Calculate its volume in cubic centimetres. 40 mm

25 mm 7.5 cm

An aquarium in the shape of a cuboid has a base with dimensions 1.2 metres by 0.4 metres. What is the volume of the tank, in cm3, if it has a height of 60 cm?

Joshua has four identical cubes of wood. The length of each side of the cube is 1.3 m. What volume of wood does he have?

90


Key words

Unit 8.2 Prisms

Prism

In this unit you will: learn how to calculate the volume of prisms solve problems involving volume.

A prism is a three-dimensional shape that has two identical end faces and a uniform cross section. The cuboid is a prism with rectangular faces while a cylinder is a prism with circular faces. The volume of a prism is found by multiplying the area of the base (the cross section) by the height. The formula is expressed as V Ah, where A is the area of the cross section and h is the height.

h

h h

A Cuboid

A

A

Cylinder

Triangular prism

A

h

Trapezium-shaped prism

Worked example Find the volume of the prism shown. Area of cross section 4.1 cm 4.1 cm 16.81 cm2 Height 9 cm 9 cm

V Ah 16.81 cm2 9 cm 151.29 cm3

4.1 cm

Exercise Find the volume of the prisms below. a)

b)

5 cm

c)

10 cm

5 cm

12 cm 8m

12 cm

6.5 cm

Volume

91

d)

e)

5.5 cm

f)

30 cm

12 cm

10 m

7m

40 cm

16 m

7 cm

14 m

Calculate the volume of the cylinders shown. a)

b)

21 cm

15 cm 50 cm 36 cm

What is the volume of

5 cm

this can to the nearest decimal place? 12 cm

The volume of a cube is 4 096 cm3. What is the length of a side of the cube?

Looking back The diagram shows the cross section of a pond. The length of the pond is 2.5 m.

1m

a) What is the volume of the pond? b) The owner of the pond decides to line the pond. What area needs to be lined?

3.5 m

A cylindrical can has a volume of 3 080 cm3. If the diameter of the base of the can is 14 cm, what is the height of the can to one decimal place?

A rectangular prism has a length of 55 mm, a width of 6 cm and a height of 14 cm. What is the volume of the prism a) in cubic millimetres

b) in cubic centimetres?

The diagram shows the aerial view of a swimming pool. If the pool is 1.2 metres deep at its deepest point, what is the volume of the pool?

92


2.4 m

10 m

Key words

Quadrilaterals Diagonals Lines of symmetry Bisects

More about shapes and solids In this chapter you will again examine quadrilaterals and solid shapes in your environment. You will group them according to various properties; use the properties of the quadrilaterals to calculate angles; observe solids from various viewpoints and draw two-dimensional representations of them.

Chapter

9

Unit 9.1 The properties of quadrilaterals In this unit you will: differentiate between diagonals and lines of symmetry use the properties of quadrilaterals to calculate angles.

Quadrilaterals are polygons with four sides. Remember that polygons are closed plain shapes bounded by straight lines.

Square

Trapezium Kite

Rectangle

Rhombus

Parallelogram

A diagonal is a line of symmetry only when one half of the quadrilateral fits exactly on the other half, when folded on the diagonal. This line of symmetry bisects (cuts in two equal parts) opposite angles.

Remember the following facts. ● The sum of interior angles of a quadrilateral 360°. ● Alternate angles are equal when lines are parallel. ● Corresponding angles are equal when lines are parallel. ● Vertically opposite angles are equal.


93

Worked examples 1. Which of the diagonals shown in these diagrams are lines of symmetry?

a. A

B

D

C

b.

Q

R

P

S

2. Calculate the value of p, q, r and s.

q

p

48°

s

r

82°

C 1. The diagonal of a rectangle is a. b. Q not a line of symmetry. When it A B is folded on the diagonal the opposite vertices (A and C) do not coincide. PR S D The diagonal of a rhombus is a line of symmetry. When it is folded on the diagonal the opposite vertices (P and R) coincide. 2. p 48° 82° 180° Sum of angles of a triangle. So p 180° 130° 50° q 82° Alternate angles. r 48° Alternate angles.

s 50°

Exercise Work with a partner to write a passage describing the diagram. You will have to use your knowledge of the special quadrilaterals (square, parallelogram etc): their sides, angles, and diagonals. Pay attention Kite to the linkages and the direction of the arrows.

Square

Trapezium

Rectangle

Rhombus

Parallelogram

Square → Rectangle means a square is a rectangle. 94


Copy and complete the table by filling in

Sq

Yes or No. Use the diagrams given to help you to describe the diagonals of the quadrilaterals. The names are abbreviated.

K

T Q

Rh

Re

P

Properties of diagonals

Quadrilateral(s) Sq

Both are lines of symmetry.

Yes

Only one is a line of symmetry.

No

Re

Rh

K

T

P

Any Q

Yes No

No

No

Yes

No

No

No

They are equal. They bisect each other.

Yes

Yes

Only one bisects the other. They are perpendicular. They bisect the opposite angles. Only one bisects opposite angles.

Calculate the value of the angles marked with

29°

letters in the rectangle. Give reasons as shown in example 2. c

Calculate the value of the angles marked with

a

d

b

letters in the trapezium. Give reasons.

104°

52°

a c


95

Calculate the value of the angles marked with letters in the rhombus. Give reasons. e a

c d

b 158°

Draw a sketch of a rectangle. Label the vertices P, Q, R and S. Draw the diagonal PR and label PRQ 35°. Calculate SPR, giving reasons for your answer.

Draw a sketch of a rhombus. Label the vertices A, B, C and D. Draw the diagonal BD and label BCD 55°. Calculate ABD, giving reasons for your answer.

Calculate the value of x clearly stating any angle facts you use. a)

b)

x

x

40°

96


30°

Key words

2-D 3-D Plans Elevations

Unit 9.2 Viewing and drawing solids In this unit you will: recognise the plans and elevations of simple solid shapes sketch the plans and elevations of simple solid shapes.

Remember 3-D drawings are the representations of solid shapes as seen in our environment. 2-D representations of shapes are plane figures representing surfaces. They have no thickness. The diagram shows a model of a barn.

The plan is the view seen when looking directly from above.

Plan view

The elevations are the views seen from the other directions, e.g. front and sides. Above

Front elevation

Front

Side elevation

Side

Worked examples 1. The diagrams show the plan view, and the front and side elevations of a solid. Draw a sketch of the solid.

Plan view

Front elevation

Side elevation

1.

Note the lines that coincide or overlap with others. Look for lines that are parallel, perpendicular or equal. More about shapes and solids

97

2. Draw the plan view, front and side elevations of this solid. Note that a dashed line shows hidden detail. Plan

Front and Side

Exercise The diagrams show the plan view, and the front and side elevations of a solid. Draw a sketch of the solid. a) Front elevation

Side elevation

b) Front elevation

Plan view

c)

Side elevation

Plan view

d) Front elevation

Side elevation

Plan view

Front elevation

Side elevation

Plan view

Draw the plan view, front and side elevations of these solids. a)

b)

Each of these diagrams shows the plan view of a solid. For each one, sketch two possible solids that the diagram could represent. a) b) c)

What 3-D shape am I? a) My front and side elevations are triangles. My plan view is also a triangle. b) My plan and my front elevations are rectangles. My side elevation is a triangle. c) My plan and my front elevations are rectangles. My side elevation is a circle.

A game for two players. Draw a sketch of a 3-D shape. Do not show it to your partner. On another piece of paper, draw the plan view, and front and side elevations and give them to your partner. Ask your partner to use them to sketch your 3-D shape. Compare the two sketches of the 3-D shape. Swap roles and repeat the process two more times.

Investigation Triangles can be made by joining three of the vertices of a cube. How many different shaped triangles can you make by joining three vertices of a cube? Record your answers using isometric paper. 98


Looking back Find the value of x in each of the following quadrilaterals. Give reasons for your answers. A

a)

M 85°

b)

75°

N x

P

c)

Q 2x

B x 95°

80°

O

85°

D

x

40°

S

P

R

C B

d) A 6x

T

e)

D

f)

2x 80° Q

P

2x

x

3x

D

E 42°

C

S

G

110°

F

x R

g)

h)

K

Q x

P

R

80° x L

N

75° 50°

T

S

M

Mike has photographed these solids from the front. Draw them in plan view.


99

Case study – Signwriter Graeme Morrison I own and run a small sign shop in a village and, even though the village is quite small, I am normally fairly busy. My business often involves using quotations and so I have to measure the surface area of the sign to base the quotation on. I need to be careful with my calculations or the project could end up costing me money! Let me first, however, explain what the art of signwriting entails. There are not many true sign “writers” left. Computers and technology have changed the way signs are made. I suppose one could call the new generation sign makers rather than writers. In my job, the first step would be a call from a customer. I would then set up an on-site meeting with them to discuss what signage they want. At the meeting (which has now cost me money in petrol), I will measure the space in which the sign will be erected. The space could be a steel plate to which we will apply adhesive vinyl, a wall that will be painted or a glass shop front. I need to use maths to first measure the space and then to convert the area into square metres. Back in the office I will design the sign on a computer and from there calculate how much material I will need. I will also need to figure out how much petrol is going to cost me. Some projects may be as far as 40 km away and may need several days work with the travelling that it entails. Any extra labour needed will also need to be calculated. So let’s do some maths and work out a quote. We have a large agricultural expo annually near to the village and I do a lot of work for a large petroleum company who is the major sponsor and advertiser. They want two double sided signs mounted in frames, measuring 1.8 m 6 m and attached to the top of a 4 m high wall. We will use white coated steel. The steel comes in some standard sizes, 1.8 900 mm being one of them. We need to know how many sheets we will need: 6 m 900 mm 6 m 0.9 m 6.6 m I will need 7 sheets per side and will trim a piece off to get to 6 m. Coated steel of this size costs $150 per sheet so 4 signs, each using 7 sheets will cost: $150 7 4 $4 200.00. Now we need to work out how much adhesive vinyl we will need: 1.8 m 6 m 10.8 m2 4 signs 10.8 m2 43.2 m2 My price for cut and applied vinyl is $250 per m2: $250 43.2 m2 $10 800 The steel for the structure that will hold the sign has to be added. To do this I need to calculate how much steel I need. After drawing up a plan I can determine how much steel I need. This would be in metres. Steel also comes in 6 m lengths, so I need to work my waste in. Steel is $20 per m, we need 30 m per frame and we are making 2 frames: $20 30 m 2 $1 200.00 Now we have to cost the following: ● ● ● ● ● ●

welding rods and electricity the salary for the guy who works for me the chemicals to clean the steel frame the undercoat and paint and thinners to paint it the drill bits and rivets and rawl bolts to assemble it the extra guys I will need to employ for a few days to erect the sign ● the truck with crane I will need to hire to transport and lift the sign onto the wall. Maths, maths and more maths! As you can see, everything around me in the running of a business involves maths. In addition to each job, I have to work out salaries, petrol costs, rent, electricity and water costs. The list is endless!

100 Jamaica Maths Connect 3

Key words

Construct

Chapter

Construction In this chapter you will use geometrical instruments to learn how to construct triangles. You will also learn to use rulers and compasses to bisect angles and construct perpendicular bisectors of lines. You will use these skills to construct some angles, using ruler and compasses only.

10

Unit 10.1 Constructing triangles In this unit you will: revise using a ruler and protractor to construct triangles. You should remember from last year that to construct a triangle accurately you need information about its sides and angles. You need at least a set of three bits of information to be sure the triangle you are drawing is the only one of its kind. In this unit we will revise two methods: ● the lengths of two sides and the angle between them (SAS), or ● two angles and the line that joins them (ASA).

Worked examples 1. Draw a triangle ABC with sides AB = 6 cm, AC = 2.5 cm and BAC = 50°. 2. Draw triangle CDE where CD = 3.6 cm, ECD = 50° and EDC = 45°. B

1. Rough sketch

Step 1

A

2.5 cm

C

6 cm 50° A

2.5 cm

C

Step 2

Step 3

B

6 cm 50° A

2.5 cm

C A

2.5 cm

C

Construction 101

2. Rough sketch

Step 1

C 50°

C

3.6 cm

3.6 cm

45° E

D

D

Step 2

Step 3

C 50°

C

3.6 cm E 50°

45° 3.6 cm D D

Exercise Construct each triangle accurately. Measure the unknown side or angle in each case and write it down.

a)

b)

c)

a

b 60°

40°

60°

8 cm

x

45°

50°

50°

9 cm

d)

e)

7 cm

10.5 cm 77°

f)

52°

m 110°

y 30°

60°

x

10.5 cm

60° 6.8 cm

g)

h)

i) 12 cm

90° l

8 cm

y

9 cm

65° 9.5 cm

30° 30°

m

10 cm

j)

x

35° 30° 8 cm


k)

30°

m

55° 90° 8 cm

90° 40°

70°

100°

Key words

Unit 10.2 More constructing triangles

Intersecting arcs Sketch Construction Hypotenuse

In this unit you will: construct triangles given the lengths of the three sides construct triangles given a right angle and the lengths of the hypotenuse and another side.

In the previous unit you used a ruler and a protractor to construct triangles. In this unit you will learn to construct triangles using a ruler and a pair of compasses. In order to construct with a ruler and a pair of compasses, you need to be able to draw intersecting arcs. You also need to be able to construct a perpendicular line at a point (to make a 90° angle). We can use our compasses to measure lines and work out where they meet. By adjusting the distance on our compasses, we can determine where two sides will meet:

A

B

Start with a line. Mark two points.

A

B

A

B

A

Draw another arc using point B as your centre.

Draw an arc using point A as your centre.

B

Use a ruler. Join A and B to the point of intersection.

We can use our compasses to construct a perpendicular on a given point:

X A

Start with a line. Draw a circle.

Y A

A

Increase the compass distance. Use the circle and line intersections (X and Y) as starting points. Make arcs above and below the line.

A rough sketch is a drawing that looks like the figure. It has no accurate measurements. It helps us to make decisions about our constructions. When we construct diagrams we make accurate drawings with the use of instruments. Do not erase your arcs, and make sure that your pencil point goes exactly through the points of intersection.

Join the intersections to draw a perpendicular line.

A

6 cm

C

12 cm

B

Construction 103

Worked example Construct triangle MNO with MN 8.5 cm, MO 5 cm and NO 7 cm. a. Do a labelled sketch first so that you are sure where to start. b. Draw a line, a little longer than MN. Sometimes it is better to start with the longest side. c. Then use your compasses to measure 8.5 cm on the ruler and mark off this measurement on the line. Label it MN. d. Open the compasses to 5 cm, use M as the centre and draw an arc. e. Open the compasses to 7 cm, use N as the centre to draw an arc to cut the first arc. Label the intersection O. f. Join MO and NO. b, c.

M 8.5 cm

N

d, e.

M 8.5 cm

O N

f.

M 5 cm O

8.5 cm

7 cm

N

In a right-angled triangle the longest side is called the hypotenuse. It is opposite to the right angle.

hypotenuse

hypotenuse hypotenuse


Worked example Construct triangle ABC with AB 7.5 cm, BC 6 cm and ACB 90°. a. Do your sketch. From your sketch, you will notice that the angle given is not the included angle but it is a right angle. The length of the hypotenuse is also given. b. Draw an accurate measurement of BC. This side has two other measurements from it. c. Use your protractor to measure 90° at C and draw the line perpendicular to BC at C. d. Open your compasses to 7.5 cm, use B as the centre and draw an arc to cut the perpendicular line. e. Label the intersection A and join AB.

b.

a. A

c, d.

e.

A 7.5 cm

7.5 cm 90° C

6 cm

B

6 cm

C

B

C

90° 6 cm

B

C

6 cm

B

Exercise Construct the following triangles. Measure and write down the sizes of the angles and sides that are not given. a) ABC, with AB 8 cm, BC 6 cm and AC 10 cm b) EFG, with EF 5 cm, FG 4 cm and EG 4 cm c) HIJ, with HI 6 cm, HJ 6 cm and IJ 6 cm d) KLM, with KL 3.5 cm, LM 7.5 cm and KM 8.0 cm e) TUV, with TVU 90°, TU 9 cm and VU 7 cm

Construct the following right-angled triangles. Measure and write down the length of the third side and the size of the other two angles. a) hypotenuse 8 cm, side 5 cm b) hypotenuse 6 cm, side 3 cm

Investigation Construct triangle ABC with CAB 50°, AB 7 cm and BC 5 cm. Try again with BC 5.5 cm and then with BC 6 cm. You will notice that sometimes the arc from B does not intersect the other arm and sometimes it intersects at two places. C2 6 cm C1 6 cm 50° B

7 cm

50° A

B

7 cm

A

Now write a statement, telling a friend what happens when you are given the lengths of two sides and an angle, but the given angle is not included between the given sides. ASS or SSA Construction 105

Key words

Unit 10.3 Bisecting angles and lines

Bisector Compasses Perpendicular bisector Mid-point Equidistant

In this unit you will: construct the bisector of an angle using a ruler and compasses construct the mid-point and perpendicular bisector of a line segment, using a ruler and compasses.

The bisector of an angle is a line that divides the angle into two equal parts. You can construct the bisector of an angle using compasses. A Q x x

B

In this diagram BD is the bisector of the angle ABC. Every point on the line BD is equidistant from the lines BA and BC.

D

D1 P C

B

The perpendicular bisector of a line segment divides the line segment into two equal parts at right angles.

A

In this diagram, BD is the perpendicular bisector of AC. It crosses AC at the mid-point (M) of the line. Every point on the line BD is equidistant from both A and C. If you join ABCD a rhombus is formed.

M

C

D Perpendicular bisector

Worked examples 1. DEF is 100°. Construct the bisector of DEF. D

D

Open the compasses and put the point on E. Draw an arc that intersects with ED and EF.

D G

F

E

E

F

E

An arc is part of a circle. You can draw arcs with compasses.

Do not change the opening of the compasses. Put the point F first on the intersection of the arc with ED and then with EF. Draw new arcs to intersect at G. Join EG.

2. The line segment PQ is 2.5 cm long. Construct the perpendicular bisector of PQ.

P

2.5 cm

Q

P

Q

P

Draw a line of 2.5 cm. Open up the compasses to over half the length of PQ.


Q

Place the point at P and draw an arc. Keep the opening of the compasses the same and repeat at Q. Join the points where the arcs intersect.

Exercise Draw angles, at least one acute, one obtuse, one right-angled, one straight, and one reflex. Construct the angle bisectors using a ruler and compasses. Check your result for each by measuring the two angles, making sure they are the same.

Draw a line 10 cm in length. Construct the perpendicular bisector of the line using only a ruler and compasses. Mark the mid-point (M) of the line.

A new water channel is to be built at the zoo. It is to be placed between two sets of cages so that the cages are equidistant from the channel. Copy this diagram and construct a red line to show the position of the water channel.

Cage 1

Cage 2

A new fence is to be built in a park. The fence is to be placed between two large trees so that the trees are equidistant from the fence. The head gardener decides to paint in a line to mark the position of the fence. Copy this diagram and construct a blue line to show the position of the fence.

Tree 1

Tree 2

Investigation Draw any triangle with sides longer than 4 cm but shorter than 8 cm. Construct the angle bisector for each of the three angles. Make sure the angle bisectors are long enough to cross each other. What do you notice? ● Is it possible to use your information to draw a circle inside the triangle? The circle should just touch the sides. ● Is it possible to use your information to draw a circle that touches the three vertices? Draw another triangle approximately the same size. Construct the perpendicular bisectors of each of the three sides. Make sure the perpendicular bisectors are long enough to cross each other. What do you notice? ● Is it possible to use your information to draw a circle inside the triangle? The circle should just touch the sides. ● Is it possible to use your information to draw a circle that touches the three vertices? Construction 107

Key words

Unit 10.4 Perpendiculars

Construct Perpendicular Arc

In this unit you will: use a ruler and compasses to construct the perpendicular from a point to a line and from a point on a line. We can use a ruler and compasses to ● construct the perpendicular from a point to a line

● construct the perpendicular from a point on a line.

An arc is part of a circle. You can draw arcs with your compasses.

Worked examples 1. Make a copy of the diagram. Using only a ruler and compasses draw a perpendicular from the point P on the line. P

P

P

Q

Q

1) Using compasses, draw two arcs from P to make two intersections on the line. From the two intersections, draw two arcs that intersect and label the intersection Q. 2) Place a ruler from P to Q and join them with a straight line. 2. Make a copy of the diagram. Using only a ruler and compasses draw a perpendicular from the point A to the line. A

A

A

B

B

1) Using compasses, draw two arcs from the point A that intersect with the line. From the two arcs draw two more arcs that intersect and label the intersection B. 2) Place a straight edge from A to B and join A to the straight line.


Exercise Make a copy of the diagram. Using a ruler and compasses, construct a perpendicular from the point D on the line. 2 cm

D

3.5 cm

Make a copy of the diagram. Using a ruler and compasses, construct a perpendicular from the point E to the line.

E

Jamina is walking her dog in the park

Jamina

when it starts to rain. She wants to take the shortest route back to the path. Copy the diagram and construct the shortest route to the path, using a ruler and compasses.

The perpendicular from a point to a line is the shortest distance from the point to the line.

Path

a) Draw an equilateral triangle, using a ruler and a protractor, with sides 5 cm in length, as shown here. Construct the perpendiculars from each vertex to the opposite side. What do you notice?

60°

5 cm

5 cm

60°

60° 5 cm

b) Draw an isosceles triangle with base 5 cm and base angles of 50°, as shown here. Construct the perpendiculars from each vertex to the opposite side. What do you notice? c) Draw a scalene triangle with base 5 cm and base angles of 70° and 30°, as shown here. Construct the perpendiculars from each vertex to the opposite side. What do you notice?

50°

50° 5 cm

70°

30° 5 cm

Construction 109

Key words

Unit 10.5 Parallel lines

Copying angle

In this unit you will: use a ruler and compasses to construct parallel lines. When lines are parallel the corresponding angles and alternate angles are equal. This means that we can copy angles to get parallel lines.

Worked examples 1. Construct a line parallel to PQ at R.

2. Construct the rectangle D ABCD with AB CD 4 cm, AD BC 5 cm. A

Q

P

R

1. Step 1. Put the point of the compasses at P and scribe an arc to cut PR and PQ at X and Y respectively. Q

Y

P

X

Q

B

Step 1. Construct with ruler and compasses the 4 cm A line AB 4 cm.

B

Step 2. Construct a perpendicular at A.

R

Step 2. Using the same radius, scribe an arc with centre R to cut the line at Z.

C

A

4 cm

B

A

4 cm

B

Step 3. Copy the 90° angle at B.

Y

P

X

R

Step 3. Open to radius XY and use that radius with centre Y Z to cut the second arc. P

Z Q

X


Z

Q

Step 4. Join R to this intersection and continue Y the line. P

R

Step 4. Open the compasses 5 cm and using A and B as centres, scribe arcs on the perpendicular lines to get D and C. D C Join DC.

5 cm

X

R

Z

5 cm

A

4 cm

B

Exercise Use ruler and compasses to construct the following. Remember to do a sketch first and put in all the information you know about the shapes, for example, parallel lines and equal angles.

Draw any line and follow the steps in example 1 again, but this time work on the other side of the line to get alternate angles equal.

Construct a parallelogram PQRS with PS PQ 5 cm, and diagonal QS 7 cm. Measure the other diagonal PR.

Construct an equilateral triangle, and then construct lines parallel to each side through the vertices. An incomplete sketch is shown in the diagram. Cut out your diagram and fold it into a solid shape.

Draw any angle. Use your compasses to cut off equal parts on one arm. Join the end points to complete a triangle. Construct parallel lines from the intersections to the other arm. What do you observe about the sections on the second arm?

The sketch shows three intersections. Challenge your friends to do other parts and talk about your results.

Construction 111

Key words

Unit 10.6 Constructing angles

Bisection Equilateral triangle

In this unit you will: construct 60° and 90° angles, with ruler and compasses only use the bisection of angles to construct combinations or parts of 60° and 90°.

You can bisect 180° to get 90° and 90° to get 45° so you can use ruler and compasses only to construct some angles.

45° 45°

90° 12 of 180°

45° 12 of 90° 1

It is possible to use this method to construct other angles such as 222°. Each angle of an equilateral triangle is 60°. We use this information to construct 60° with ruler and compasses only. Z

Z

60° X

Y

X

Y

X

Y

Put the centre at the point for the vertex X of the angle, scribe an arc about a quarter of a circle to cut the line (arm of the angle) at Y. Use the same radius with centre Y to cut the first one at Z. Since XY YZ XZ then XYZ is an equilateral triangle and ZXY 60°. Bisecting 60° gives 30°.

60°


30°

Worked examples 1. Construct, with ruler and compasses only, angles of a. 150° 1 b. 672° 2. Using a ruler and compasses only, construct triangle EFG with EF 5 cm, E 60° and F 45°. 1. a. 150° 120° 30° or 90° 60°

60° 120°

150°

b.

45° 1

1

67 21 °

135°

1

672 45 222 or 2 of 135 Remember, do not erase your construction lines. You may use any appropriate combination. 2. Use your compasses and ruler to construct EF 5 cm.

E

5 cm

F

At E construct an angle of 60°. Remember the equilateral triangle. E

At F construct an angle of 90°, then bisect 90°. G is where the two lines from E and F intersect.

5 cm

F

G

Be careful to use the acute angle. E

5 cm

F

Construction 113

Exercise Use a ruler and compasses only to construct the following angles. a) 15°

b) 75°

1

d) 222°

c) 135°

e) 105°

Construct the parallelogram ABCD with AB 6 cm, AD 4 cm and A 60° Remember your sketch and the properties of a parallelogram.

The diagram shows straight roads

Cinema

from Peter’s house to the bank and to the cinema. Use a scale of 1 cm to 250 m to do a scale drawing of the location and distance of the three places. Use a ruler and compasses only. What is the actual distance between the bank and the cinema?

1.5 km

Bank 120°

1 km

Peter’s House

Use 1 cm to represent 1 m to do a scale drawing of the side of a building. The width is 5 m, the height 3 m and the slant heights make angles of 30° with the horizontal.

30°

30°

3m

5m

Looking back Draw several lines and angles and then bisect them. Draw a line of 7 cm. Bisect it and use it to construct a square. Construct an equilateral triangle with sides of 8.5 cm. Construct an angle of 22.5° without using a protractor. Draw line AB of length 5 cm. Construct a 45° angle at A and use it to draw line CD AB.


Construct these triangles (all measurements in cm). a)

b) 12

5

6 65° 7

13

c)

d) 4 70°

30° 7

9

e)

f) 6 112°

120°

34° 6

9.5

g) ABC with AB 10 cm, BC 8 cm and AC 4 cm h) JKL with JK 90 mm, J 50° and K 60° i) MNO with MN NO MO 45 mm

Construct these shapes. a)

b)

5 cm

8 cm 8 cm

40° 60°

8 cm 30°

7 cm

Construction 115

R3

Revise and consolidate Using a ruler and compasses only, make a rough sketch and then construct: a) Triangle ABC with AB 8 cm, BC 7 cm, AC 6 cm. Measure the smallest angle. b) Triangle PQR with PQ 7.5 cm, QR 5.6 cm, PR 6 cm. Measure the smallest angle. c) Triangle LMN with LM 8.4 cm, LMN 60°, MN 6.2 cm. Measure LN. d) Triangle FGH with FG 6 cm, FGH 90°, FH 9 cm. Measure GH. e) Triangle JKL with JK 8 cm, JKL 60°, LJK 30°. Measure KL and LJ.

Use your ruler and compasses only to construct an equilateral triangle. Measure the angles to make sure they are all equal.

Copy and complete the following table. Cash price

Deposit (% of cash price)

Repayment term (years)

Monthly payment

a)

$500

12

1

$62.50

b)

$550

5

2

$30.00

1 72

2

$30.00

1 12

$42.25

c) d)

$700

9

e)

$600

15

f)

$400

g)

$900

h) i)

2 5

$650

10 1 92

$3 000

d) Guy$

e) EC$

$25.00

$690.00

$21.00

$544.00 $1 435.50

1 22

$845

1

$3 855

will be exchanged for US$500 in each of these currencies? b) BD$

$756.00

3

Given these foreign exchange equivalents, how much a) TT$

Total HP

Exchange rates (per 1 US$)

c) JA$ TT$6.25 JA$61.50

Exchange 1 000 dollars of each Caribbean currency into US dollars using the rates in the table.

BD$1.99 EC$2.67 Guy$178.50

How much money would you get if you changed a) US$20 into BD$

b) US$100 into TT$

c) BD$60 into US$

d) TT$110 into US$

e) TT$250 into BD$

f) TT$175 into JA$?

Find the volume, in litres, of a cylinder of height 66 cm and diameter 20 cm.


Find the volume of the objects shown below. Give your answers correct to two significant figures.

a)

b)

2 cm

c) 10 cm

5 cm 4 cm

12 cm 2 cm

2.4 cm 8.7 cm

7 cm

d)

e)

8 cm

f)

35 mm 80 mm

12 cm

100 mm

2.8 cm

Copy and complete this flow diagram to show the relationships between different quadrilaterals. Write the names of the correct quadrilaterals in the empty boxes. Does it have at least one pair of parallel sides?

NO

Quadrilateral

YES

Does it have two pairs of parallel sides?

NO

YES

Are all the sides equal?

Does it contain right angles?

YES

YES

NO

NO

No

Are all the sides equal?

YES


Write a word to match each definition given below. a) to cut exactly in half b) angles that add up to 180° c) lines that never cross d) angles that are equal e) angles that form an F shape on parallel lines f) a line that crosses two parallel lines g) an angle of 90°

Draw the front and left side view of this solid. The plan view and oblique view are given here. 6.1 m

7.2 m

6.3 m

9.6 m


left side front side

Key words

Discrete Continuous Class Grouped frequency

Statistics: graphs and diagrams In this chapter you will review previous work on graphs and diagrams. You will work with different methods of organising and representing data in order to make sense of it, interpret and identify trends and draw conclusions.

Chapter

11

Unit 11.1 Frequency tables In this unit you will: distinguish between discrete and continuous data construct grouped frequency tables. You should remember from Year 1 and 2 that when we conduct a survey or an investigation, the information collected is called data. Data may either be discrete or continuous. Discrete data is data which can be counted and is recorded as whole numbers, such as the number of children in a group or the number of houses on a street. Continuous data is data which is usually measured and has values which can lie within a given range, such as the height of a plant or the mass of a brick. Sometimes when there is a large amount of data, it can be put into groups or classes. If the data is shown in a table, it is known as a grouped frequency table.

Worked examples 1. Describe each of the following as discrete or continuous data. a. weight b. number of pets c. time taken to eat breakfast d. volume of water drunk by a person in a day e. number of glasses of water drunk by a person during the day 2. Joe is collecting information from 15 of his friends about the distance they live from the library, in kilometres to one decimal place. 6.3 5.2 6.5 5.7 6.2 2.7 1.9 7.8 2.8 4.1 10.1 3.6 1.2 3.3 8.2 Show this information in a frequency table. 1. a, c and d are continuous, b and e are discrete. Distance (km) 2. There are 15 different values to be recorded, so a grouped frequency table is more appropriate. Note 1x3 that the shortest distance is 1.2 km and the furthest 3 x 5 is 10.1 km. Let the distance be represented by x. 5x7 Since the data is continuous and approximated to 1 decimal 7x9 place, we must select classes that would include all possible 9 x 11 values. 1 x 3 implies that the distance is more than or equal to 1 km but less than 3 km, while 3 x 5 implies that the distance is more than or equal to 3 km but less than 5 km.

Frequency 4 3 5 2 1

Statistics: graphs and diagrams 119

Exercise Classify each of the following as discrete or continuous data. a) b) c) d) e) f)

time taken to complete your maths homework the number of correct answers in your maths homework the number of pupils in your school the distance pupils travel to get to school shoe size foot length

The pulse rates of 20 athletes are given below. Construct a frequency table with equal sized classes to show the rates. 67 76 71 74

70 65 80 66

68 75 69 78

66 72 73 71

69 68 70 79

The number of books read by the students in Form 3 during the summer vacation is shown below. 6 3 8 10 4

4 1 6 12 4

10 2 7 6 4

5 2 1 5 9

4 9 8 3 7

3 5 8 10 6

Construct a frequency table for this data using classes starting with 13, 46 and so on.

The heights of 20 boys in centimetres are 147 133

167 152

157 148

131 159

151 142

158 164

159 135

144 155

156 145

138 163

Show these heights in a frequency table using equal sized classes.

Investigation Think about the data that could be collected about you. Copy the table. Record at least 12 types of data in your table. The table has been started for you. Quantitative (can be counted or measured) Qualitative Discrete

Continuous

Number of brothers Height


Eye colour

Key words

Histogram Frequency polygon

Unit 11.2 Graphs and charts In this unit you will: construct graphs and diagrams to represent data interpret graphs and diagrams.

Data collected in statistics can be presented in a diagram, where it can be easily analysed. It is important to choose the most appropriate diagram for a set of data. Pictographs use a shape or symbol to represent a given frequency, where the symbol is repeated to represent larger frequencies or divided to represent smaller frequencies.

BICYCLES SOLD BY EACH STORE Travel Mart Bikers Cycle Town

Pie charts and compound bar charts are usually used to show parts or a fraction of a whole amount.

Shopping

11 10 9 8 7 6 5 4 3 2 1 0

Boys Girls 7A

7B 7C Form group

A bar graph is best used to represent discrete data especially when the variable consists of categories instead of numbers. The frequency axis may be either horizontal or vertical. It is also possible to use two or more graphs on the same diagram to compare variables.

7D

40 Price

Sleeping

Frequency

Reading

30 20 10 0 0

Line graphs are good for showing trends or how data varies over a time period.

1

2 3 Weeks

4

Distance (km)

Distance travelled by a car during one hour 60 50 40 30 20 10 0 0

5 10 15 20 25 30 35 40 45 50 55 60 Time (minutes)


A frequency polygon also shows the frequency distribution. It is drawn by plotting the mid value of each class and joining the points to form a straight line.

Frequency

A histogram is a graph of a frequency distribution. It looks like a bar graph but there are no spaces between the bars and the area of each rectangle represents the frequency.

18 16 14 12 10 8 6 4 2 0

Number of hours pupils use the Internet

0

2

4 6 8 Time (hours)

10

12

Worked examples 1. Jasmine collects some data about a cereal. Show this information as a. a pie chart b. a vertical bar graph. Nutritional content

Amount per portion (g)

Protein

12

Carbohydrate

74

Fibre

10

Fat

24 Total 120 g

2. The frequency table below shows the heights of 15 plants. Show this information on a. a histogram b. a frequency polygon. Height (cm) Frequency 10–14

2

15–19

5

20–24

3

25–29

4

30–34

1

Note that although the class is 10–14, the heights in the class may be as short as 9.5 cm and as tall as 14.4 cm. So these values will be used on the graph.

Fibre Protein Fat Carbohydrate

Amount per portion (g)

1.

Nutritional content in cereal

80 60 40 20 0

Fibre

Protein Fat Carbohydrate Nutritional content

Note that the pie chart shows more clearly the proportions of the content in relation to each other. 122 Jamaica Maths Connect 3

2. For the frequency polygon, we plot the mid values of each class. Height of plants

Frequency

5

Mid value (cm)

4

12

2

3

5

17

5

1

22

3

0

27

4

32

1

2

9.5 14.4 19.5 24.4 29.5 34.4 Height (cm)

Height of plants

6 Frequency

Number of plants

6

4 3 2 1 0

9.5 14.4 19.5 24.4 29.5 34.4 Height (cm)

Exercise The number of people sitting in a school hall for a concert is shown on this graph. 140 120 100 80 60 40 20 0

a) b) c) d)

7:30

7:45

8:00

8:15

8:30

8:45

9:00

9:15

9:30

9:45

At what time did the concert start? What might have happened between 8:45 and 9:00? How many people watched the concert? What time did the concert end?

The table below shows the price paid for petrol measured in litres. Petrol Price ($)

2

5

8

12

3.00

7.50

12.00

18.00

a) Draw a line graph to show this information. b) Use your graph to determine i) the price of 4 litres of petrol ii) the price of 15 litres of petrol iii) the amount of petrol that can be bought for $15.00.

Mrs Dash took her children to a fair. The pie chart shows the money spent by the family at the fair. a) If the family spent $20 on games, how much money was spent on food? b) What fraction of the money was spent on books and plants?

Games 120°

Food

Books and plants


A survey was conducted among a group of 90 people to determine their favourite West Indies cricketer of all times. The results were Cricketer

No.

Brian Lara

25

Sir Vivian Richards

12

Sir Garfield Sobers

34

Courtney Walsh

19

The bar graph shows the number of books borrowed by a group of children. a) How many children borrowed four books? b) How many children were there in the survey? c) How many books were borrowed altogether?

Number of children

Show these results on a pie chart. Borrowed books 8 6 4 2 0

1

2 3 4 5 Number of books

6

The scores of 20 students in a Science test were as follows: 5 7 6 8

8 7 7 5 10 6 9 8 7 5 4 6 5 10 8 7 a) Show the scores in a frequency table. b) Draw a bar chart to illustrate this information.

The distances that 50 students travel each day are recorded in the table below. Show this data on a) a histogram

b) a frequency polygon.

Distance (km) Frequency

0–4

5–9

7

15

10–14 15–19 20–24 6

13

9

The times taken by 24 students to run the 100 metre dash at a sports meet are given in the table. Show this information on a) a histogram Time (sec) Frequency


b) a frequency polygon.

15–17 18–20 21–23 24–26 27–29 2

5

6

8

3

Key words

Unit 11.3 Other statistical diagrams

Scatter Stem-and-leaf diagram

In this unit you will: plot stem-and-leaf and scatter diagrams Interpret diagrams and make inferences.

The scatter graph is used to plot measurements that have been taken in pairs. The graph shows the relationship between the measurements.

positive relationship

negative relationship

no relationship

In a stem-and-leaf diagram it is easy to see how the data is distributed, without losing the detail that was present in the raw data. In the diagram below, the ‘stem’ represents tens and the ‘leaves’ represent units. 1 2 3 4

2 0 2 0

2 1 3 0

3 1 3

4

5

Key: 1 2 means 12

Worked examples 1. The ages of a group of 14 people going to a wedding are 45 18 31 55 11 13 31 25 29 40 48 9 a. Draw a stem-and-leaf diagram for this data. b. Calculate statistics from the diagram. 2. Jessica asks ten of her teachers about their journey to school.

42 21

Distance from school (km)

7

2

14

22

16

9

28

11

4

10

Journey time (minutes)

11

4

23

31

26

15

42

11

4

23

Age of car (years)

5

1

7

2

4

2

5

9

6

4

a. Draw scatter graphs showing i. distance from school against journey time ii. distance from school against age of car. b. Describe the relationships shown on each graph.


1. a. 0 1 2 3 4 5

9 1 1 1 0 5

Key: 1 1 means 11 3 8 5 9 1 2 5 8

You will revise mode, median and mean in chapter 12.

14 1 b. Mode 31; median 7.5th item of data 30; mean 29.86 2 2. a. i

ii

Scatter graph showing distance from school against journey time

10

40

9

30

8 Age of car (years)

Journey time (minutes)

50

Scatter graph showing distance from school against age of car

20 10 0

5

10 15 20 25 Distance from school (km)

30

7 6 5 4 3 2 1 0

5

10 15 20 25 Distance from school (km)

b. Scatter graph i. shows that the distance and journey time are related. The greater the distance, the longer the time. Graph ii. shows no relationship between distance and age of the car.

Exercise A group of pupils measure their handspans in centimetres. Draw a stem-and-leaf diagram for this data. 17.2 16.8

16.8 15.7

17.9 16.9

15.4 18.6

18.1 17.7

19.0 19.4

The marks scored by 15 students in a Mathematics tests were 82 48

54 67

42 59

35 63

70 85

a) Show this data in a stem-and-leaf diagram. b) What is the modal mark for the test?


76 74

92 67

78

30

The table shows the engine size and miles per gallon for a range of cars. Draw a scatter graph to represent this information. What type of relationship does your graph show? Engine size (cc)

1100

1400

1600

1800

2000

2200

2300

2800

4000

Miles per gallon

46

44

38

36

34

29

28

26

21

Some information about a group of children is shown in the table below. Age (years)

6

11

14

8

2

9

16

4

13

12

Height (cm)

115

145

155

130

90

135

170

105

155

155

1

2

1

2

1

3

0

0

2

1

No. of siblings

a) Draw a scatter graph showing age against height. b) Draw a scatter graph showing age against number of brothers and sisters. c) Describe the relationship shown on each scatter graph.

The average temperatures (°C) each month in different cities are shown below. Month

J

F

M

A

M

J

J

A

S

O

N

D

Wellington

17

17

16

14

12

10

9

10

11

12

14

16

New York

0

1

5

11

16

22

25

24

20

15

9

2

16

16

18

22

26

28

28

28

27

25

21

17

City

Hong Kong

a) Draw scatter diagrams showing the temperatures of i) Wellington against New York ii) New York against Hong Kong. b) Draw a stem-and-leaf diagram for New York. c) Use your scatter graphs to say whether the temperatures of any of the cities are related.


Looking back A group of students take two tests. Maths

63

41

80

37

52

61

75

44

40

87

37

50

48

62

51

46

Science

58

46

68

45

55

64

52

70

43

62

74

67

74

84

71

44

a) Draw a scatter graph for their marks. b) Draw a stem-and-leaf diagram for each subject. c) Use the two diagrams to decide which test was more difficult for the students.

The marks scored by 10 students in their Mathematics and Science examinations are given in the table below. Student

A

B

C

D

E

F

G

H

I

J

Mathematics

68

50

62

45

38

89

75

50

45

42

Science

55

50

65

55

30

74

66

55

40

46

a) Display this information using i) a single bar chart ii) two separate stem-and-leaf plots iii) a scatter diagram plotting Mathematics scores on the vertical axis and Science scored on the horizontal axis. b) Determine the mean, mode and range for each examination. c) In which subject did the students perform better? d) Which diagram was most effective in comparing the scores? e) Write a short paragraph on your findings based on one or more of the diagrams.


Key words

Statistics: analysing data

Mean Mode Median Range

This chapter deals with some of the measures of central tendency (averages) that we use to analyse statistics and also the measure of spread that is found in data (the range). You will revise previous work on averages and learn how to apply your skills to work with grouped data in order to work with more complex data sets.

Chapter

12

Unit 12.1 Processing data In this unit you will: calculate the mode, median and mean for a set of data determine the range for a set of data. Once you have collected data you need to process it and make sense of it. This often involves calculating statistics such as averages and the range. sum of all data The mean is the . number of items in the data The mode is the category or outcome with the highest frequency. The median is the middle item once the data has been ordered. The range is not an average, but it shows how the data is spread. The range is the highest value minus the smallest value in the data. If you collected your data in ungrouped form, you can also put it into groups or classes. Class intervals should be of equal size and there should be no gaps or overlaps.

Worked example The table below shows the number of matches in 20 boxes. Number

25

26

27

28

29

30

Frequency

2

1

4

2

4

7

Determine the a. mode

b. median

c. mean

d. range.

a. The modal number of matches per box is 30. b. The median, or middle value, is 29. c. The mean is calculated as (2 25) (1 26) (4 27) (2 28) (4 29) (7 30) 566 28.3 20 20 d. The range is 30 25 5.

Statistics: analysing data 129

Exercise The times in seconds that it takes a group of Grade 4 pupils to recite the 7 times table are given below. Girls

24

31

19

28

35

37

32

29

20

25

Boys

28

32

24

29

22

30

31

25

29

32

Use the mean, median and the range to compare the times of boys and girls.

In a competition, two archers from a team fire ten arrows at a target and the number of hits are counted. The results are shown in the table. Round

1

2

3

4

5

6

Archer A

0

10

9

0

8

0

Archer B

5

1

4

3

5

6

Calculate the mode, mean and range of the scores of each archer. Only one archer can go through to the next round. Which one should it be? Why?

The longest jumps (in metres) of two athletes are given below. Athlete A

8.79

8.76

8.76

8.72

8.71

8.71

8.68

8.67

8.67

8.65

Athlete B

8.95

8.70

8.66

8.64

8.63

8.62

8.61

8.59

8.58

8.57

a) Calculate the mean, median and range for each athlete. b) Write a sentence comparing the two athletes.

The mean of x, 2x and 3x is 8. What is the value of x? The mean of x, x 3, 3x and 2x is 9.5. What is the value of x? The number of times a group of 15 children went to the movies during the vacation is shown in the table below. Times

1

2

3

4

5

Frequency

3

1

4

5

2

a) Determine the mode of the data. b) Calculate the mean numbers of times the children went to the movies.


Exercise Andrew finds the mass of some packets of crisps. 147 149

150 151

152 148

150 146

150 150

148 145

151 149

146

a) Is ‘average contents 150 g’ a fair way to describe the mass of the crisps? b) Which measure of central tendency did you use to decide your answer to a)?

A teacher keeps a record of the number of pupils attending basketball practice each week. 12 13

13 12

12 16

10 14

14 12

12 10

16 13

16 12

13 16

12 14

a) Calculate the mean of this data. b) How many weeks saw lower than average attendance at practice?

The goals scored by 15 basketball players in a shooting competition were Goals

10

12

15

17

20

21

25

Frequency

3

2

2

2

1

2

3

a) Calculate the mean number of goals per player. b) What is the modal score?

The number of sweets in 25 bags is recorded in the table below. Number

10

11

12

13

14

15

16

Frequency

6

3

4

5

2

4

1

Determine the mean, mode and median for the data.

Looking back Karim throws three dice. He throws another dice. The mean increases by 1. What is the number on the fourth dice?

Jake and Melissa record the number of points

Jake

7

10

11

12

scored, in order, when playing a game. Melissa 9 11 They each have the same mean number of points, but Melissa’s range is twice Jake’s. What are Melissa’s missing scores?

The ages of five children are 3 years, 6 years, 4 years, 8 years, 4 years. a) Find the mean, median, mode and range of their ages now. b) Find the mean, median, mode and range of their ages in one year’s time. c) Which of the four answers in b) have remained the same? How have the others changed? d) Use your answers from part c) to write down the mean, median, mode and range of their ages i) in two year’s time and ii) one year ago. Statistics: analysing data 131

Project The graphs below show you the number of threatened species on some Caribbean islands. One method of protecting biodiversity is to protect the habitats (living areas or environments) of plants and animals. Carry out an investigation to find out how much land is set aside for conservation in your country. Collect your data from the Internet and local environmental or governmental sources. Prepare a report on your findings which includes at least one graph. Threatened Species, Jamaica, 2002-03

Threatened Species, St Lucia, 2002-03

Higher plants

Higher plants

Mammals

Mammals

Breeding Birds

Breeding Birds

Reptiles

Reptiles

Amphibians

Amphibians

Fish

Fish 0

50

100

150

200

250

0

1

2

3

4

5

6

Number of Species

Number of Species

Threatened Species, St Vincent & Grenadines, 2002-03

Threatened Species, Grenada, 2002-03

Higher plants

Higher plants

Mammals

Mammals

Breeding Birds

Breeding Birds

Reptiles

Reptiles

Amphibians

Amphibians

Fish

Fish 0

1

2

3

4

5

0

2

3

4

5

Number of Species

Number of Species

Threatened Species, Barbados, 2002-03

Threatened Species, Antigua and Barbuda, 2002-03

Higher plants

Higher plants

Mammals

Mammals

Breeding Birds

Breeding Birds

Reptiles

Reptiles

Amphibians

Amphibians

Fish

Fish 0

1

7

1

2 Number of Species

3

4

0

1

2

3

4

5

Number of Species

Graphs from : World Resources Institute, 2005. EarthTrends: The Environmental Information Portal. Available at http://earthtrends.wri.org.


6


R4

Rent

Sa vin gs

Mr John earns $1 000 per week. The pie chart shows how he budgets his money.

Recreation 54°

Expenses

The amount Mr John saves each week, in dollars, is a) 100

b) 125

c) 250

d) 500

The table shows the scores of students in a test question. Marks

0

1

2

3

4

5

Frequency

2

2

3

1

2

x

If the median is 3 marks, the value of x is a) 3

b) 4

c) 5

d) 7

The heights in centimetres of 15 Form 3 students are 151

151

153

140

144

135

140

140

154

145

143

147

148

140

150

The median height is a) 135

b) 144

c) 145

d) 154

A man, firing at a target, obtains the following scores. Score

0

1

2

3

4

5

Frequency

4

8

13

15

7

3

The mode of this distribution is a) 2

b) 3

c) 15

d) 25


14 13 12 11 10 Frequency

9 8 7 6 5 4 3 2 1 0

1

2

3

4

5

6

7

8

Number of peas per pod

In the bar graph shown above, the total number of peas is a) 36

b) 40

c) 100

d) 200

The following scores were obtained by 50 students in a mathematics test. 8 9 10 10 7

7 6 5 3 4

4 3 2 4 6

1 9 4 5 7

5 3 6 8 9

8 5 3 3 2

4 2 5 1 8

5 9 10 5 6

4 2 2 7 8

7 6 3 9 10

a) Use a tally column to organise the data in an ungrouped frequency table. b) Draw a bar graph to represent the data.

Fifty students scored the following marks in a science test. 20 25 31 17 6

35 28 16 3 29

39 26 14 32 18

37 23 17 19 24

8 16 22 34 40

36 33 30 20 14

15 28 32 17 9

36 32 27 21 12

a) Copy and complete the grouped frequency table shown. b) Draw a histogram to display the information. c) Find the mode, median and mean of the data.

9 31 24 11 4

10 40 39 37 35 Mark 1–5 6–10 11–15 36–40


Tally

Frequency

Key words

Equations Substitute

Simultaneous equations and inequalities

Chapter

13

There are times when two or more variables and two or more equations are necessary in order to solve real-world problems. In this chapter you will learn how to calculate the solutions when you have two variables in two equations at the same time – this is known as solving simultaneous equations. You will also learn how to solve simple inequalities.

Unit 13.1 Solving simultaneous equations by elimination In this unit you will: solve simultaneous equations using the elimination method.

Let us examine the following situation. Three cup cakes and a box of drink cost $1.50, but two of the same cakes $1.50 and a box of drink cost $1.10. What is the cost of a cup cake and the cost of a box of drink? The difference in the two purchases is 1 cup cake. $1.10 The difference in the total cost is 40 cents. Therefore 1 cup cake cost 40 cents. We can the use this to find the cost of a box of drink. This would be easier if we write the equations and solve them. Let 1 cup cake cost p cents and 1 box of drink cost k cents. $1.50 150 cents Then 3p k 150 Equation i) 2p k 110 Equation ii) $1.10 110 cents The difference between the two situations is one cup cake and 40 cents. p 40 So 2 40 k 110 80 k 110

Subtracting like terms in ii) from i), k is illuminated. Substitute (put) 40 for p in ii).

k 110 80

Simplifying.

k 30

Transposing 80.

Therefore a cup cake costs 40 cents and a box of drink costs 30 cents. Checking: 3 cakes and a drink cost 3 40c 30c (120 30) c 150 cents $1.50 (i) 2 cakes and a drink cost 2 40c 30c (80 30) c 110 cents $1.10 (ii) p 40 and k 30 satisfy both equations.

Simultaneous equations and inequalities 135

Worked example Solve the following pairs of equations. a. 2x y 4 and 3x 3.5 y b. 5x 2y 9 and 2x 3y 8 a. 2x y 4 Eq. i) Rewrite the equations so that like terms are in line. 3x y 3.5 Eq. ii) 5x 7.5 Eliminate y by adding both equations, (y y 0). x 7.5 5 x 1.5 Substitute 1.5 for x in Eq. i), 2 1.5 y 4 3y4 y431 Solution is x 1.5 and y 1 b.

5x 2y 9 Eq. i) Multiplying Eq i) by 2 — Eq. iii). 2x 3y 8 Eq. ii) Multiplying Eq ii) by 5 — Eq. iv). 10x 4y 18 10x 15y 40 Subtracting iii) from iv) to eliminate x. 11y 22 Dividing by 11. y2 Substitute 2 for y in Eq ii), 2x 3 2 8 2x 6 8 2x 8 6 2 x 221 Solution is x 1, y 2 Checking 5 1 2 2 5 4 9 Eq. i) 2 1 3 2 2 6 8 Eq. ii)

Exercise Solve the following pairs of equations. You may have to rewrite one or both equations. a) 6b 4c 2 and 7b 4c 11 c) 5r s 0 and 7r s 2 e) p 1 2q and 2p 2q 8

Find the solution for each set of equations. a) 2x y 5 3x 3y 9 d) 6x 2y –18 2x 3y 1


b) 4x 3y 4 and 2x 3y 7 d) x y 7 and y 4x 8 f) w 3v 1 and 8 w 12v

b) c 2d 6 2c 3d 26 e) 3j 4k 5 2j 6k 8

c) 4p 8q 14 3p 4q 10 f) 2g 3h 9 g 2h 1 2

Key words

Substitution Subject of an equation

Unit 13.2 Solving simultaneous equations by substitution In this unit you will: solve simultaneous equations using the substitution method.

Earlier you learned how to substitute numbers for variables or unknowns and to write any letter as the subject of the equation. You will now use subject of the equation with the substitution of terms or expressions as another way of solving simultaneous equations.

Worked examples 1. Solve the following equations simultaneously. w 10 3y 4w 2y 0 2. Find the value of t and r if 3t r 7 and 2t 3r 1. 3. Solve the equations 2x y 6 and 4x 2y 8. 1. w 10 3y Eq. 1) 4w 2y 0 Eq. 2)

In Eq. 1) w is the subject of the equation and equals 10 3y. Therefore 10 3y can replace (substitute for) w in Eq. 2)

4(10 3y) 2y 0 40 12y 2y 0 10 y – 40 y (– 40) (– 10) 4 So w 10 3(4) w 10 12 – 2 Solution is w –2, y 4 Therefore w –2, y 4 satisfies both equations. 2. 3t r 7 Eq. 1) 2t 3r 1 Eq. 2) In Eq. 1), 3t 7 r In Eq. 2), 2t 3(3t 7) 1 2t 9t 21 1 11t 1 21 22 t 2 So r 327 r 1 r 1 and t 2 r 1 and t 2 satisfies both equations.

There is now one simple equation to solve. Moving the brackets. Simplify and transpose. Substituting in w 10 3y.

Putting r as the subject. Substituting 3t 7 for r. Remove brackets. Simplify and transpose. Dividing by 11. Substituting 2 for t in 3t 7.


3. y 6 2x y 2x 4 So 6 2x 2x 4 10 4x 1 1 x 22 and y 6 2(22) 1

You can also put y as the subject in both equations. Transposing 2x. Transposing 2y and dividing throughout by 2.

Exercise Use the substitution method to solve the following pairs of equations simultaneously. a) p 3q 1 2q p 6 b) x 2y 11 4x 3y 4 c) 2 7d e 2e 4d 4 y d) x 2 4x y 20 e) 4x 2y 15 3x 6y 15 f) 8h 6 2k 12 5h 3k

Go back to Unit 13.1 and use the substitution method to find the values of the unknown for the equations in questions 1 and 2.

State the questions that were more challenging when the substitution method was used. Tell a partner why this was the case.

Investigation Research the use of words you learn in Mathematics, writing the common use and the mathematical use of simultaneous, eliminate, and substitute. Explain to a partner when it is best to a) add or subtract equations to eliminate an unknown b) multiply both equations or only one equation to eliminate an unknown c) select the elimination method or the substitution method.


Key words

Unit 13.3 Solving inequalities

Solution set Continuous Discrete

In this unit you will: solve simple inequalities.

To solve simple equations you transposed by using inverse operations. You investigated the use of multiplying and dividing both sides of an inequality. We will now investigate further to see if inequalities may be treated as equations. ● 4 5 12 Subtracting 5 from both sides: 4 12 5 ● 642 Adding 4 to both sides: 6 2 4 ● 524 ● 12 3 6

Dividing both sides by 2: 5 4 2 Multiplying both sides by 3: 12 6 3

Try other positive numbers with the four operations to see if the inequalities are still true. Now let’s try negative numbers: ● 4 (5) 2 Subtracting (5) from both sides: 4 2 (5) ● 6 (4) 2 Adding (4) to both sides: 6 2 (4) ● 5 (2) 4 Dividing both sides by (2): 5 4 (2) The sign must be reversed to make the statement true: 5 4 (2) ● 12 (3) 6 Multiplying both sides by (3): 12 6 (3) The sign must be reversed to make the statement true: 12 6 (3) Try other negative numbers with the four operations to see if the inequalities are true or false. ● 6 4 10 Interchanging sides: 10 6 4 ● 634 Interchanging sides: 4 6 3 The sign must be reversed for the statement to be true: 4 6 3 The inequality remains the same for all the operations except: ● when multiplying or dividing both sides by a negative number ● when the sides are interchanged. The following number lines show how the solution set for a given inequality can be displayed. 5

4

3

2

1

0

1

2

3

5

4

3

2

1

0

1

2

3

4

5

6

7

x 1

x is greater than or equal to 1. x is less than 3.

x3

5

4

3

2

1

0

1

2

3

4

5

6

7

5

4

3

2

1

0

1

2

3

4

5

6

7

x 7

x is a whole number and is less than or equal to 7.

4 x 4

x lies between 4 and 4. It is not equal to 4 but can be equal to 4.


Worked examples 1. a. Solve the inequality 3(2x 4) 6. b. Show the solution on a number line. 2. a. Find the solution set (from the set of whole numbers) for the inequality 17 3x 1. b. Show the solution on a number line. Remove the brackets. 1. a. 6x 12 6 Add 12 to both sides. 6x 6 12 x 18 6 Divide both sides by 6. x 3 Check by putting x 5 (because 5 3): LHS: 3(2 5 4) 3(10 4) 3 6 18 RHS: 6 18 6 Try other numbers that are greater than 3. All numbers greater than 3 will satisfy the inequality. b. 32 1 0 1 2 3 4 5 6 7

Subtract 17 from both sides. 2. a. 17 3x 1 Divide both sides by 3. 3x 1 17 x

18 (3) The inequality sign is reversed. x

6 Check by putting x 4 (because 4 6). LHS: 17 3(4) 17 12 5 RHS: 1 5 1 6 will satisfy the equality part and all numbers less than 6 will satisfy the inequality. b. 5 43 2 1 0 1 2 3 4 5 6 7

Note the differences in the number lines: ● x 3 has a continuous line from 3 but an unshaded circle at 3 because x 3.

means ‘is not equal to’

● x 6 has discrete points at 6 and other whole numbers less than 6.


Worked example a. 10 5(x 26) 0 represents the number of students in a class who like mathematics. Solve the inequality. b. List some possible values in the solution set. a. 10 5(x 26) 0 10 5x 130 0 5x 140 0 5x 0 140 x

140 (5) x

28 b. The solution set is {0, 1, 2, 3, 4, 5,…, 28}.

Remove the brackets, times is . Collect like terms. Transpose 140. Divide both sides by 5; the sign is reversed.

Exercise From the box, choose all the numbers which satisfy these inequalities. 2

0.1

2

4

8

10

3

4

1

1.25

1 4

1

42

a) b 4

b) r 2

c) m 10

d) w 5

e) 3 r

f) 7 q

g) 8.5 t

h) 2.1 z

i) p 3.1

j) k 0

k) w 2

l) 1.5 y

a) Write down all the integers between 10 and 10 which satisfy the following inequalities. i) x 3.2 ii) 5.6 m iii) 9.1 t iv) d 2.61 b) Write down inequalities like the ones in part a) which are satisfied by the following sets of integers between 10 and 10. i) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ii) 3, 4, 5, 6, 7, 8, 9, 10 iii) 9, 10 iv) 10

Solve the following inequalities. a) 3x 15 c) x 7 15 e) 4a 6 2 g) c 5 7c 5c 1

b) 6 x 2 x d) 5 16 f) 12 9b 6 3z 1 h) 0.25 4z 4 2 Simultaneous equations and inequalities 141

Draw number lines to show the solution sets for question 3. Solve the following inequalities: a) 6 3(p 2) 2p 18 b) 4(w 6) 3( 2w 15) c) 4(6f 3) 2(f 17) 2f 5(8 2f) d) 3(h 9) 4(h 0.5) 5(2h 20)

Find the solution sets for : 2c 5c 6 a) 3 4 5x 6 x b) 7 2 4x 5 x c) 6 4 3x 7x 13 1 d) 4 2 2

Mr Green can afford only 2 rolls of wire to fence a rectangular part of his land to put goats. Each roll is 100 metres long. He prefers to have the length 4 metres longer than the width. a) Write an inequality to represent the perimeter of the land he may fence. b) What is the longest length he may use?

Mr Smart invested his money in three ways. He invested $m in bonds, a half of that m 200 in stocks and $ in a savings account. His total investment was not less than 6 $5 400. a) Write an inequality for his total saving. b) What is the least amount he invested in each way?

Mrs Green spends no more than $250 at the supermarket each week. One week she spends $10 more on vegetables than on meats. The total cost of the other items is 3 times that of the meat. What is the most amount she spent on meat that week?


Looking back Solve these simultaneous equations using the method you find easiest in each case. a) 8x 3y 70 2x 5y 3 d) 2x 3y 12 3x 4y 1 g) 3x 11 8y 2x 6 5y

b) 14x 4y 34

c) 15x 12y 15

10x 12y 6

21x 36y 117

e) 6x 5y 23

f) 2x 7y 31

4x 3y 14

16 3y 5x

12x 21y h) 22 2 3

i) 2(9x 4) 16

27y 8x 29 3

6x 5y 3

Mrs Gaines bought 3 table cloths and 2 tray cloths. She paid $26. Ms Smythe bought 4 table cloths and 1 tray cloth. She paid $28. The cost of a table cloth was x dollars, the cost of a tray cloth was y dollars. a) Use the information to form two equations in x and y. b) Solve the equations to find the cost of one table cloth.

The sum of two numbers is 72. Their difference is 18. Find the numbers. The cost of tickets to a jamboree for 2 adults and 3 children is $36. The cost for 3 adults and 5 children is $57. Find the cost of an adult ticket.

The total cost of 3 blank CDs and 4 pens is $98. The total cost of 2 blank CDs and 5 pens is $91. a) Write equations to represent this information. b) Solve the equations to find the cost of a CD. c) Substitute to find the cost of a pen.

Solve each inequality: a) p 6 18

b) d 12 3

c) 14 m 3

d) 3 t 1

e) 8 w 3

f) 13 k 10

For each statement, write an inequality and then solve it. a) b) c) d) e) f)

7 more than a number p is less than 30. k minus 14 is less than or equal to 30. 5 increased by x is less than 12. t decreased by 8 is greater than 0. 4 subtracted from y is greater than or equal to 1. 9 added to a is less than 14.

Draw a number line to show the solution set for each inequality. a) 13m 52 t d) 4 7

b) 3x 5 10 e) 3a 5 7

n c) 4 5 f) 5k 6 11 Simultaneous equations and inequalities 143

Chapter

14

Key words

Algebraic graphs This chapter deals with graphs in algebra. You will revise the work on functions, line graphs and intersection of lines that you covered last year and investigate other properties of these concepts. You will learn how to use graphs to solve inequalities, simultaneous equations and real-world problems.

Function Intercepts Straight-line graph Quadratic curve

Unit 14.1 Graphs of functions In this unit you will: associate equations with their graphs and tables determine the equation of straight lines given the table or the graph differentiate between straight-line and quadratic graphs.

Remember that a one : one (read as ‘one to one’) mapping and a many : one mapping are functions. Consider the following mappings:

7 4 1 2 5 8 11

A 3 2 1 0 1 2 3 x

B 3 2 1 0 1 2 3

y

C 3 2 1 0 1 2 3

4 0 2

10

x

y

x

12 5 0 3 4

D 3 2 1 0 1 2 3

7 5 3 1 1 3 5 y

x

y

You will notice that in all the mappings each value of x maps to one and only one y-value. Therefore all are mappings of functions, but B and D have more than one x mapping to one y-value. A and C are ‘one : one’ functions, B and D are ‘many : one’ functions. The following tables represent the mappings above. A

x

3

2

1

0

1

2

3

y

7

4

1

2

5

8

11

B

y increases by 3 as x increases by 1. The equation is y 3x 2.

C

3

2

1

0

1

2

3

y

7

5

3

1

1

3

5


3

2

1

0

1

2

3

y

4

0

2

2

0

4

10

y decreases, then increases but not by the same amount. The equation is y x2 x 2.

x

y decreases by 2 as x increases by 1. The equation is y 1 2x.

x

D

x

3

2

y

12 5

1

0

1

2

3

0

3

4

3

0

y increases, then decreases but not by the same amount. The equation is y 3 2x x2.

Look at the following graphs and the zeroes in the table. A

B

y 10 8

8

6

6

4 ( 23 ,

2

4

6

6

6

8

8

10

10

D

6 4

(0, 1)

2 1

2

3 x

2

3 x

(0, 2)

4

(3, 0)

2

3 2 1 0 2 4

1

2

3 x

6 8 10

6 8

1

y 6 (0, 3) (1, 0)

8

(1, 0)

3 2 1 0 2 4

8 10 x

y 10

3 2 1 0 2 4

2

(2, 0)

108 6 4 2 0 2 4

C

4

(0, 2)

2

0)

y 10

( 12 , 0)

10

12 14 16

Remember x 0 on the y-axis and y = 0 on the x-axis. Graphs A and C are straight lines. Remember, the equation for a straight line is in the form y mx c, where m gradient and c y-intercept. An intercept is where lines meet or cross. Therefore x 0 is the intercept on the y-axis. Graphs B and D are called parabolas. They look like a bowl turned up or down. These are graphs of quadratic functions or quadratic curves. A quadratic function is an expression which includes a square as the highest index.

Algebraic graphs 145

Worked examples 1. Which of the following are graphs of functions? i) ii) y 10 8 6 4 2 32 1 0 2

1 0 1 2 3

1 2 3 x

y 5 4 3 2 1

iii)

5432 1 0 1 2 3 4 5

y 3 2 1 1 2 3 4 5 6 7 8 9 10 x

y 5 4 3 2 1

iv)

1 2 3 4 5 x

32 1 0 1 2 3 4 5

2. Which of these graphs are graphs of quadratic functions? 3. Match each of the following equations to one of the graphs. a. y 2x 1 b. y x2 1 c. y 4 x2

1 2 3 x

d. x y2

1. i), iii) and iv) are graphs of functions, ii) is one : many. It is not a function because there are x-values that have two y-values. 2. i) and iv) are graphs of quadratic functions. 3. i) to b. It is a parabola that cuts the y-axis at (0, 1). ii) to d. Take any y-value and square it to get the x-value. iii) to a. A straight line. The coordinates at the y-intercept are (0, 1). iv) to c. An inverted parabola because of the negative sign before the x2.

Exercise Put the following equations in the correct column of the table. a) y x2

b) y 3x 3

c) y x

d) y x2 4

e) y 7 3x

f) y x2 x 2


Straight line

Quadratic curve

Match the following graphs to the equations in question 1. i)

ii)

iii) y 5 4 3 2 1

y 10 8 6 4 2 3 2 1 0 2 4 6 8 10

y 5 4 3 2 1

5 4 3 2 1 0 1 2 3 4 5

1 2 3 x

iv)

5 4 3 2 1 0 1 2 3 4 5

1 2 3 4 5 x

v) y 5 4 3 2 1

5 4 3 2 1 0 1 2 3 4 5

vi) y 16 14 12 10 8 6 4 2

y 2 3 2 1 0 2 4 6 8 10 12 14 16

1 2 3 4 5 x

1 2 3 4 5 x

1 2 3 x

3 2 1 0 2 4

1 2 3 x

Complete the following tables. a)

b)

x

5 4

3

y 7 3x

22

19

16

x

5 4

3

yx 4 2

21

12

2

2 0

1

1

0

0

1

1

2

2

3

4

2

5

3

4

3

12

Write the coordinates for the y-intercept of the graphs given in questions 1 and 3. Explain to a friend how to find these coordinates from a) the table

b) the graph

c) the equation.


Key words

Unit 14.2 Straight lines and their gradients

Linear Gradient Constant

In this unit you will: draw graphs of straight lines calculate the gradient of a straight line.

A straight line graph is usually referred to as a linear graph. To draw a straight line we need only two points but we use three to be sure. Remember the gradient of a straight line is constant. It is the same throughout its length.

y 5

y 4x 1

4 D

y

4x 4

3 2 1

6 5 4 3 2 1 0 1 2

1

2

3

4

5 x

3 4 5

y 3 4x

6

BA

Gradient can be defined as the vertical distance divided by the horizontal distance.

y 2 4x C

Using a right-angled triangle.

The gradient of line A 82 4 or 41 4. The gradient of line B is also 4, the gradient of line C 4 and the gradient of line D 14. You will notice that: ● A is parallel to B

Both gradients 4, the coefficient of x in the form y mx c.

● D is perpendicular to A and B

The product of the gradients 1, [4 (14) 1].

● the gradients of A and B are positive ● the gradients of C and D are negative.


4, as x increases y increases. 4 and 14, as x increases y decreases.

Worked examples 1. If the gradient of a straight line is 5 and the y-intercept is (0, 3), write the equation of the line and use it to draw the graph of the line. 1. m 5 is the gradient. At (0, 3) x 0, y 3 c. y y 5x 3 24 Therefore the equation is y 5x 3. 22 Choose two other values for x and substitute to 20 find the y-value. 18 E.g. x 1, y 5 (1) 3 2. The point is 16 (1, 2) and x 4, y 5 (4) 3 23. 14 The point is (4, 23) and the given point is (0, 3). 12 Using the three points (0, 3), (1, 2), (4, 23), 10 8 plot and connect the points to get the straight 6 line. 4 2 5 4 3 2 1 0 2 4

2. Draw the graphs of y 2x 32 and y 21x on the same set of axes. Substitute three values for x to find the value of y for each equation. Plot the points and draw the lines. y 10 2. In 2y 4x 3 8 x 0, y 121 6 1 x 4, y 92 4 x 4, y 621 2 In y 21x 5 4 3 2 1 0 x 0, y 0 2 x 4, y 2 4 x 4, y 2 6

1

2

5 x

3 4

2y 4x 3

y 21 x 1

2

3 4

5 x

8 10


Exercise Select the equations that represent parallel lines. You may have to rearrange an equation to get the form y mx c.

t 1 Remember 3 2 2 3 and t. 2 2

a) y 7x b) y 8 7x c) y 7 x d) y 4 7x e) y 7x 2 x f) y 2 7

Select the equations that are for perpendicular lines. a) y 8x 3

b) y 8 8x

x c) y 2 8

d) y 6x 7

e) y 2 6x

f) y 2 16 x

Use the same grid and axes to draw the lines that are parallel

Same coordinate plane.

in question 1.

Use the same coordinate plane to draw the lines that are perpendicular in question 2. Draw the graphs of the following equations for 5 x 5. a) y x 47


b) f(x) 4x 23

Key words

Solution Continuous Discrete

Unit 14.3 Solving real-world problems using graphs In this unit you will: identify points or regions on graphs that represent real-life situations draw graphs to represent real-life situations.

Some of our problems encountered in our daily lives involve equations or inequalities. Remember ● when equations are represented graphically the solutions lie on lines, in regions or at vertices ● the solution for more than one situation falls on the intersection of lines or regions ● a discrete variable does not allow fractions e.g. the number of people is discrete. If fractions are allowed then the variable is continuous.

Worked examples 1. One day Roy had $150 and decided to share lunch with friends. They wanted pizza at $30 each and a bottle of drink at $15 each. No fraction of a pizza or a bottle of drink is sold. a. If each friend got a pizza and a bottle of drink, what is the most number of friends Roy may treat that day? b. How much would he spend for himself and his friends? c. Roy decides to spend all his money. What other item(s) could he buy? 1. Let the number of pizzas be x and the number of drinks be y. Roy spent 30x 15y. It is an inequality because he can spend no more than $150. Therefore 30x 15y 150. He must buy at least one drink and one pizza, therefore x 1 and y 1. In 30x 15y 150, when x 0, y 10 and Dividing by 15 gives y 10 2x, y-intercept and gradient may be checked. when y 0, x 5. Connecting the points (0, 10) and (5, 0) gives the line 30x 15y 150. y All the points represented by dots are The values of x and 10 y are discrete. No possible solutions. 9 fraction of a bottle Since each person should get a pizza and a 8 of drink or fraction 7 drink then the solution narrows down to the of a pizza is sold. 6 dots on the red line (1, 1), Roy only, (2, 2), 5 Roy and a friend, or (3, 3), Roy and two friends. 4 3 a. He can treat, at the most, two friends. 2 b. He would spend $(30 3 15 3) $135. 1 c. He would have $15 to purchase one other 0 1 2 3 4 5 x bottle of drink. This would be the point (3, 4) which lies in the region.


2. Mr Kelly is told he can use the following graph to decide on the A 9 area to plant some seedlings. 8 A represents the area, in square metres, of the rectangular 7 2 6 plot of land. The formula for the curve is A 6x – x and x is 5 one of the sides of the rectangle. 4 3 a. What is the largest area he could have? 2 What is the value of x for this area? 1 b. Write the inequality for the shaded region and explain it 0 1 2 3 4 5 6 x for Mr Kelly. c. If he has seedlings for only 8 square metres, what would be the lengths of one side of the rectangle? 2. a. The largest area is 9 m2, and the value The line A 9 touches the highest point at x 3. of x would be 3 m. b. A 6x x2. He will not be able to get the rectangular plot if he uses a side 6 m or greater. The area 0 at x 0 and 6. The area is negative when x is less than 0 and greater than 6, absurd. c. The line A 8 cuts the curve where x 4 and 2, A A9 9 so the lengths would be 4 m or 2 m.

A8

8 7 6 5 4 3 2 1

0

1 2 3 4 5 6 x

Exercise area with a chemical. A and O represent the top and foot of the building, respectively. B represents the farthest distance from the building that he can spray. Use the graph to answer the following questions. a) What is the farthest distance from the building the spray will reach? b) Approximately how high is the building? c) What is the highest height that the spray will go? d) Make a sketch of the graph and shade in the section that represents a safe area out of the reach of the spray.


Height (metres)

The graph represents a man on a building spraying an

90 80 70 60 50 A 40 30 20 10

B O 20 40 60 80 100 Distance (metres)

Danny got $600 to purchase T-shirts and shorts. A T-shirt costs $100 and a pair of shorts costs $90. He has to purchase at least one set. a) Write the information as inequalities. b) Represent the information graphically. c) Use your graph to find the most sets he can purchase. d) What will his change be, if any?

Mrs Prince and Mrs King met at a sale. Mrs Prince spent $50 and bought 5 pens and 2 books. Mrs King spent $45 and bought 6 of the same pens and 1 copy of the same book. a) Write the equations to show their purchases. b) Show this information on a graph. c) Use your graph to find the cost of a book and a pen.

The graph represents the number of hours per day that

y 14

Jack and James are able to work on a site. James can work for no more than 10 hours per day. Jack can do 5 to 12 hours per day. x represents the number of hours per day for James and y represents that for Jack.

12 10 8 6

a) Explain the meaning of the vertices of the shaded portion for their employer.

4

b) Which point would give a maximum total time of work?

0

2 2

4

6

8 10 x

c) What would be the total time? d) How many hours would each work?

Jane is to make small rectangular table mats that have a width 3 cm less than the length. The area must be greater than 40 cm2 and the length must not be more than 10 cm. Given that the length is x cm and the area is A cm2: a) write an expression in terms of x for the width b) write an equation in terms of x for the area. This graph shows the information. c) Explain what the points of intersection of the lines represent for Jane. Which sections of the graph could be omitted because they are unrealistic? d) What is the smallest possible length that may be used to conform to all the given conditions? e) What is the largest possible area and the length for this area?

x 10

A 70 60 50

A x2 3x A 40

40 30 20 10 0 10

2

4

6

8 10 x


Looking back Sketch the following sets of graphs on the same axes. Say what each set has in common. a) y 3, y 2 and y 4 b) x 4, x 5 and x 3 c) y x 2, y x 4 and y x 6 d) y x 3, y x 2 and y x 6 e) y 12x, y 12x 2 and y 12x 3

What is the gradient of a line? Solve each of the following pairs of equations by graphical methods. Before drawing the graphs, check whether a solution exists. a) x y 5 xy1 b) x 2y 2 y x 7 c) x y 1 x 2y 1 d) x 2y 5 x y 12

The length of a rectangle is (x 7) cm and its breadth is 6 cm. If its area is less than 67 cm2, write an inequality and solve it. Show the solution on a diagram.

Write down three inequalities, in terms of x and y, whose solution is shown by the shaded area in the graph below. y 5

x2

4

y 23 x 2

3 2 1 5 4 3 2 1 0 1 2


1

2

3

4 x

y 1


R5

Solve each pair of simultaneous equations. a) 5a 3b 4 2a b 5 b) 4x 5y 1 7x 4y 3 c) 8x 6y 14 10x 4y 32 x d) 3y 4 2 3x 2y 8 e) 6x 5y 11 3x 6y 4 f) 3x 2y 4 7x 5y 11 g) 3x 4y 4 2y x 1

Mrs Gordon bought 3 rugs and 2 cushions. She paid $260. Ms Jones bought four rugs and 1 cushion. She paid $280. The cost of a rug was x dollars, the cost of a cushion was y dollars. a) Use the information to form two equations in x and y. b) Solve the equations to find the cost of one rug.

The sum of two numbers is 97. Their difference is 19. Find the numbers. The cost of tickets to a movie for 3 adults and 2 children is $52. The cost for 2 adults and 5 children is $64. Find the cost of an adult ticket.

The total cost of 3 shirts and 4 pairs of shorts is $73.50. The total cost of 2 shirts and 2 pairs of shorts is $43. a) Write equations to represent this information. b) Solve the equations to find the cost of a shirt. c) Substitute to find the cost of a pair of shorts.

Solve the following inequalities for the set of integers. a) 7x 5 5x 6

b) 3y 4 1 4y

c) 2x 5 23 4x

d) 11 4y 3y 17

e) 4(x 3) x 3

f) 3(x 1) x 5

g) 2(x 9) x 21

h) 3(2x 4) 4x 14

If 2 is subtracted from x the difference is less than or equal to 1. Find the possible values of x if x is a whole number. Revise and consolidate 5 155

The result of subtracting 2 from 3 times a number x, is less than 11. Find the possible values of x if x is a whole number.

Which of the following mappings show a function. (There might be more than one.)

a)

x

y

b) x

y

1

a

1

2

b

3

c

c)

x

y

d) x

y

a

1

a

1

a

2

b

2

b

2

b

3

c

3

c

3

c

Which of the following sets of ordered pairs represent a function? (There might be more than one.) a) (1, 2), (2, 3), (3, 4), (4, 5) b) (1, 4), (5, 12), (3, 8), (1, 2) c) (4, 6), (2, 6), (1, 9), (0, 14) d) (6, 4), (4, 2), (6, 2), (2, 5) Use the graph to answer questions 11, 12 and 13. y 4 2 4

2

0

2

2 4

What is the gradient of the line? a) 2

b) 2

c)

1 2

d) 12

What is the x-intercept of the line? a) 2

b) 0

c) 4

d)

1 2

What is the y-intercept of the line? a) 2

b) 0


c) 4

d) 4

4

x

Use the following to answer questions 14 and 15. x1 f : x → and g : x → x2 x 2x

The value of g3(1) is a) 8

b) 27

c) 42

d) 216

c) fg (1)

d) fg (2)

Find the value of a) f (2)

b) g (2)

e) fg (2)

On 1 January 1996 Bill had $180 in a savings plan. Each month he saved $20 more. a) Complete the table and draw a graph to show his pattern of saving. Number of months after 1 Jan 96 (N ) Amount in $ (A)

0

1

2

180

200

220

5

10

b) From your graph answer the following: i) How much was in the plan on 1 January 1997? ii) On which date was $300 first to be found in the plan? iii) Write an equation linking A and N.

On the same axes plot the lines y 2x 6 and 2x y 2 for 4 x 3. Write down the coordinates of the point at which these lines intersect.

a) On the same axes draw the graphs of y x 2 and y x2 for 3 x 3. b) Write down the coordinates of the points where your graphs intersect.


Chapter

15

Key words

Right-angled triangles Many right-angled triangles are found in our environment and their distinctive properties can be used to calculate lengths or distances, and angles or directions. Trigonometry is a branch of mathematics which deals with the relationship between the sides and angles of triangles. In this chapter you will explore the properties of right-angled triangles, the relationship of the sides that Pythagoras discovered and the three basic trigonometric ratios (tangent, sine and cosine). Then you will apply the information to solve real-world problems involving triangles.

Hypotenuse Pythagoras’ theorem Squares Square roots

Unit 15.1 Pythagoras’ theorem In this unit you will: describe Pythagoras’ theorem use Pythagoras’ theorem to calculate sides of right-angled triangles. Pythagoras was a Greek mathematician and philosopher. He discovered that in any rightangled triangle, the square of the hypotenuse equals the sum of the squares on the other two sides. This is known as Pythagoras’ theorem.

F

We can draw grids on each side and count the squares or square the lengths of the sides to illustrate his theorem. The hypotenuse is the longest side of a rightangled triangle. It is always opposite the right angle.

e

d f

D

E

Side d is the hypotenuse; it faces the right angle D. e2 f 2 62 82 36 64 100 square units d2 100 square units d2 e2 f 2 d 100 10 units We use this information to calculate the lengths of sides of any right-angled triangle.

Worked example Find the length of a and b in the following diagrams. 12 cm a. b. 6 cm

a 20 cm

b

2.5 cm

a. a2 62 2.52 36 6.25 42.25 a 42.25 6.5 cm


b. b2 122 202 b2 202 122 400 144 256 b 256 16 cm

Exercise Review squares and square roots before you do these problems.

Calculate the lengths of the unknown sides in the following figures. a)

b) 24 cm

a

4m

c

c)

7 cm

13 cm

b

12 cm

3m

d)

e)

d

12.5 m

15 cm

12 m

12 cm e

The sides of a square are 10 m. Find the length of a diagonal. The sides of a rectangle are 15 cm by 20 cm. What is the length of a diagonal? What is the length of the sides of a rhombus if the diagonals are 7 cm and 24 cm? Which of the following measurements are incorrect for right-angled triangles? Explain your answers. a)

b)

3

2

5

1

3

4

c)

4

d) 4

5

4

8

3

e)

9

12

f)

6.5 2.5

6

15

Right-angled triangles 159

A ladder is 8 m long and Mark needs to go on top of a wall 6 m high. The top of the ladder is almost in line with the top of the wall. a) Use this information to make a sketch of a right-angled triangle, putting in the appropriate measurements. b) Calculate the approximate distance between the wall and the foot of the ladder?

The diagram shows four flag poles at the vertices of a square.

metres. The longest distance between two of the feet of the flag poles is 50 a) Sketch the triangle that can be used to find the shortest distance between two of the flag poles. b) Calculate this shortest distance. c) If a pole is 3 metres high, what is the approximate length of the wire that joins the top of one pole to the foot of another i) along a side ii) across the middle of the square?

Investigation Do research to find out more about Pythagoras and his contributions to mathematics. There are sets of three numbers called Pythagorean triples. Find some and explain them to a friend.


Key words

Unit 15.2 The tangent ratio

Hypotenuse Opposite side Adjacent side Tangent Degrees Minutes

In this unit you will: determine the tangent of an acute angle use a calculator or table to find an angle, given the tangent ratio and vice versa use the tangent ratio to calculate angles or sides of a right-angled triangle.

In any right-angled triangle the hypotenuse is the longest side. It faces the right angle. The side facing an angle is the opposite side. There are two adjacent sides to an angle. But the hypotenuse already has its special name. In the diagram below AC is the hypotenuse, it faces the right angle B. AB is adjacent to A and BC is opposite to A. C

opposite

B adjacent

hypotenuse A

The ratio of the opposite side to the adjacent side gives a fraction that is called the ‘tangent of the angle’. 3 QR For example, the tangent of P 0.3. QP 10 That is, tan P 0.3. In short, tan P R 3 cm Q 10 cm P

Angles are measured in degrees. There are 60 minutes in 1 degree. We can use a calculator or a tangent table to find the value of the angles. Using tables The tables sometimes give the values as degrees and fractions of degrees and sometimes as degrees and minutes. Look in the body of the table for the number or the one closest to what you need e.g. 0.300.


Minutes Degrees

0’ 0.0°

6’ 0.1°

12’ 0.2°

18’ 0.3°

24’ 0.4°

30’ 0.5°

36’ 0.6°

42’ 0.7°

48’ 0.8°

54’ 0.9°

0

0.000

0.002

0.004

0.005

0.007

0.009

0.011

0.012

0.014

0.016

1

0.018

0.019

0.021

0.023

0.025

0.026

0.028

0.030

0.031

0.033

2

0.035

0.037

0.038

0.040

0.042

0.044

0.045

0.047

0.049

0.051

3

0.052

0.054

0.056

0.058

0.059

0.061

0.063

0.065

0.066

0.068

4

0.070

0.072

0.073

0.075

0.077

0.079

0.081

0.082

0.084

0.086

5

0.088

0.089

0.091

0.093

0.095

0.096

0.098

0.100

0.102

0.103

9

0.158

0.160

0.162

0.164

0.166

0.167

0.170

0.171

0.173

0.175

10

0.176

0.178

0.180

0.182

0.184

0.185

0.187

0.189

0.191

0.192

15

0.268

0.270

0.272

0.274

0.275

0.277

0.279

0.281

0.283

0.285

16

0.287

0.289

0.291

0.292

0.294

0.296

0.298

0.300

0.302

0.304

6.31

6.39

6.46

6.56

6.61

6.70

6.77

6.86

6.94

7.03

19.1

19.7

20.5

21.2

22.0

22.9

23.9

24.9

26.0

27.3

57.3

63.7

71.6

81.9

95.5

114

143

191

287

573

Degrees

81 87 89

That is P 16.7° or 16°42 17° Using the scientific calculator Input

.

3 , then inverse , then tan . The answer is 16.699.

Calculators differ so learn how to use your calculator efficiently.


Worked examples 1. What is the tangent of a. 15° b. 10.5°

c. 87° 24?

Look at the purple squares in the tan table.

2. Find the angle that has a tangent of a. 0.158 b. 6.460

c. 0.277.

Look at the blue squares in the tan table.

3. In the following diagrams, find a. A b. MN.

a.

b.

A

O 36°

8m N

15 cm

C

4 cm

B

M

1. Using the table a. 15° in the first column, under 0° in the 2nd column ‘0.268’ b. 10° in the first column, under 0.5° in 8th column ‘0.185’ 24 c. 87° in the first column, under 24 or 0.4° in the 7th column ‘22.0’. 24 (60)° 0.4°. Using the calculator Remember calculators a. Input 15 then tan or tan then 15 function differently. b. As for a. but input 10.5. c. For 87° 24 you may have to change it to 87.4°. 2. a. 9° b. 81.2° or 81° 12 c. 15.5° or 15° 30

See tables, or use the calculator as above.

4 opp 3. a. tan A 0.267 adj 15 So A 15°. ON 8 b. tan 36° MN MN 8 So 0.727 MN 8 MN 11 m. 0.727

Transposing.


Exercise Find the tangent of the following angles. a) 19°

b) 74°

c) 25°

d) 60°

e) 59°

Find the values of the following angles if the tangents of the angles are a) 0.719

b) 63.7

c) 1.402

d) 0.845

e) 0.007.

d) 25.6°

e) 7°54

Calculate the tangent of the following angles. a) 11.5°

b) 47.8°

c) 55.9°

Work out the angles marked a, b, c in the following diagrams. 16 cm a

0.25 m 1.9 m

20 cm

c

25.8 m

b 5m

Calculate the sides marked n, p, q in the following diagrams. 10 cm n 50° 7m


q p

28° 70° 25 cm

f) 45°

Key words

Unit 15.3 The sine and cosine ratios

Sine Cosine

In this unit you will: distinguish between the sine and cosine of an acute angle use tables or calculators to find the sine or cosine of an angle use tables or calculators to find an angle, given the sine or cosine determine the use of sine or cosine to find the values of angles or sides of right-angled triangles. Now we will examine ratios that involve the hypotenuse. opposite The sine of an angle, in a right-angled triangle, is the ratio . hypotenuse adjacent The cosine of an angle, in a right-angled triangle, is the ratio . hypotenuse P

hypotenuse

opposite

Q

adjacent

R

r PQ QR p sine R or , cosine of R or q PR PR q Remember the lowercase letter represent the side opposite the angle with the corresponding uppercase letters e.g. R faces side r. Practice using your tables and calculators to find the sine and cosine of angles. These tables are used the same way as you use the tangent tables. You input sin or cos instead of tan on the calculator.

Worked examples 1. Calculate the value of R and X in the figures.

a.

R

b.

T

S

5 i.e. sin R 0.5 10

20 cm

5 cm

10 cm

opposite O 1. a. sin R hypotenuse H

Z

Y

10 cm

X

In short, sin R.

R 30° Right-angled triangles 165

adjacent A b. cosine X hypotenuse H 10 i.e. cos X 0.5 20

In short, cos X

X 60° 2. Use the sine or cosine ratios to calculate the unknown sides. O AB 2. sin C H AC c sin 40° 50 c So 0.643 50

A

B

50 m

Substituting the values.

40°

Simple equation.

C

50 0.643 c c 32.2 m A BC cos C H AC a cos 40° 50 a So 0.766 50 50 0.766 a a = 38.3 m

Exercise Use a table or calculator to find the sine of a) 50°

b) 45°

c) 72°

d) 60°

e) 88°

f) 90°.

Use a table or calculator to work out the sine of a) 35.4°

b) 27.5°

c) 72°48

d) 61°12

e) 8°30.

d) 90°

e) 30°

d) 36°24

e) 45°42

Use a table or calculator to find the cosine of a) 17°

b) 5°

c) 59°

What are the cosines of the following angles? a) 70.1°

b) 5.9°


c) 57°18

f) 45°.

Use the sine or cosine ratio to find the values of the angles a to e in the following figures.

a)

c)

b) b

30 cm

c

2 cm

2.5 m

5 cm a 20 cm 1m 6m

d)

e)

e

36 cm

48 cm

10 m d

Use the sine or cosine ratio to calculate the value of the sides p to t in the following diagrams.

a)

q

b)

48°

c)

62°

r 12 m

p

100 m

16 cm

d)

31°

e)

5 cm

t

s

44 cm

54°

70°


The two equal sides of an isosceles triangle are 15 cm. Two angles are 45° each. Do a sketch and put in the given information. a) What is the length of the line of symmetry? b) Work out the length of the third side.

The slant height of a cone is 8 cm. The angle between the radius and slant height is 65°.

65°

a) Calculate the radius. b) Find the vertical height.

8 cm

Investigation Compare the three sets of values, tangent, sine and cosine, for angles from 0° to 90°. Which ones increase or decrease?

Compare the sine and cosine values of 60°, 30°, 45°, 15° and 75°. Describe any pattern you observe and use the pattern to check other sets of angles.

What is the tangent of 45°? Explain this result to a classmate. Explain the relationship between the tangent of an angle and the tangent of a line to a classmate.


Key words

Angle of elevation Angle of depression Bearings

Unit 15.4 Using trigonometry to solve real-life problems In this unit you will: determine when and how to use a specific trigonometric ratio and/or Pythagoras’ theorem.

Let us summarise what we know about right-angled triangles, using triangle ABC, right-angled at B. b2 a2 c2 c sin C b

a cos C b

c tan C a

O S is H

A C is H

O T is A

A

b

c

B

a

C

Let us review what we know about bearings and other angles in real life. ● Bearings are always taken from the north line. ● Angles of elevation and depression are always taken from the line of sight or eye level. Eye level

N

Angle of depression

N

Angle of elevation Eye level

Heights, widths and depths in real life are usually measured perpendicularly.

Height

Depth Width

Worked example A ferocious dog is tied to a post that is 5 metres from a path. The length of its chain is 6 m. a. Calculate the length of the side of the path that may not be safe from the dog (to the nearest metre). b. What is the angle through which the dog would turn to cover this distance?

Path 6m Dog

5m


The dog can cover a circle if not restricted anymore. Part of the path is within his reach because his chain is longer than the distance he is tied from the path. Let the required diagram be the isosceles triangle ABC. The length BC is the distance along the path and A is the angle that the dog turns. A y 6m 5m

B

2x

C

a. Let BC 2x Using Pythagoras’ theorem. x2 52 62 2 2 2 x 6 5 36 25 11 x 11 3.32 BC 6.64 7 7 m is not safe on one side of the path.

b. Let y be 21 of A cos y 65 0.833 y 33.6° A 2 33.6 67.2° The dog turns about 67°.

Exercise The slant height of the ramp in the diagram is 3 m. The angle between the slant height and the vertical height is 50°. Calculate the vertical height. 50°

3m

The angle of elevation is taken 12 m from a tower and is 65°. What is the height of tower?

The top of a ladder is resting 10 m up a wall. The angle between the wall and the ladder is 25°. How far from the wall is the foot of the ladder?

A man on a hill sights an object about 40 m out at sea. From the foot of the hill the angle of depression he used was 15°. a) What was the vertical height of the man from sea level? b) How far was he from the object?


Pat lives 2 km north of Mark’s house. Rupert’s house is 1.5 km east of Mark’s house. a) Calculate the distance between Pat’s house and Rupert’s house.

Pat

b) What is the bearing of Rupert’s house from Pat’s house?

Rupert

Mark

A rope 16 m long is tied to the top of a vertical pole and anchored on the ground. The angle of elevation from the point that it is anchored to the top of the pole is 30°. a) What is the height of the pole? b) How far from the foot of the pole is the rope anchored?

Sissy and Joan are on the same bank of a river that runs north to south. Sissy is 24 m north of Joan and sees Mr Lindsay’s house is east of her. The house is on a bearing of 72° from Joan. How far is the house from each girl?

Calculate the angles and the sides of the kite ABCD

A

if the line of symmetry is 10 cm and the shorter diagonal is 6 cm. X is the intersection of the diagonals. 7 cm 3 cm D

3 cm X

B

3 cm C

Investigation Make a list of five careers in which trigonometry is used in different ways, describing briefly how it is used in each case.


Looking back Find the unknown lengths in each of these right-angled triangles. a)

b)

16 cm

b

11.9 m

9.1 m

c)

c

41 mm

24 cm 59 mm

119 mm

d

10.2 cm

a e

A square tin of biscuits has sides of length 34 cm. It is decorated with a ribbon across one diagonal. What is the length of the ribbon?

For the triangle shown write down in terms of a, b and c

C

each of the following. a) sin B

b) cos B

c) tan B

d) sin C

e) cos C

f) tan C

A

For the triangle shown, calculate

a

b

c

A

a) BC 10 cm

b) AB.

30° B

C

In a triangle PQR, Q 90°, PR 60 mm and PQ 30 mm. Calculate a) R

b) P.

In a triangle ABC, AB 84 cm, C 84° and B 90°. Calculate the length of AC. Jessica is standing on a grandstand 15 m high. She is looking at the top of a shed roof which has height of 8 m. She works out that the angle of depression of the top of the roof is 8°. The grandstand and the shed are both on level ground. a) Draw a diagram to show this information. b) Work out the distance from the grandstand to the roof of the shed.


B

Chapter

Key words

Enlargement Scale factor Centre of enlargement Ratio

Transformations In this chapter you will examine enlargement which changes size but not shape. You will use your knowledge gained in enlargement to calculate side lengths of figures.

16

Unit 16.1 Enlargement In this unit you will: understand and use the language and notation associated with enlargement enlarge 2-D shapes given a centre of enlargement and a positive whole number scale factor. Enlargement is a type of transformation. When a shape is enlarged, the lengths of all the sides are multiplied by a scale factor. The angles stay the same during enlargement. Every enlargement has a centre of enlargement. Lines joining equivalent points on the object and image meet at the centre of enlargement. When a shape is enlarged, the ratios of corresponding sides on the image and the object are equal to the scale factor. For any enlargement, if the scale factor is n, then the ratio of the corresponding sides on the object and the image is 1 : n. A The diagram shows a triangle ABC that has been enlarged by scale factor 3 with centre of enlargement at O.

A O

The lengths of the sides of triangle ABC are three times the lengths of the corresponding sides of triangle ABC. For example, AB 3 AB.

C

B B

C

Worked example y 10

Describe this enlargement.

The side lengths of the image ABC are three Scale factor 3 Centre of enlargement (0, 0) times the size of the side lengths of the object ABC, so the scale factor is 3.

A

9 8 7 6 5 4 3

A C

Lines joining equivalent points on the object and image meet at (0, 0), so this is the centre of enlargement.

B

2 1 0 0

C

B 1

2

3 4

5 6

7

8 9 10 x

Exercise Trace each shape and enlarge it, using the centre of enlargement O, by the given scale factor.

a)

b)

c) O

O

scale factor 3

O

scale factor 4

scale factor 2 Transformations 173

Describe this enlargement: ABCD → ABCD

y 8 7

B

A

6 5 D A

4 3

B

2 D

C

1 0

C 0

1

2

3

4

5

6

7

9 x

8

Describe this enlargement.

O A1

A2 and A3 are enlargements of A1. a) Trace the diagram and find the centre of enlargement that maps i) A1 onto A2 ii) A2 onto A3 iii) A1 onto A3.

A3

b) What do you notice about the three centres of enlargement?

A2

a) What centre of enlargement maps

A

triangle ABP onto triangle ACK? What is the scale factor of this enlargement? b) What centre of enlargement maps triangle JKL onto triangle JAD? What is the scale factor of this enlargement?

B O

P

I

D

N

L

K J

C

E

M

H

F G

a) Mark the following points on squared paper or using ICT. A(1, 1), B(3, 1), C(3, 3) and D(1, 2). Join up the points to make a quadrilateral. b) Enlarge ABCD by scale factor 3 using centre (0, 0). c) On the same diagram, enlarge ABCD by scale factor 4 using centre (0, 0). d) What is the relation between the coordinates of the vertices of ABCD and each of its enlarged images? 174 Jamaica Maths Connect 3

Key words

Similar Ratio Scale factor Simplest form Proportion

Unit 16.2 Scale factors, ratio and proportion In this unit you will: understand and use ratio related to enlargement of 2-D shapes identify the scale factor of an enlargement as a ratio of any two corresponding line segments reduce a ratio to its simplest form use ratio and proportion and link to fraction notation.

If a shape is enlarged, the image is mathematically similar to the object. This means that all the angles are the same and that the ratios of corresponding sides on the image and the object are equal to the scale factor. Rectangle ABCD is an enlargement of rectangle ABCD by scale factor 3. The ratio of the sides AB : AB is 1 : 3. The ratio of the sides CD : CD is 2 : 6, which cancels to give 1 : 3 in its simplest form. Two quantities are in proportion if their ratio stays the same. Since ABCD was enlarged by scale factor 3, we know that the dimensions of ABCD are three times the dimensions of ABCD. This means that the sides of rectangle ABCD and rectangle ABCD are in proportion.

O

B

C

A

D

B

C

A

D

Since ABCD was enlarged by scale factor 3, each point on the image ABCD is three times as far away from the centre of enlargement as the equivalent point on ABCD. For example, OA is three times the distance OA.

Worked example Triangle ABC is an enlargement of triangle ABC. The side lengths of ABC are in proportion to the side lengths of ABC. a. Write the ratio of the corresponding sides AB and AB in its simplest form. A Repeat for BC and BC. 5 cm b. Write the side lengths of ABC as a 3 cm fraction of the side lengths of ABC. C C B 4 cm c. Find the scale factor of the enlargement. d. Find the size of the missing length AC. a. The ratio of AB to AB is 3 : 9 1 : 3 and the ratio of BC to BC is 4 : 12 1 : 3. b. Side lengths of ABC are 31 of ABC. c. Scale factor is 93 3. d. The length AC 5 cm so AC 5 3 15 cm.

A

?

12 cm

9 cm

B

The ratio of corresponding lengths in the object and the image is equal to the scale factor. Since the scale factor is 3, the side lengths of ABC are 3 times the lengths of the corresponding sides on ABC.

Transformations 175

Exercise ABCD is to be enlarged by scale factor 2, where D is the

A

centre of enlargement.

4 cm

B

5 cm

a) Calculate the lengths of A’B’, B’C’, C’D’ and D’A’ on the enlarged shape.

5 cm

D

C

10 cm b) Check your answers to part a) by drawing ABCD accurately and enlarging it by scale factor 2, using D as the centre of enlargement.

In each of these diagrams, the side lengths of shape A are in proportion to the side lengths of shape B.

i)

ii)

10 cm

iii) 12 cm

36 cm

A 7 cm

A 8 cm

A 20 cm

16 cm

B

6 cm

35 cm

B

5 cm

B 20 cm ?

?

?

For each enlargement: a) write the ratios of the corresponding sides of A and B in their simplest form b) write the lengths of shape A as a fraction of those of shape B c) find the scale factor of the enlargement d) find the size of the missing length.

In each of these diagrams, the side lengths of shape X are in proportion to the side lengths of shape Y. Calculate the missing length for each pair of shapes.

i)

ii)

8 cm X 4 cm

?

Y

iii)

3 cm X

? ?

2 cm

Y

X 50 cm

6 cm 16 cm

16 cm

30 cm

Are the side lengths of these triangles in proportion? Explain your answer. 7 cm

A

21 cm

B

11 cm 44 cm

Investigation Investigate the proportions of metric paper sizes. a) Can you divide up a sheet of A3 paper to make sheets of i) A4 paper

ii) A5 paper

iii) A6 paper?

b) Is an A3 piece of paper an enlargement of an A4 piece of paper? How can you prove this? 176 Jamaica Maths Connect 3

Y

Key words

Similar Ratio Proportion

Unit 16.3 Problems involving enlargement In this unit you will: solve problems involving ratio and proportion.

When a shape is enlarged, the image is mathematically similar to the object. The angles on the object and the image are the same, and the ratios of corresponding sides on the image and the object are equal to the scale factor. An object and its enlarged image are in proportion. We can use this information to solve problems to do with enlargement.

Worked example Ali has a photograph that measures 10.5 cm by 15.5 cm. He takes the photo to be enlarged so that its sides are four times their original size. Allowing for a 2 cm border all round, what size frame should he buy? The photograph and its enlargement are proportional. If the original photograph measures 10.5 cm by 15.5 cm and is enlarged to four times the size, the ratios of the sides will be 1 : 4. The enlargement will therefore measure 42 cm by 62 cm, so the frame needs to measure 46 cm 66 cm.

Exercise The dimensions of two pictures, A and B, are

B

in proportion. Picture B is an enlargement of picture A. a) Calculate the width of picture B. b) Calculate the perimeter of picture A and of picture B. Write the ratio of the perimeters in its simplest form.

60 cm

A

c) Explain why the ratio of the perimeters is the same as the ratio of the lengths.

20 cm

12 cm

?

a) Plot the following points on a coordinate grid, A(2, 1), B(3, 1), C(3, 3) and join them up to make a right-angled triangle. b) Measure and write down the size of the angles. c) Enlarge the triangle by scale factor 2, centre (0, 0). Label the image ABC. Measure the angles of the enlarged triangle. d) What can you say about the angles of ABC and ABC? e) Explain why angles remain unchanged when a shape is enlarged. Transformations 177

Investigations During their holiday in London, Alf and Sue take one of their photos to be enlarged. The original photo measures 5 cm by 8 cm and costs 40p. The enlargement measures 10 cm by 16 cm and the chemist wants to charge £1.60 for it. Alf thinks the enlargement should cost 80p. How do Alf and the chemist calculate the cost of the enlargement? How much do you think the enlargement should cost?

We use microscopes to give us enlarged images of very tiny objects. Investigate other pieces of equipment that are used to enlarge images.

Shape A

Shape B

You will need isometric paper. All the triangles in this diagram are equilateral. The small triangles are the same size. a) Write down the ratio of the lengths of the sides of shape A to shape B. b) Write down the ratio of the perimeter of shape A to the perimeter of shape B in its simplest form. c) Compare the areas of shapes A and B. d) Investigate the ratios of the lengths, perimeters and areas for other enlargements made of small triangles. What do you notice?

Looking back Copy these shapes and the given centre of enlargement, O. Enlarge them by the given scale factor in each case.

a)

O

b)

c) O O Scale factor 3

Scale factor 2

d)

e) O O Scale factor 2.5


Scale factor 3.5

Scale factor 2

Project T

W

P

P

T

W

W

P

T

T

W

P

P

T

W

W

P

T

T turquoise P purple W wine

You can draw a design of this double bed quilt as a tessellation of squares. This is how you might think about making and measuring a quilt: ● Material is 115 cm wide. Each completed square needs to be 15 cm across (150 mm 150 mm). ● Add 6 mm to each side of a square for the stitching, so each square is 16.2 cm across. ● Measure a strip of 16.2 cm wide across the width of the material. Make sure the lines are parallel, i.e. that your whole strip is 16.2 cm wide, before cutting. ● Use this strip to measure and cut the next strip. You will get six strips out of one metre. ● Cut a square template out of cardboard 16.2 cm 16.2 cm. Use this to cut out your squares. You will get seven squares out of one strip, 42 squares altogether. You can now work out how much material you have to buy. For this double bed quilt, use 9 squares across and 14 for the length. That is 126 squares so you need 3 metres of material. As the material is highly patterned you should choose a plain fabric for your edging or border. So that the border does not have a join down the length of the quilt, measure the completed length of your quilt and add double the width of the border. Buy this length of fabric. You can always use the extra for other quilts or share with a friend. Cut 10 cm width for the edging. This allows 2 6 mm for stitching and to fold over the side batting (stuffing). You can calculate the cost of this double bed quilt as follows. 2 m wine velvet 2 m purple velvet 4 m turquoise velvet 4.5 m calico (inside pockets) Threads 10 m batting 1 sheet Consumables

$43 $43 $86 $36 $12 $108 $39 $31

Total cost of materials

$398

Mark up for labour (100%)

$398

Total cost

$796

Note: shops would mark up 150–200% for labour, and quilts involving less work and made of cheaper fabrics sell for $1 000 upwards in craft shops locally. Transformations 179

Design your own quilt using one of the following patterns and the materials listed. Calculate the amount of material needed, and how much it will cost to make.

a) Star of the West

Material

Cost

Velvet

$21.50/m

Cotton

$8/m

Wool felt

$18/m

Flannel

$12/m

Calico

$10/m

Batting

$10/m

b) 8 pointed star

10 cm

10 cm

c) Twisted star

d) Churn dash

20 cm


Case study – Quilt maker Susan Ross-Morrison I am a retired teacher. I started making quilts as a hobby because I’ve always loved sewing and I hate to sit around doing nothing. Anyone with an imagination can design a quilt. Whether you think of a pattern of strictly repeating designs, or whether you let your design take its own course – it doesn’t matter, you should let your imagination run. What matters is that you use basic techniques and get your measurements right.

People normally use 100% cotton, but you can also make a work of art by combining different textures and thicknesses. You should try to use the correct equipment if possible, but to experiment and try out this line of art, you can start with what is available to you, for example, rulers, cutting mats and scissors. Lay out the whole quilt on the floor or a big board to see the completed effect before you sew it. Once you are happy, stitch the first row of squares together (6 mm from the edge). Do the same with row 2 and then join row 1 to row 2 and so on. Stitch a border on to the four edges of your quilt using mitred corners.

Mitred corners Cut the batting 4 cm bigger than the quilt on all sides and tack it onto the whole back of the quilt. For the batting to stay in place, either a) stitch through the quilt and the batting along a few strips on the right side of the quilt, following the line between the strips b) stitch beads or buttons onto the quilt in a design of your own by hand. Turn the quilt over and now work on the back. Fold the border over the sides of the batting. Place a backing on the quilt. I use a sheet of the same width as the quilt so that the quilt does not slide off the bed. Plain flannelette or other fabric is also suitable. Tack this in place and then stitch right around it. Remember to autograph your quilt. For example: To Lulu, with lots of love Susan, 25 December 2005 I do mine in embroidery on the backing sheet.

Transformations 181

R6

Revise and consolidate How could you prove that a triangle with sides of 25 cm, 60 cm and 65 cm is right-angled?

Find the value of x in each of these triangles: 9

a) 7

b) 14

10

x

x

c)

15

x

x

Find the value of y in each case. a)

b) y

45 15

y

y 30

15 cm 12 cm

The Leaning Tower of Pisa in Italy is 150 m tall. When a stone is dropped from the top, it lands 5 m from the base because the tower is leaning over. What is the height of the tower from ground level? (Give your answer correct to two decimal places.)

Measure the lengths of the sides of this triangle to the nearest millimetre. Use your measurements to calculate the values of: a) sin 28°

b) cos 28°

c) tan 28°

d) sin 62°

e) cos 62°

f) tan 62°

62° 64 mm

28° 56 mm


Find the values of a) to e). 7.65 cm

a)

b)

145 m

c)

16°

c

a

22 cm

b

250 m

43°

d)

e) 45.5 cm

750 cm 23°

66°

e

d

A kite is attached to a string 15 m long. If the string makes an angle of 26.1° to the horizontal, how high is the kite?

Jess and John are arguing about whether it is shorter to walk diagonally across a field or to walk around the two sides. D

A

38°

36 m

C

B

a) Calculate the length of the diagonal distance across the field. b) Calculate the distance from A to B to C. c) Which route is shorter? d) Will this always be the case? Why?

Draw an enlargement of each shape using the scale factor given.

Scale factor 2

Scale factor 3

Scale factor 2.5


Chapter

17

Probability In this chapter you will deal with the branch of statistics known as probability. You will extend your knowledge of probability terms and learn how to express probability in numbers. You will also carry out investigations to determine probability of events and compare experimental and theoretical probability.

Key words

Event Outcome Theoretical probability Mutually exclusive

Unit 17.1 Probability In this unit you will: use the language of probability learn that the sum of probabilities of all the mutually exclusive outcomes of an event is 1. An event, such as rolling a dice, can have several outcomes. Throwing a six is one possible outcome. Probability is a measure of the chance of an outcome happening. It should be written as a fraction, decimal or percentage only. The theoretical probability of an outcome is shown as: number of ways the outcome can happen P(outcome)= total number of outcomes Outcomes such as throwing a 1, throwing a 2 are examples of mutually exclusive outcomes, as they cannot occur at the same time. The outcomes throwing an even number, throwing a 2 are not mutually exclusive, as they can both occur at the same time. The sum of all mutually exclusive outcomes of an event is 1. If the probability of an outcome happening is p, the probability of the outcome not 3 happening is 1 p. For example, if the probability of an outcome happening is 11, the 3 8 probability of the outcome not happening is 1 11 11.

Worked example A breakfast cereal contains one of four gifts: sticker, badge, pen or key ring. P(sticker) 0.2, P(badge) 0.35, P(pen) 0.3 a. A new packet of cereal is opened. Calculate the probability of the gift being a key ring. b. Guy has collected 20 gifts. How many pens is he likely to have? c. Meera has three stickers. How many packets is she likely to have opened? a. 1 [0.20.350.3] 0.15 b. 20 0.3 6 pens c. Number of packets 0.2 3 Number of packets 3 0.2 15 packets


Exercise A computer database categorises people

Age

into one of six groups. The probability of a person being chosen from a group is 0–14 15–64 shown opposite. A person is chosen at random. Calculate the probability 65 that the person is a) male b) aged 15–64 c) not 65 or over d)

Male

Female

0.1

0.08

0.34

0.32

0.07

0.09

female and aged 0–64.

A factory packs bags of crisps using three different machines. A bag of crisps will be underweight, overweight or the correct weight. The manager calculates the probabilities for each machine for each category. Machine

Underweight

Overweight

A

0.04

0.23

B

9%

39%

C

1 6

1 5

Correct weight

a) Calculate the probability that a bag will be the correct weight for each machine. b) If machine B packs 300 bags of crisps, approximately how many will be overweight? c) If machine C packs 15 underweight bags, how many bags did C pack in total?

Grace has two packets of sweets. Each packet contains the same number of sweets. The probability of Grace choosing a sweet that she likes is 0.5 from one packet, and 0.1 from the other packet. The two packets are emptied into one bag. What is the probability now that Grace will choose a sweet that she likes? Work out the new probability if each packet contains 10 sweets.

a) Which of the three shapes is the spinner most likely to land on? b) Calculate the probability of the spinner landing on each shape . c) The spinner is spun 120 times. Estimate the number of times the spinner will land on each shape.

60° 45° 45° 70° 50°

You need to round your answers.

Ella says that each of the eight triangles on her spinner has an equal chance, as each triangle has the same area. Is Ella correct? Justify your answer.

Probability 185

Key words

Unit 17.2 Estimating probabilities

Theoretical probability Experimental probability

In this unit you will: estimate probabilities from experimental data.

The data from an experiment can be used to find an estimate of the theoretical probability. For some investigations, such as the one in the example, this may be the only type of probability that can be calculated for an experiment. The experimental probability (or estimated probability) is calculated in the same way as the theoretical probability. number of times an outcome happens Experimental probability number of times the experiment was carried out The sum of the estimated probabilities will equal 1 if the table is recording all possible outcomes of the experiment.

Worked example Jenny is performing an experiment to explore the hypothesis ‘toast always lands butter side down’. She uses a piece of card with the word ‘butter’ written on one side to simulate the experiment. Estimate the probabilities of the two outcomes from her data. Direction of toast

Butter facing up

Butter facing down

Frequency

53

47

Direction of toast

Butter facing up

Butter facing down

Frequency

53

47

Estimated probability

53 100

0.53

4 7 100

0.53 0.47 1

0.47

Exercise Some pupils throw 3 coins, recording the number of heads showing each time. Pupil

Number of throws

0 heads

1 head

2 heads

3 heads

A

30

4

13

10

3

B

120

13

48

44

15

C

50

5

19

20

6

a) Which pupil’s results should be the most reliable? b) Use all of the above results to find the total frequency of the 4 outcomes for the experiment being carried out 200 times. Number of heads

0

1

2

3

Total frequency Estimated probability

c) Find the estimated probabilities for each of the number of heads being thrown. d) Find the sum of the estimated probabilities. 186 Jamaica Maths Connect 3

This question requires digit cards 0, 1, 2 and 3. a) Use cards 0 and 1. Shuffle the cards and deal face up. Are they in ascending numerical order? Record the result in a copy of the table opposite.

In order

Not in order

b) Repeat the experiment 40 times. Use your data to calculate the probability of the two cards being dealt in order. Repeat the above experiment using digit cards 0, 1, and 2, then digit cards 0, 1, 2 and 3. Each time calculate the experimental probability of the cards being dealt in order.

This question requires three six-sided dice. a) Throw the three dice. The numbers showing will be one of the outcomes below. Outcome

3 numbers the same

Only 2 numbers 3 consecutive the same numbers (eg 2, 3, 4)

Other outcome

Frequency

b) Copy the table and record the outcomes of 40 throws. c) Calculate the estimated probabilities for each outcome from your table. d) Find the sum of the estimated probabilities.

Helen throws 10 matches, and counts the number that land completely between two lines. She repeats the experiment 60 times. No. of matches completely between lines

0

1

2

3

4

5

6

7

8

9

10

Frequency

1

2

12 17 15

7

2

1

3

0

0

a) How many throws of a match were made in total? The answer is not 10! b) How many matches landed completely between the two lines? c) Use your answers above to calculate the probability of one match out of 10 landing between the two lines. 2

The answer is not 60!

Investigation Question 4 is similar to a famous experiment called ‘Buffon’s needle’, which is related to the calculation of . Search the Internet to find a simulation of Buffon’s needle to find out more about this investigation. Try out the simulation. Write a short report on what you have found out.

Probability 187

Key words

Unit 17.3 Comparing probabilities In this unit you will: compare experimental and theoretical probabilities in a range of contexts.

Random event Theoretical Experimental

A dice is fair if every number has an equal chance of being thrown. When a fair dice is thrown, it is a random event, so we do not know beforehand what number the dice will show. We can throw the dice a number of times and use the results to calculate the experimental probability of, for example, throwing a 6. We can also find the theoretical probability of throwing a 6. We do not expect these two values to be equal, but it can be interesting to compare them.

Worked example A coin is thrown a different number of times and the number of tails counted. This is used to calculate the probability of throwing one tail. Number of throws

20


40

0.45

60

80

100

120

140

160

180

200

0.4 0.483 0.525 0.52 0.492 0.507 0.4875 0.494 0.505

a. Draw a diagram to show the estimated probability of a tail as the number of throws increases. b. Comment on the difference between the estimated probabilities and the actual probability of the number of tails. c. The experimental probability after 1000 throws is 0.487, so it still does not equal 0.5. Explain why this does not mean that the coin is biased. a.

0.55 0.5 Estimated 0.45 probability 0.4 0.35

20

40

60

80 100 120 140 160 180 200 Number of throws

b. The theoretical probability is 0.5. As the number of throws increases, the experimental probability comes closer to this value. c. As we do not expect the experimental probability to have exactly the same value as the theoretical probability, the coin is not necessarily biased. ‘Biased’ means ‘not fair’.


Exercise A tetrahedral dice, numbered 1 to 4, is thrown and the sum found of the three uppermost faces. This is repeated and the two totals are added together and recorded. a) Show that there are 4 possible totals when the dice is thrown once. b) Samir carries out this experiment, recording his results in a frequency table. Calculate the experimental probability for his table. Total Frequency

12

13

14

15

16

17

18

4

12

17

19

14

9

5

Experimental probability

c) Copy and complete the sample space diagram for this experiment. d) Use the table above to compare the experimental and theoretical probabilities.

6

7

8

9

6 7 8

You will need to find the theoretical probability for each of the 7 possible outcomes.

9

e) How could the experimental probabilities be made more accurate?

This question requires coins.

1

The probability of a baby being a girl is approximately 2. Use a coin to simulate the gender of a baby. A baby will be a girl if it lands heads up and a boy if it lands tails up. For example, throwing three coins will simulate the number of boys and girls in families with three children. a) Copy the table opposite. Number of girls 0 1 2 Throw two coins. Record the Frequency number of ‘girls’ the two coins show in the table. b) Use your results to find the experimental probability of different numbers of girls in families with two children. c) Draw a tree diagram to find all possible outcomes and the theoretical probabilities for each number of girls. d) Compare the results for the two types of probability. e) Repeat the experiment using three coins to simulate families with three children.

This question requires dominoes. a) Use systematic working to find all possible outcomes when the number of spots on each end of a domino is added. Calculate the theoretical probabilities of each total. b) Draw a data collection sheet to record all the possible totals when the total number of spots on a domino is found. c) Put the box of dominoes in a bag. Remove a domino and record its total on your sheet. Decide on an appropriate sample size for this experiment, and repeat the experiment this number of times. Use your results to calculate the experimental probabilities for each total. Compare the results for the two types of probability. d) Draw a sample space diagram to show all possible totals when two dice are thrown together. Compare the theoretical probabilities for the totals of a domino and two dice. Why are they not similar? Probability 189

Looking back The probability of choosing a toffee from a box of chocolates is 0.2, and the probability of choosing a cream is 0.35. The remainder of the chocolates are truffles. a) What is the probability of choosing a truffle? b) What is the probability of choosing a cream or a toffee? c) If there are 20 chocolates in the box how many are toffees? d) If I have eaten 18 truffles altogether, what is the least number of boxes of chocolate that I must have had?

60 raffle tickets are sold numbered 1–60. The winning ticket is chosen at random. What is the probability that the winning ticket is a) even b) less than 10 c) not less than 10 d) odd and divisible by 5 e) odd and divisible by 2 f) not number 6? Give your answers as percentages.

A CD contains 15 tracks. The tracks are played at random. What is the probability that the first track played is a) number 7

b) not number 7?

Copy and colour this spinner (twice) so that the probability of it landing on: a) green is 13, not blue is 56 and not yellow is 12 b) green is 13, not blue is 56 and not yellow is 23. Note: you will need four colours.

Joseph is trying to find out the probability of a drawing pin landing on its head (point up) when dropped. He performs an experiment with the following results: Direction

Heads

Point down

Frequency

14

26

a) Estimate the probability of the two results from his data. b) What is the sum of the two outcomes? c) Tom says ‘I did this 100 times and my results for a head was 0.28.’ Which result was the most accurate? Explain your answer. d) The drawing pin manufacturer says the theoretical probability of a drawing pin landing on its head is 0.25. Does this mean there is something wrong with Tom’s drawing pins? Explain.


Test your knowledge In all tests choose the correct answer for multiple choice items. Working must clearly be shown for all other questions.

Test 1 The next term in the series {1, 2, 6, 24, …} is a) 48

b) 72

c) 120

d) 144

If N {natural numbers}, Q {rational numbers} and Z {integers}, then a) N Q Z

b) N Z Q

c) Q N Z

d) Q N Z

24 million expressed in standard form is a) 24 107

b) 24 106

c) 2.4 107

d) 2.4 106

If U {letters in the alphabet}, S {letters in the word ‘Spanish’ and F {letters in the word ‘French’} then n(S F)

a) 10

b) 11

c) 13

d) 16

b) 7

c) 19

d) 38

c) 8

d) 64

In base 10, 100112 a) 6

The value of 100 36 is a) 2

b) 4

All the students in a class of 28 play cricket or tennis or both. Twenty play cricket and 16 play tennis. a) If x students play both games how many play i) cricket only

ii) tennis only?

b) By first forming an equation in x, find out how many play both games. c) Illustrate the problem by means of a Venn diagram.

Simplify, giving answers in index form. a) 23 22

b) 35 32

c) 24 32 25

d) 54 53

e) 74 72

c) 24 000

d) 0.0053

e) 0.708

Express in standard form: a) 436

b) 3 million

a) Convert these base 10 numbers to the base indicated in brackets. i) 27 (2)

ii) 568 (5)

b) Calculate in the given base: i) 726 ii) 4102 3648 24305

iii) 3261 (8) iii)

10111 1012 Test your knowledge 191

Test 2

1 40

a) 0.025

b) 0.04

c) 0.4

d) 2.5

How many of the following statements are true? 50 22.4

500 22.4

a) 0

5000 224

b) 1

c) 2

d) 3

b) 748

c) 848

d) 868

1428 568 a) 648

A rope of 20 m is divided into seven equal parts. Correct to two significant figures the length of each part is a) 2.6 m

b) 2.8 m

c) 2.85 m

d) 2.9 m

A soccer team played 40 matches and lost 18. What percentage did it win? a) 512%

b) 8.8%

c) 22%

d) 55%

Charles from the UK changes £200 into BD$ at £0.32 BD$1.00. If he spends BD$480, how many BD$ remain? a) 145

b) 280

c) 310

d) 625

c) 1

d) 1

If 3(3 2y) 4(y 4) 3, then y a) 215

b) 123

If Can$1.00 EC$2.20, how much is EC$110 worth in Canadian dollars? a) 2 420

b) 500

c) 242

d) 50

x and y are sets: n(x) 12, n(y) 9, n(x y) 3, n(x y) a) 24

b) 21

c) 18

d) 15

c) 7.3 102

d) 73 103

Written in standard form 0.073 a) 7 101

b) 7.3 102

Calculate a) i) 313 234

ii) 513 113

b) i) 14.2 3.4

ii) 32.43 6.9

The probability that the school’s netball team wins the next match is 15. What is the probability that it will not win?

Find which is cheaper: a shirt bought in San Fernando for TT $85.50 or one bought in Georgetown for Guy $1 862.00. [US $1.00 is equivalent to TT $5.70 and to Guy $133.]

Jim and Ali set out from Deal and St Antonio and walk towards each other at speeds of 6 km/h and 5 km/h respectively. If the distance between the towns is 33 km, how long will it be before the two men meet? 192 Jamaica Maths Connect 3

In seven examination papers Val scored 38, 72, 19, 45, 57, 61 and 44. a) What was his average score? b) To gain promotion he has to have an overall average score of 50. What is the minimum he must gain in the eighth examination for this to happen?

Ten paperback books were weighed. The number of pages and the weight of each book is given in the table. Number of pages

160

210

218

250

268

272

288

294

328

342

Weight in grams

175

210

220

220

210

230

240

230

270

270

a) Draw a graph, with the number of pages on the horizontal axis. b) Draw the line of best fit and find its equation.

Mary’s older sister was driving on a highway to Montego Bay. She travelled at an average speed of 60 km/h. Complete the following table and draw a graph to show the distance travelled in 3 hours. Time (h) Distance (km)

1 2

1

112

30

60

90

2

212

3

A candle was 24 cm long and it burned at a rate of 2 cm/h. Complete the table below and draw the graph to show how the length of the candle decreased. No. of hours

0

1

2

Length (cm)

24

22

20

3

4

5

6

The numbers in the regions of this Venn diagram show the number of members in that region. Find a) n(X)

b) n(Y)

c) n(X Y)

d) n(X Y)

e) n(X)

f) n(Y)

g) n(X Y )

h) n(X Y )

7

8

9

U X

Y 14

32

17 29

Test your knowledge 193

Test 3 Q

P

The shaded region in the Venn diagram is the set a) (P Q)

b) (P Q)

c) (P Q)

d) (P Q)

Given that a * b denotes a 12b, the value of 8 * 6 is a) 48

b) 14

c) 11

d) 2

Two coins are tossed. If H means ‘heads’ and T means ‘tails’, the possible outcomes are a) {T, H}

b) {HT, TH, H, T}

c) {HT, TH, HH, TT}

d) {HTH, TTH, THH, HHH}

A car travels for 2 hours at 100 km/h. How long will it take to travel the same distance at 80 km/h?

If x 3, (1 x)(1 3x) a) 32

b) 16

c) 16

d) 40

If 2a 7b 13 and 2a 3b 1 simultaneously, then b a) 3.5

b) 3

c) 1.4

d) 1.2

A ship sails 14 nautical miles due west, then 8 nautical miles due north. Its distance in nautical miles from its starting point is then approximately a) 4.7

b) 6

c) 11.5

d) 16.1

The diagram shows a running track with two equal straights and two semicircular ends. The diameter of each semicircle is 70 metres and the perimeter of the track is 400 metres. Using 272, the length of each straight is a) 40 m

b) 90 m

c) 145 m

70 m

d) 180 m

A boxer weighs 200 lb. What is his weight, in kilograms, if 1 lb 0.454 kg? a) 908

b) 45.4

c) 9.08

d) 90.8

In the diagram PQ is a diameter of the circle.

If the two angles at P are equal, the angle x a) 70°

b) 75°

c) 80°

d) 90°

x 70° Q

P


The factors of (x y)2 (x y)2 are a) 0

b) 2x 2y

c) (x y)(x y)

d) 2(x y)(x y)

For printing a school magazine, a printer charges a fixed amount plus a certain amount per copy. If he charges $2 000 to make 500 copies and $3 000 to make 1 000 copies, what is the fixed charge and the charge per copy?

Given that 25 m/s is equivalent to 90 km/h, draw the graph to show this information. Use the graph to change a) 7 m/s to km/h

b) 60 km/h to m/s

d) 32 km/h to m/s

e) 13 m/s to km/h

c) 20 m/s to km/h

Construct triangle PQR with PQ 7 cm, QR 11 cm and PQ R 60°. By constructing and measuring a suitable perpendicular, calculate the area of triangle PQR. (Use ruler and compasses only.)

a) Two fields, one a rectangle and the other a square, are equal in area. If the length of the rectangular field is 169 m, and the width is 81 m, what is the length of the side of the square field? b) A circular Dominican rug, radius 2 m, partly covers a living room floor, 5.7 m by 4.8 m. Calculate the area of floor not covered by the rug.

Plot the graph of the numbers 1 to 10 against their squares. From your graph find a) the square of 3.6

b) the square of 7.8

c) the square root of 70

d) the square root of 94.

A stereo system costing $2 840 can be bought on hire purchase: 25% down payment and the balance at 10% interest to be repaid in equal monthly instalments over 1 year. a) How much more does the stereo cost on the HP agreement? b) How much has to be repaid each month?

US$1.00 BD$1.96 TT$7.48 Guy$12.33 a) Using these figures, calculate, to the nearest $, the amounts received when you change US$20 into i) BD$ ii) TT$ iii) Guy$ b) How many US$ do you get for i) BD$392 ii) Guy$600?

Construct accurately and label a) triangle XYZ with XY 10 cm, YZ 9 cm, XZ 7 cm. Measure and record the sizes of the angles.

b) triangle LMN with LM 8 cm, LMN 50° and LN 8 cm. Measure and record the

length of MN and the sizes of LN M and ML N.


Test 4 The value of (49 25 )2 is a) 4

b) 24

c) 48

d) 576

c) 8

d) 9

If 43n 57n 122n, then n is a) 6

b) 7

In a sample of 100 students at Mount Joy school, 8 are studying music. How many can you expect to be studying music in the entire school of 1 200 students? a) 8

b) 12

x2 2 a) 3

b) 1

c) 96

d) 800

c) 1

d) 3

x 6

If 0, then x

The diameter of the front wheel of a tractor is 40 cm, and the diameter of the back wheel is 60 cm. When the front wheel turns 180 times, the number of times the back wheel turns is a) 13.5

b) 80

c) 120

d) 270

The gradient of the line joining the points (1, 2) and (3, 4) is a) 3

b) 1

c) 12

d)

1 3

c) 9

d) 15

If f(x) 2x 1 and g(x) x2 then fg(2) a) 3

b) 7

The number of goals scored in 10 football matches are 0, 3, 1, 1, 0, 4, 2, 1, 5, 3. The mode, median and mean are a) 1, 1.5, 2

b) 1, 2, 1.5

c) 2, 1, 1.5

d) 2, 1, 2

a) Solve the simultaneous equations 3a b 12, a b 2. b) Draw the line that passes through the points (1, 3) and (3, 7). Write down

i) the gradient

ii) the intercept

iii) the equation of the line.

a) Solve 6y 11 2y 9 where y I. b) Simplify 3x2y 5xy 4xy2 2x2y x2y2 6xy2.

Given that US$1.00 EC$2.67 and US$1.00 Guy$141.40 a) change US$40 into i) EC$ and ii) Guy$ b) change i) EC$500 and ii) Guy$6 000 into US$, giving answers to the nearest cent where necessary.


The masses of 120 students in Form 3 of a secondary school are shown in the table below. Mass (kg)

Frequency

25–34

7

35–44

23

45–54

39

55–64

23

65–74

19

75–84

9

Total

120

a) Draw a frequency polygon to represent the data. b) Calculate the median and mean of the data.

The marks gained by the students at St Mary’s have to be converted into percentages. The highest mark possible is 75. Draw a graph which will convert marks out of 75 into percentages. Use your graph to change a) 10 marks

b) 27 marks

c) 33 marks

e) 50 marks

f) 61 marks

g) 68 marks

d) 45 marks

h) 72 marks into percentages, to the nearest whole number.

If n is an integer, solve the following: a) 3n 4 2n b) 2n 1 n 5 c) 4n 9 12 3n

a) Triangle XYZ is isosceles, XY YZ 7 cm. XYZ 70°. YM is the axis of symmetry of triangle XYZ. Make an accurate drawing from these facts. b) Which triangles are congruent?


Test 5 38.2 0.041 0.19

A reasonable rough estimate of is a) 80

b) 8

c) 0.8

d) 0.08

16 25 10 a) 0.04

b) 0.4

c) 4

d) 40

is 2

The cash price of a table is $800. It can be bought on Hire Purchase (HP) for a deposit of $100 and 24 monthly payments of $35. How much more does the table cost if bought on HP instead of paying cash? a) $20

b) $40

c) $120

In the diagram the area of the triangle as

2p

a fraction of the area of the rectangle is a)

4 9

c)

2 9

b)

1 3

d)

1 6

d) $140

p

2q q

The kite LMNP is symmetrical about the diagonal LN. L is (1, 2), M is (7, 10) and N is (13, 8). P is a) (7, 6)

b) (11, 2)

c) (15, 2)

d) (19, 6)

Which of the relations shown in the arrow diagrams is a function? a) 4

4

b) 2

4

c) 4

d) C

3

3

1

3

3

2

2

2

0

2

2

1

1

1

1

1

1

0

2

0

0

1

F B

A

If h (x) x2 3x 2, then h (1)


b) 4

c) 0

S E

2

a) 6

M

d) 2

H

Number of pets in a household Number of households

0

1

2

3

4

5

7

8

14

20

8

2

The table shows the number of pets in households in a certain town. The mean number of pets per household is a) 2.3

b) 2.4

c) 9.3

d) 23.2

b) i) 4.2 0.37

ii) 55.2 6.9

Calculate a) i) 314 225

ii) 258 170

a) The figures in the diagram indicate the

U A

number of elements in each subset of U. Find i) n (A C) ii) n (B C) iii) n (A B )

B 1

8 5 8

3

5 1

9 C

b) At St Jude’s, students must take at least one of the languages French, Spanish and German. In a group of 100 students, 14 take all three subjects, 18 take Spanish and German only, 16 take French and Spanish only and 10 take French and German only. Of the 100 students, x take French only, x take Spanish only and x 6 take German only. Draw a Venn diagram, find x, and hence find the number taking French.

a) If US $1.00 JA $64.00, find in JA $ i) US $3.00 ii) US $5.00 b) Draw and label axes using scales of 1 cm to represent US $1.00 horizontally and 1 cm to represent JA $50.00 vertically. Plot the points given by your answers to a) and draw a straight line through these points. c) Use your graph to find i) US $6.50 in JA $ ii) JA $300 in US $

iii) JA $550 in US $

a) Solve for x: 7(3x 2) 12 40 b) Make x the subject of the formula a 12 bx c) To find the size of the interior angle x of a regular polygon, divide 360° by the number of sides n and then subtract your answer from 180°. i) Form an equation in n. ii) Find the interior angle when n 24.

a) Construct a triangle LMN with LM 8 cm, LN 6 cm and MN 9 cm.

b) Construct the bisector of MLN. c) Construct the line through N perpendicular to LN and mark the point P where this line meets the bisector MLN.

d) Measure the lengths of NX and MX.


Of the 2000 candidates in one territory last year, the following grades were awarded in a chemistry examination. Grade 1: 280, Grade 2: 605, Grade 3: 436, Grade 4: 162, Grade 5: 128, Unclassified: 389 a) Draw a bar chart to represent this information. Use 2 cm to represent 100 candidates. b) The information is represented on a pie chart. Calculate the angle required for the sector representing Grade 2. Give your answer to the nearest degree.

a) i) State the rule for each of these relations. x

p(x)

x

q(x)

x

r(x)

0

1

2

4

8

2

1

3

1

3

9

3

2

5

0

0

10

4

3

7

1

12

5

2

ii) Which of the relations are functions? b) If g : x → 2x 3 and h : x → x2 2, write down i) g (2) ii) h (2) iii) g (0)


iv)

hg (1)


Answers Page 9

Chapter 1 Page 2 1 a) 32 b) 17, 41 c) 6 d) 49 e) 294 2 36 3 402 300, four-hundred-and-two-thousand, three hundred 4 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 5 122 6 1, 19 19 is a prime number 7 $88.40 8 a) 27 b) 54 c) 69 d) 6 Page 4 1 a) base 11; index 5 b) base 2; index 4 c) base 3; index 1 d) base 7; index 0 e) base 5; index 3 f) base 4; index 8 g) base 1; index 9 h) base 34; index 6 i) base 4; index 2 j) base 13; index 5 2 a) 5 5 5 b) 4 4 4 4 c) 2 2 2 2 2 2 d) 3 3 3 3 3 3 3 e) 8 8 f) 1 1 1 1 1 g) 2 2 2 2 2 2 2 2 h) 10 10 10 i) 11 11 11 11 j) 6 6 b) 34 c) 75 d) 23 3 a) 62 4 5 1 e) 5 f) 4 g) 8 h) 27 i) 96 j) 43 4 a) 16 b) 81 c) 64 d) 144 e) 32 f) 125 g) 1 h) 128 i) 75 j) 72 Page 6 1 a) 2 b) e) 9 f) i) 8 j) 2 a) 20 b) e) 60 f) i) 210 j) 3 a) 30 b) e) 70 f) b) 25 4 a) 98 5 7 cm 6 100 m 7 a) Square b) e) Not f) 8 a) 272 b) e) 1560 f) Page 7 1 a) 2 2 a) 6

11 12 100 18 80 600 20 600 c)

1 3

Square Not 1640 132

b) 125 b) 7

c) 5 c) 2


c) 6 g) 3

d) 7 h) 4

c) 60 g) 22

d) 21 h) 180

c) 50

d) 14 d)

5 4

e)

1 a) 18.39 b) d) 194.97 e) 2 a) 22.3 b) d) 39.66 e) 3 a) 10.12 b) d) 3.28 e) 4 13.225 m 5 27.5 cm 6 a) 3.49 b) 7.2 7 a) 25.34 b) 15.34 8 Answers will vary. 9 Answers will vary.

14.42 37.91 4.6 0.78 1.87 4.83

c) f) c) f) c)

c) 1.78 c) 25.34

23.845 43.905 23.92 2.2 15.41

d) 13.2 d) 37.64

e) 14.84

Page 11 1 b) 3.8 4.6 38 46 100 c) 7.21 5.2 721 52 1000 d) 32.4 4.3 324 43 100 e) 9.82 7.6 982 76 1000 2 a) 28.512 b) 17.48 d) 139.32 e) 74.632 3 25.97 m2 4 $417.05 5 $4 212 6 $19.27 7 13.2 5.4 or 1.32 54 8 a) 91 b) 0.91

c) 37.492

c) 0.091

Page 13 1 b) 83.4 0.3 834 3 c) 36.12 2.1 361.2 21 d) 4 0.32 400 32 e) 0.02106 0.065 21.06 65 2 a) 8.2 b) 278 c) 17.2 d) 12.5 e) 0.324 3 No 4 23 5 a) 17 b) $4.05 6 a) 12.5 7 c) 8 All correct 9 a) 125 b) 1.25 c) 125 d) 1.25 Page 15

10 11

c) Not

d) Square

c) 156

d) 240

1 2 3 4 5 6

. .. . . . a) 0.5 b) 0.6 3 c) 0.416 d) 1.16 e) 0.46 b) 19 c) 23 d) 16 e) 79 a) 13 a), b), d) and e) No .. Ali 0.18, James 0.18, James had the biggest answer. a) False b) True c) False a) True b) False

Page 17 d) 5 d) 5

e) 64 e) 3

1 a) 60 millions c) 12.89 millions e) 0.5 millions

b) 9.9 millions d) 1.342 millions f) 0.234 millions

2 a) 106 b) 102 c) 104 d) 105 e) 101 1 2 3 a) 4.9 10 b) 5 10 c) 1.8 103 1 5 d) 1.42 10 e) 1.65 10 f) 7 101 g) 9.5 103 h) 4 102 i) 8.32 104 j) 7.6 105 k) 6 109 l) 3 106 4 a) 1200 b) 300 c) 800 000 d) 2 590 000 e) 91 000 f) 0.5 g) 0.000473 h) 0.036 i) 0.0018 j) 0.0000207 5 Afghanistan, 2.9 107 The Bahamas, 2.97 105 Barbados, 2.77 105 Canada, 3.2 107 Cuba, 1.1 107 Dominica, 6.96 104 Fiji, 8.685 105 Iceland, 2.81 105 Japan, 1.27 108 New Zealand, 3.95 106 6 a) Earth, 5 980 000 000 000 000 000 000 000 kg Sun, 1 970 000 000 000 000 000 000 000 000 000 kg b) Students’ own answers c) approximately 1023 7 Various answers Page 20 1 a) binary b) seven c) 8 2 a) 0, 1, 2 b) 0, 1, 2, 3, 4, 5 c) 0, 1, 2, 3, 4, 5, 6, 7 d) 0, 1, 2, 3, 4, 5, 6 e) 0, 1, 2, 3, 4, 5, 6, 7, 8 3 a) 13, 23, 103, 113, 123, 203, 213, 223, 1003, 1013 b) 15, 25, 35, 45, 105, 115, 125, 135, 145, 205 c) 16, 26, 36, 46, 56, 106, 116, 126, 136, 146 d) 17, 27, 37, 47, 57, 67, 107, 117, 127, 137 e) 12, 102, 112, 1002, 1012, 1102, 1112, 10002, 10012, 10102 4 a) 10 000 b) 16 c) 9 d) 8 e) 16 f) 18 g) 24 Page 21 1 a) 3334 e) 30078 i) 5509

b) 11112 f) 10002 j) 1111002

c) 1145 g) 346

d) 110023 h) 637

b) 23 f) 16 j) 49

c) 27 g) 215 k) 3383

d) 130 h) 62 l) 1845

Page 21 1 a) 26 e) 167 i) 5 Page 22 1 a) d) 2 a) d) 3 a) d) 4 a) d) 5 a) d)

22115 222014 306 673 2516 215 103 11 11102 231124

b) e) b) e) b) e) b) e) b) e)

1022213 533238 322 22227 12467 100014 475 257 101023 200045

c) f) c) f) c) f) c) f) c) f)

101112 177769 23 12300 1112 400005 7 625 20108 366667

Looking back (page 23) 1 24: 1, 2, 3, 4, 6, 8, 12, 24 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 325: 1, 5, 13, 25, 65, 325 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 2 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 3 Yes, 21 21 421 4 a) 144 b) 64 c) 100 000 d) 512 e) 20736 f) 6859 e) 89 5 a) 36 b) 28 c) 30 d) 161 6 a) 4 (4 8) 48 b) 18 (3 2) 3 c) 7 (6 3) ÷ (3)2 = 9 . . . 7 a) i) 0.5 ii) 0.518 iii) 0.4 iv) 3.14 (2 d.p.) b) i) and ii) 8 a) 19 b) 12 c) 106 d) 2 e) 5 9 Antigua and Barbuda, 6.79 104 Australia, 1.9732 107 Belize, 2.66 105 Brazil, 1.82033 108 Haiti, 7.5 106 10 France, 547 000 km2 Grenada, 342 km2 India, 3 288 000 km2 Ireland, 70 000 km2 Mauritius, 20 400 km2 11 a) 2115 b) 121013 c) 111012 12 a) 78 b) 82 c) 75 b) 10336 c) 46478 13 a) 1001002

Chapter 2 Page 25 1 a) True b) False d) True e) False 2 Various answers 3 256 sets 4 a) 32 b) 64 d) 1024 e) 128 5 Various answers 6 a) {Prime numbers} b) {Multiples of 11} c) {Caribbean capital cities} d) {Units of currency} e) {Regular shapes} 7 a) {5, 6, 7, 8, 9, 10, 11} b) {8} c) 7 d) 8 e) {1, 2, 3, 4, 9, 10, 11, 12} f) {1, 2, 3, 4, 5, 6, 7, 12} 8 a), d) and e)

c) True f) True

c) 16

Page 27 1 a) A B {table, bed, bookcase, chair, desk, stool} b) A B {St. Lucia, Barbados, Trinidad, Grenada, Dominica}

Answers 203

2

3

4

5

6

7

c) A B {E, Q, U, A, L, X, T R} d) A B {David, Betty, Jim, Brenda, Cathy, Sharon, Thomas} e) A B {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} a) {D, I, T} b) {D, O, I} c) {P, S, E} d) {P, O, S, E, T} e) {D, O, I, T} f) {D, I} g) {D, I,} h) {D, I, T, O} i) {O} j) {D, I, T, P, E, S} a) {F, C, S} b) {I, O, N, S} c) {R, A, T} d) {R, A, T, I, O, N, C, F} e) {F, C, I, O, N, S} f) {S} g) {S} h) {F, C, S, I, O, N} i) {I, O, N} j) {F, R, A, C, T,S} a) {T, W, Th, Sa} b) {M, T, W, Th, F} c) {Su} d) {M, F, Sa, Su} e) {M, T, W, Th, F, Sa} f) {T, W, Th} g) {T, W, Th} h) {M, T, W, Th, F, Sa} i) {M, F} j) {T, W, Th, Sa, Su} a) {Feb, Apr, Jun, Sept, Nov} b) {Feb, Mar, Apr, May, Aug, Sept, Oct, Nov, Dec} c) {Jan, Jul} d) {Jan, Mar, May, Jun, Jul, Aug, Oct, Dec} e) {Feb, Mar, Apr, May, Jun, Aug, Sept, Oct, Nov, Dec} f) {Feb, Apr, Sept, Nov} g) {Feb, Apr, Sept, Nov} h) {Feb, Mar, Apr, May, Jun, Aug, Sept, Oct, Nov, Dec} i) {Mar, May, Aug, Sept, Nov} j) {Jan, Feb, Apr, Jun, Jul, Sept, Nov} a) {1, 2, 3, 5, 7} b) {3, 6, 7, 9} c) {4, 8, 10} d) {1, 2, 4, 5, 6, 8, 9, 10} e) {1, 2, 3, 5, 6, 7, 9} f) {3, 7} g) {3, 7} h) {1, 2, 3, 5, 6, 7, 9} i) {6, 9} j) {1, 2, 3, 4, 5, 7, 8, 10} a) A B (A B) b) A B (A B)

d) i) ii) iii) iv) e) i) ii) iii) iv) f) i) ii) iii) iv) g) i) ii) iii) iv) h) i) ii) iii) iv) i) i) ii) iii) iv) j) i) ii) iii) iv)

{1, 2, 5, 10, 15, 20, 25} {1, 2, 4, 5, 9, 10, 16, 25} {25} {1, 2, 4, 5, 9, 10, 15, 16, 20, 25} {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 5, 7} {3, 5} {1, 2, 3, 4, 5, 6, 7} {1, 5, 7} {1, 2, 3, 6, 7, 14} {1, 7} {1, 2, 3, 5, 6, 7, 14} {5, 10, 20, 25, 30, 40, 50} {5, 10, 20, 25, 30, 40, 50} {10, 50} {10, 20, 25, 30, 40, 50} {1, 2, 3, 4, 6, 9, 10, 12, 15, 18} {1, 2, 3, 4, 6, 8, 9, 12, 18} {1, 3, 6} {1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 18} {2, 3, 5, 7, 11, 13} {1, 2, 3, 5} {3} {1, 2, 3, 5, 7, 11, 13} {6, 8, 10, 12, 14} {6, 8, 12} {12} {6, 8, 10, 12, 14}

Page 31 1 a)

U A

B

3 5

9 1

7

2

Page 29 1 a) A B C { }; A B C {chair, table, stool, bed, dresser} b) A B C {O, E, R}; A B C {B, O, T, H, E, R, L, D, Y, U, N, G} c) A B C {E, L, I, A}; A B C {D, E, L, I, A, S, H, O, P} d) A B C {6}; A B C {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12} e) A B C { }; A B C {Donna, Sarah, Sharon, David, Jonathan, Sandra, Ricky} 2 a) i) {B, I, O, L, G, Y, H, S, T, R} ii) {H, I, S, T, O, R, Y, C, E, M} iii) {I, Y} iv) {B, I, O, L, G, Y, H, S, T, R, C, E, M} b) i) {G, R, A, P, E} ii) {P, E, A, R, L} iii) {A, P, E} iv) {G, R, A, P, E, L} c) i) {0, 2, 3, 4, 6, 8, 9, 10} ii) {0, 1, 2, 3, 4, 6, 9} iii) {6} iv) {0, 1, 2, 3, 4, 6, 8, 9, 10}


C

b)

4 6 8 10

U A

B

M E A

N

C

I H C

c)

U A W

R

B U

E

O G H C

N Y

d)

Looking back (page 34)

U B

A Sat Tues Mon Wed Fri C

e)

Thur Sun

U A

10

14 16 20

B

15 18 12

C 11 13 17 19

f)

1 a) A B C {T, A, E}; A B C {T, A, B, L, E} b) A B C {Ken}; A B C {Harry, Dean, Ken, Fran, Joe, Bob} c) A B C { }; A B C {2, 3, 4, 6, 7, 8, 9, 12} d) A B C {pear, orange}; A B C {apple, pear, orange, grapes, strawberry} 2 a) A B C {3}; A B C {1, 2, 3, 5, 6, 7, 9, 10} b) A B C {T, E, L}; A B C {T, E, L, V, I, S, O, N, P, H, A, Y} c) A B C { }; A B C {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} 3 a) U A

U A 10

7

B 5

15

4 8 1 3 9 2 6

4 8 1

9 5

7 10 11 13 14

C

H 2 1 0

C

b)

U A

g)

B

2

6 12 3

U

6 9

B

A 15

1

1

4 16 2

B 10 12 14

8

5 3 7 11 13 17 19 C

2 a) i) ii) b) i) ii) c) i) ii) d) i) ii) e) i) ii) f) i) ii) g) i) ii)

C 9

A B C {1, 2, 3, 5, 7, 9} (A B C) {4, 6, 8, 10} A B C {M, A, C, H, I, N, E} (A B C)’ { } A B C {H, U, G, R, O, W, E} (A B C) {N, Y} A B C {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} (A B C) {Sunday} A B C {10, 12, 14, 15, 16, 18, 20} (A B C) {11, 13, 17, 19} A B C {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (A B C) {2, 1, 0} A B C {1, 3, 5, 7, 11, 13, 15, 17, 19} (A B C) {9}

Page 33 1 11 4 23 7 a) 18

2 26 5 20 b) 63

c) 13

3 41 6 7 d) 32

4 32 5 a) 55

b) 87

3 5 7 11 13 15

c) 18

d) 14

Chapter 3 Page 36 1 a) 175.8 cm b) 31.4 cm c) 62.8 cm d) 241.8 cm 2 a) 43 cm b) 21 cm c) 85.98 m 3 a) 49.98 cm b) 32.56 cm c) 234.85 m 4 a) Johnathan 188.4 m; Jeremy 157 m b) Johnathan, by 628 m 5 64.3 cm Page 39 1 2 3 4 5 6

a) 32 cm2 a) 96 cm2 32.4 cm2 35.4 cm2 4.6 cm 0.16 m2

b) 90 cm2 b) 61 cm2

c) 45 cm2 c) 84 cm2

d) 118.8 cm2

Answers 205

Page 41 b) 1 a) 5 m2 d) 0.825 m2 e) 2 a) 5 cm2 b) d) 82.54 cm2 e) 3 a) 500 mm2 b) d) 670 mm2 e) 4 a) 70 000 cm2 b) d) 5 000 cm2 e) 5 a) 25 mm, 18 mm ii) b) i) 4.5 cm2 6 a) 52 cm2 b) 7 a) 33 cm2 b) d) 323.96 cm2 8 128

25 m2 0.0575 m2 25 cm2 3.75 cm2 6 500 mm2 22 mm2 750 000 cm2 3 475 cm2 450 mm 48 cm2 75 cm2

c) 3.57 m2 c) 725 cm2 c) 275 mm2 f) 30 mm2 c) 35 000 cm2

15 a) d) 16 a) 17 a) 18 a) 19 a) 20 a)

1 200 0.0022 304 181 30416 1112 122223

21 a)

U

b) b) b) b)

b) 3 456 000 e) 0.0009034 b) 100002 22 c) 11 62647 4205 130226

A

B

10

2

2

c) 11.25 cm2 c) 59.99 cm2

75 3 1 11 1323 21 15 17 19

9

6

15 25 C

Page 44 1 a) 880 cm2 2 a) 1 536 cm2 3 a) 102 cm2

b) 96 cm2 b) 224 cm2 b) 384 cm2

c) 136 cm2 c) 504 cm2 c) 967.74 cm2

Looking back (page 45) 21.99 cm 2 72.25 cm2 5 206 times 5 11 cm P 714 m, A 27 850 m2 a) i) 35.03 cm ii) 88.5 cm b) i) 69.08 cm2 ii) 196.25 cm2 9 4 558 tiles 10 8 cm 1 4 7 8

3 30 cm2 6 314 cm2

Revise and Consolidate 1 Page 49 1 a) 12 b) 35 c) 38 d) 8 e) 53 f) 32 2 a) 32 b) 64 c) 0.0625 d) 248 832 e) 0.111 f) 1 3 a) T b) F c) T d) T e) F f) T g) F h) T i) F j) F k) F l) F d) 45 4 a) 3 b) 5 c) 89 5 2, 3, 11, 17, 19, 23, 29, 37, 47, 59, 61, 73, 101 6 2, 4, 6, 8, 10; 3, 6, 9, 12, 15; 5, 10, 15, 20, 25 7 a) 1, 2, 4 b) 1, 2, 3, 4, 6, 12 c) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 8 36 9 $528.15 10 27.5 11 a) 21.373 b) 169.17 c) 5.065 d) 135.22 e) 134.152 f) 42.746 g) 425.96 12 a) 0.012 b) 0.016 13 a) 0.125, not recurring . b) 0.2. , recurring . c) 0.4. 28571, not recurring d) 1.3,. recurring . e) 2.318, recurring b) 6.758 107 c) 8.1 109 14 a) 2 106 3 7 d) 4 10 e) 9.8 10 f) 2.34 107


c) 80 f) 0.00007 c) 1568 d) 2020

4 8 12 14 16 18 20 22 24

b) {1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 19, 21, 23, 25} c) {3} 22 27.7 cm 23 9 cm 24 30.9 m2

Chapter 4 Page 51 1 Students’ own answers 2 Students’ own answers 3 a) Jason: 15 blinks per minute Benita: 14 blinks per minute Delia: 12 blinks per minute b) Students’ own answers c) Students’ own answers 4 a) 80 beats per second b) 288 000 times 5 Yes, Jessica can type 60 words per minute. 6 Students’ own answers 7 a) i) 8 m/s ii) 6 m/s iii) 7 m/s iv) 17 m/s v) 2 m/s vi) 4 m/s b) Victor, Sandra and Indira were runners; Ravi was a cyclist; Sarah was a swimmer and Francis was a rower. Page 55 1 a) 571.4 km/h c) 93.6 km/h e) 206 km/h 2

b) 116.8 km/h d) 24 km/h f) 43 km/h

Speed

Distance

Time

100 m/s

200 m

2 seconds

4 km/h

12 km

3 hours

2 m/s

200 m

100 seconds

15 km/h

400 km

26 hours 40 mins

5 cm/day

49 cm

9 days 19 hours 12 mins

400 km/h

200 km

1 2

5 m/s

7 500 m

25 mins

12 km/min

1 080 km

12 hours

3 10 km

hour

1

Distance (kilometres)

4 48 km 5 a) 55 miles b) 90 miles c) 0 miles d) 30 minutes e) stop for 12 minutes f) Between 1 hour and 12 minutes and 1 hour and 30 minutes. 6 3 2 1

Page 59 1 a) b) 2 a) b) 3 a) b) 4 a) b)

$18.20 4 $16.40 15 4 40 4 hours 45 km/h

i) i) i) i) i) i) i) i)

$46.80 24 $114.80 40 10 1 33 5 hours 50 km/h

Looking back (page 60) 1 2 3 4

0 3.30 3.40 3.50 4.00 4.10 4.20 pm pm pm pm pm pm Time

804.25 km/h a) 292 km a) 7 hours a) 750 m b)

Page 58

800

Direct proportion None Inverse proportion None Direct proportion i) $5 ii) $10

Distance (metres)

1 a) b) c) d) 2 a) b) c)

ii) ii) ii) ii) ii) ii) ii) ii)

y 7

400 200 0 0

5 3 2 1 0 0 10 20 30 40 50 60 70 80 90 100 110 x

d) $5 e) You would have to extend the graph. f) i) Students’ own answers ii) $7 3 a) Inverse proportion b) 10 km/h c) 5 hours d) 1 X

1

0.5

0.1

Y

40

20

4

10 20 Time (minutes)

See graph i) $147.20 ii) i) 6.5 m ii) Direct proportion i) 96 kg ii) i) 9 ii)

30

$80 12.8 m 56 kg 14

y 120 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x

e) y

e) i) 32 kg

40

ii) 15

Chapter 5

30

Page 62

20 10 0 0

c) 5 a) b) 6 a) b) c) d)

Aisha Deborah

600

6 4

b) 182.5 km b) 15 minutes

0.5

1

1 x

1 a) 3x4 d) 10n5

b) 2m7 e) 8q3

2 a) x

b) 4a4

d) 2n

2 e) 3 y

c) 20z6 f) 8a6 6x4 c) w3 f) 7

Answers 207

3 a) t12

b) k3

c) p2g2 3 4

d) y3z2 4 a) d) 5 a) 6 a) 7 1 8 a)

e) 8ax+y

36 729 50 1

b) 4 096 e) 144 b) 420 c) a0 b) 1 c) 1

9

b) 8

qr f) 4 c) 4 608 f) 118 098 d) y0 d) 1

6 8, 14, 20, 26… The number of seats available is equal to 6 times the number of tables plus 2. 7 a) various answers e.g. 0.5, 1.5, 2.5… b) various answers e.g. 0.25, 1.25, 2.25… c) various answers, e.g. 0.2, 1.2, 2.2… d) various answers e.g. 0.1, 1.1, 2.1… Page 68

c) h

1

Sequence

Difference between consecutive terms

Ascending or descending?

a) T(n) n 4

1

Ascending

b) T(n) 7 n

1

Descending

c) T(n) 9 3n

3

Ascending

d) T(n) 4n 12

4

Descending

e) T(n) 84 2n

2

Descending

f) T(n) 8 5n

5

Descending

Page 64 1 a) d) 2 a) d) 3 a) d)

4 20 2 6 6.325 1.3 1 5

4 a) 243 1 3

d) h 1 5 a) 5 a 1 d) 1 492 6 a) d5 d) q3

b) e) b) e) b) e)

40 13 5 8 12.649 4.111

c) 4 c) 2.530 1 2

1 10

b) b

e) 100

c) 2

c) g 1 6

b) p2q3

1 c) 3 c

c4 e) 3 b 1 b) 22 f

1 c) 8 y

e) r5s

f) 128u7

Looking back (page 65) 1 a) d) g) j) m) 2 a)

x11 2a6 3x5y4 24x4yz5 27a6 x2

d) a7 3 a) d) 4 a) d)

m 12 m 23 16 3x

b) e) h) k) n) b) e) b) e) b) e)

y5 30d6 36a4b5 21m2n2 64x12 2a 4 2 x m0 m3 x21 83b

c) f) i) l) o) c)

4x5 45s7 x2y2z2 x8 2m9 5xy2

c) m 3 5

3

c) m2n2

Chapter 6 Page 66 1 a) T(1) 4. The term-to-term rule is ‘add 7’. The last term is 20. b) T(1) 9. The term-to-term rule is ‘subtract 2’. The last term is 5. c) T(1) 8. The term-to-term rule is ‘add 20’. d) T(1) 0. The term-to-term rule is ‘subtract 4’. 2 a) T(10) 28 b) T(10) 80 c) T(10) 115 d) T(10) 6 3 a) T(20) 88 c) T(20) 65 4 a) i) 185 ii) 12 iii) 81 iv) 50 b) i) 455 ii) 60 iii) 71 iv) 140 5 a) T(1) 1.5. The term-to-term rule is ‘add 0.5’. b) i) 6 ii) 8.5 iii) 13.5 c) The decorator multiplies the number of bags of paste by 0.5, and then adds 1.


2 a) T(1) 3, T(2) 2, T(3) 1, T(4) 0, T(5) 1 b) T(1) 6, T(2) 5, T(3) 4, T(4) 3, T(5) 2 c) T(1) 12, T(2) 15, T(3) 18, T(4) 21, T(5) 24 d) T(1) 16, T(2) 20, T(3) 24, T(4) 28, T(5) 32 e) T(1) 82, T(2) 80, T(3) 78, T(4) 76, T(5) 74 f) T(1) 13, T(2) 18, T(3) 23, T(4) 28, T(5) 33 3 a) T(10) 10, T(12) 12, T(99) 99 b) T(10) 5, T(12) 7, T(99) 94 c) T(10) 25, T(12) 31, T(99) 292 d) T(10) 9, T(12) 11, T(99) 98 e) T(10) 90, T(12) 88, T(99) 1 f) T(10) 4, T(12) 5, T(99) 48.5 4 a) T(n) 3n 6 b) T(n) 16 4n c) T(n) n 8 d) T(n) 6n 4 5 a) 3n 1; each time 3 dots are added b) 2n 1; each time two squares are added c) 11 n; each time a star is taken away d) 8 2n; each time two diamonds are taken away 6 a) The increase in cost per km is $0.10 with the cost for travelling 1 km $2.10, so the general term T(n) 2 0.1n b) 25 km 7 a) N

1

2

3

4

5

T(n)

4

7

10

13

16

b) T(n)

4

4 (1 3) 4 (2 3) 4 (3 3) 4 (4 3)

n 1 times 3(n 1) T(n) 4 3(n 1) T(n) 3n 1 Finding the general term in this way gives the same answer as looking at differences. 8 a) 2n 5 b) 3n 2 c) 4n 3 c) d) e) f) g)

Page 70 1 a) b) c) d) e)

T(1) 4, T(2) 7, T(3) 12, T(4) 19, T(5) 28 T(1) 11, T(2) 14, T(3) 19, T(4) 26, T(5) 35 T(1) 4, T(2) 1, T(3) 4, T(4) 11, T(5) 20 T(1) 6, T(2) 3, T(3) 2, T(4) 9, T(5) 18 T(1) 1.5, T(2) 4.5, T(3) 9.5, T(4) 16.5, T(5) 25.5

2 Term number sequence

1

2

3

4

5

a) T(n) n

1

4

9

16

25

b) T(n) n 1

2

5

10

17

26

c) T(n) n 2 2

3

6

11

18

27

d) T(n) n 2 1

0

3

8

15

24

e) T(n) n 2

1

2

7

14

23

2 2

2

3 a) T(1) 2, T(2) 8, T(3) 18 b) T(1) 7, T(2) 13, T(3) 23 c) T(1) 3, T(2) 15, T(3) 35 d) T(1) 5, T(2) 35, T(3) 85 e) T(1) 3, T(2) 21, T(3) 51 4 T(2) 25, T(3) 130, T(4) 147, T(5) 256 5 a)

ab 4 a) 2

Page 77 1 a) m t 5

b) m w 2

d) m 2y

e) m x 5

g) m z 2 z a) t 2 a) z 4t a) x y m v d) q r 3 a) r (5y)

h) m 7p w b) t 5 b) z 8m b) t bz

3 4

1

2

3

4

Number of white squares

1

4

9

16

Number of green squares

3

3

3

3

Total number of squares

4

7

12

19

5

F2 d) K

1 a) 9 2 a) 156 1 36

a) 5 b) 3 c) 2 d) 4 a) 3 b) 1 c) 12 d) 5 a) 1 b) 11 c) 7 d) 8 a) 0.5 b) 0.316 c) 0.3 d) 0.5 a) i) x 5 ii) 2x 10 iii) 3x 15 b) 7x 30 c) Patricia 4; Sheila 9; Anne 18; Winifred 27 6 a) i) p 1 ii) 2p 2 iii) 2p 5 b) Katie 5, Alex 11, Matthew 14 7 1 8 Students’ own answers

3 a) d) 4 a) d) 5 a) b) c)

Page 75

Page 80

1 2 3 4 5

3 a) 18 d) abc

6ef b) 5 y5 b) 5 zx b) 150 e) a2b2

c) 2j5g wvx c) 26 yr

y c) t 6 c) z 6p c) p z r y f) x z

e) g b x (t s) b) k 4

c) h 12g

3

e) e

c) 39

b) 1

f) 2q

be d) cd b4 d) 25 c) 6p2 f) e2f 2

d)

1 a) b) c) 2 a) d) 3 a) b)

e) 5

d) 2

b)

1 8

c)

e)

6 49

2 f) 2

Page 73

7b 1 a) 2 2a c 2 a) q

r c) m 3 y f) m 12

Page 78

d) T(n) n2 3 c) n2 2 6 T(n) 2n

3

2t d) 3

c) x

2b 3a 22a c 5 a) b) ab 2c 3dy ex 6 y2 c) d) xy xy 5e 3d 3xz 4x 6 a) b) de 4z wp x 7d bd c) d) p2 ab 7a 65a a a 7 a) a b) 8 24 2 3 4x 2y 2x y 8 a) b) y x y x 4x2 2y2 c) Perimeter , Area 2 xy

2

b) Pattern number

pq b) 9

1 3

17 b) 39 c) 7 1 e) 3 f) 8.6 11 b) 4 c) 0.29 4 e) 4 f) 1 4 * 1 10, 1 * 4 10 Yes The two numbers are added together, and adding can be done in any order without changing the result.

3 1.2 3 1 3 0.2 3.6 1.6 50 1 50 0.6 50 50 6 5 80 2.3 300 2 300 0.3 300 690 160 b) 92 c) 73.5 690 e) 15.6 f) 1380 228 4 72 so 288 16 18 345 5 69 so 345 15 23

Answers 209

4 5

6

7

8

c) d) e) a) d) a) c) e) a) c) e) a) d) g) j) a) b) c) d) e)

9 a) b)

3150 10 315 so 3150 30 105 126 6 21 so 126 18 7 735 7 105 so 735 35 21 41 b) 37 c) 0.7 28 e) 7.1 f) 14 m(x z) b) 8m(2n 3p) 2abc(17d) d) 7ef(3g 2h) 3x(yz 6y 2z) f) 3a(2x 1) b) x2(x y x2z) a2(a2 a 1) 2 2 d) 2x2(a 3b 4cx) m n (m n) 2 f) k2m2(k 2) p q(21p 14q 1) 18.2 b) 4830 c) 43.2 48.3 e) 318.24 f) 95.4 0.66 h) 4.16 i) 92.82 2273.6 260 30 200 30 60 30 6.66 2 8.66 930 4 800 4 130 4 200 32.5 232.5 810 11 770 11 40 11 70 3.64 73.64 17.6 8 16 8 1.6 8 2 0.2 2.2 430 13 390 13 40 13 30 3.08 33.08 i) 12 50 12 40 ii) 12 (40 50) $1080

Looking back (page 81) Next two terms 26, 31. T(n) 5n 4, T20 96 Next two terms 28, 32. T(n) 4n +4, T20 84 Next two terms 23, 27. T(n) 4n 1, T20 79 Next two terms 37, 43. T(n) 1 6n, T(20) 121 e) Next two terms =5.5, 8. T(n) 7 2.5n, T(20) 43 2 a) 3 b) Descending c) T(1) 1 3 a) Quadratic b) 4, 7, 12, 19, 28 1 a) b) c) d)

25x 4 a) 6 nx d) my 5 a)

13 6

d) 2196

b) t

c) 35d2

y4 e) 25

n5 f) 5 pm

b) 17.5

c) 3.6

e) 395

(tr) (v x) 6 a) a b) a mn t (pk w) c) a x 7 length 23 m, width 16 m 8 a) 9 b) 19 c) 3 d) 1 e) 1 f) 3.8 9 a) 3 b) 1 c) 3 d) p e) 3xy 10 a) 5(2x 7) b) 7(2d 1) c) y(5x 2) d) 2y(12xz 5) e) 2a(2 3b 7c)

Revise and Consolidate 2 Page 82 1 1 500 km/h 2 8 km


3 a) i) US$37.50 b) i) 6 hours 4 a) 5 people 5 a) x9 d) 7a6 g) 24x3y4 j) 32x4yz5 m) 64a6 6 a) x d) a3 7 a) 36 d) 8 a) b) c) d) e) 9 a) 10 a)

b) b) e) h) k) n) b) e) b)

3

ii) US$150 ii) 13 hours 45 days c5 36d6 30a4b5 4m2n2 8x12 5a2 1 x2 3 y2 3

c) f) i) l) o) c)

12x5 30s7 x2y2z2 y8 3m9 4y2

c) x3y2

6x e) 12 b 11, 13; 2n 1; 39 0.28, 0.32; 0.04n 0.04; 0.84 25, 29; 4n 1; 81 37, 43; 6n 1; 121 55, 80; 70 25n; 430 2 b) descending c) 3 quadratic sequence b) 5, 8, 13, 20, 29

11 a) x 143 d) x

34

1

b) x 545

c) x 2

e) x 276

(b k) (x y) 12 a) a b) a mn t (mj l) c) a x 13 width 19 m, length 36 m 14 a) 0 b) 14 c) 6 d) 4 e) 5 f) 6.4 15 a) 4 b) 5 c) 3c d) pq e) 30x 16 a) 12(x 3) b) 7(3p 2) c) 2n(5m 1) d) 2c(6ab 5) e) m(5n 3p 2np) f) 2a(3 2b 6c)

Chapter 7 Page 85 1 a) $490.25 b) $527.35 c) $1722.50 2 a) EC$4 050 b) KYD$1 640 c) B$500 d) T&T$25 413.80 e) BD$696.50 f) EC$405 3 a) $203.70 b) $370.37 c) $46.30 4 a) $2 150 b) $2 236 c) $537.50 5 US$32.50 6 a) EC$270; $1 590; $2 142; $3 000; $1 026 b) EC$264.60; $1 558.2; $2 099.16; $2 940; $1 005.48 Page 87 1 2 3 4 5 6 7 8

$105 $2 934 a) $5 800 a) $2 305 a) $6 000 a) $1 200 a) $6 080 a) $800

b) b) b) b) b) b)

$2 800 $177.08 $1 500 $10 200 $2 280 $4 000

c) c) c) c)

3313% $2 200 $380 100%

Looking back (page 88) 1 a) $3 780 2 a) US$944.85 3 a) US$65

b) $1 380 b) US$45.85 b) US$760.50

c) US$110.50

4 a) i) iv) b) i) iii) c) i)

3 a 61° (angles in a right angle) b 61° (sum of angles in a triangle) c 29° (alternate angles) 4 a 76° (supplementary angles) b 104° (alternate angles) c 128° (sum of angles in a quadrilateral) 5 a b c d 11°, e 158° (vertically opposite angles) 6 SPR 35° (alternate angles) 7 62.5° 8 a) x 70° b) x 255°

$9 762 ii) $25 452 iii) $11 625 $1 100 ii) €32.26 Y 4 545.45 US$38.41 iv) £22.10 ii) €48.39 Y 6 818.18

Chapter 8 Page 90 1 a) 5.25 m3 c) 95.7 m3 e) 0.0075 m3 2 a) 5 cm3 c) 72.5 cm3 e) 0.075 cm3 3 a) 3 000 mm3 c) 2 500 mm3 e) 270 mm3 4 a) 7 000 000 cm3 c) 6 500 000 cm3 e) 327 580 cm3 5 a) 1.5 cm, 2.5 cm, 3.2 cm c) 12 cm3 6 75 cm3 7 288 000 cm3 8 8.788 m3

b) d) f) b) d) f) b) d) f) b) d) f) b)

0.754 m3 0.0825 m3 0.00145 m3 8.534 cm3 0.825 cm3 0.012 cm3 25 000 mm3 1425 000 mm3 600 mm3 3 547 000 cm3 500 000 cm3 99 000 000 cm3 12 000 mm3

Page 98 1 a)

b)

c)

d)

2 a)

Page 91 1 a) 600 cm3 d) 462 cm3 2 a) 20763.25 cm3 3 942 cm3 4 16 cm

b) 512 m3 c) 195 cm3 e) 28 260 cm3 f) 1 344 m3 b) 25 434 cm3

Plan view

Front elevation

Side elevation

Plan

Front elevation

Side elevation

b)

Looking back (page 92) 1 2 3 4

a) 8.75 m3 20.0 cm a) 462 000 mm3 22.6 m3

b) 20.75 m2

3 Answers will vary 4 a) Pyramid with triangular base b) Prism with triangular cross section c) Cylinder 5 Students’ own answers

3

b) 462 cm

Chapter 9 Page 94 1 Allow pairs to check each others paragraphs. Check for accuracy of properties, then check connections e.g. square is a rectangle with equal sides, rhombus is a parallelogram with equal sides. 2 Properties of diagonals

Quadrilaterals

Sq

Re

Rh

K

T

P

Any Q

Both are lines of symmetry

Y

N

Y

N

N

N

N

Only 1 is a line of symmetry

N

N

N

Y

N

N

N

They are equal

Y

Y

Y

N

N

N

N

They bisect each other

Y

Y

Y

N

N

Y

N

Only one bisects the other

N

N

N

Y

N

N

N

They are perpendicular

Y

N

Y

Y

N

N

N

They bisect the opposite angles

Y

N

Y

N

N

N

N

Only one bisects the opposite angle

N

N

N

Y

N

N

N

Answers 211

Looking back (page 99) 120° (sum of angles in a quadrilateral) 140° (sum of angles in a quadrilateral) 60° (supplementary angles) 30° (sum of angles in a quadrilateral) 60° 42° (vertically opposite angles) 115° (sum of angles in a quadrilateral, equal angles in a kite) h) 105° (alternate angles, supplementary angles)

1 a) b) c) d) e) f) g)

Revise and Consolidate 3 Page 116 1 a) 42° b) 48° c) 7.6 cm d) 6.8 cm e) KL 4 cm, LJ 7 cm 2 Students’ own constructions 3 Cash price

2

Chapter 10 Page 102 1 a) 7 cm d) 15 cm g) 9.5 cm j) 5 cm

b) e) h) k)

8 cm 51° 15 cm 12 cm

c) 5 cm f) 6.8 cm i) 12 cm

Page 105 1 a) Pupils’ own constructions. CAB 37°, BCA 53°, ABC 90° b) Pupils’ own constructions. EFG 51°, FGE 78°, GEF 51° c) Pupils’ own constructions. HIJ 60°, IJH 60°, JHI 60° d) Pupils’ own constructions. KLM 85°, LMK 26°, MKL 69° e) Pupils’ own constructions. TUV 39°, VTU 51°, VT 5.7 cm 2 a) Students’ own constructions. 6.2 cm, 51°, 39° b) Students’ own constructions. 5.2 cm, 60°, 30°

Deposit Repayment Monthly Total HP (% of term (years) payment cash price)

a) $500

12

1

$62.50

$810.00

b) $550

5

2

$30.00

$747.50

c) $480

712

2

$30.00

$756.00

d) $700

9

112

$42.25

$823.50

e) $600

15

2

$25.00

$690.00

f)

10

2

$21.00

$544.00

5

3

$38.63

£1 435.50

h) $650

10

212

$26.00

$845

i)

912

1

$3 855

$3 855

$400

g) $900

$3 000

4 a) TT$3 125 b) BD$995 c) JA$30 750 d) Guy$89 250 e) EC$1 335 5 TT$1 000 is US$160; BD$1 000 is US$502.51; JA$1 000 is US$16.26; Guy $1 000 is US$5.60; EC$1 000 is US$374.53 6 a) BD$39.80 b) TT$625 c) US$30.15 d) US$17.60 e) BD$79.60 f) JA$1 772 7 20.73 l b) 170 cm3 c) 210 cm3 8 a) 40 cm3 3 3 d) 600 cm e) 310 000 mm f) 120 cm3 9

One pair of parallel sides?

Students’ own constructions

Two pairs of parallel sides?

Page 109

All sides equal?

Students’ own constructions Students’ own constructions, PR 7.1 cm Pyramid with triangular base The sections are equal.

Students’ own constructions Students’ own constructions 2.18 km Students’ own constructions

Looking back (page 114) Students’ own constructions


Trapezium Parallelogram

YES

NO

Page 111

1 2 3 4

NO

Rectangle

1 to 3 Students’ own constructions 4 The perpendiculars intersect at a single point.

Page 114

Quadrilateral

YES

Page 107

1 2 3 4

NO

10 a) c) e) g)

YES

NO

Right angles?

NO

All sides equal?

YES

YES

Square

Rhombus

bisect parallel lines corresponding right angle

b) supplementary d) congruent f) transversal

11

Front

Left side

4

Chapter 11

Courtney Walsh

Page 120 1 a) continuous b) discrete c) discrete d) continuous e) discrete f) continuous 2 For example: Pulse rate

Frequency

65–67

4

68–70

6

71–73

4

74–76

3

77–79

2

80–82

1

Books read

Frequency

1–3

7

4–6

12

7–9

7

10–12

4

5 a) 6 6 a)

Height (cm)

Frequency

130–139

4

140–149

5

150–159

8

160–169

3

c) 72

Score

Frequency

4

1

5

4

6

3

7

5

8

4

9

1

10

2

5 4 3 2 1 0 4

5

6

7

8

9 10

Score

7 a) and b) 16 14 Frequency

Page 123 8:00 There was an interval. 120 9:30

12 10 8 6 4

18

2

16

0

14 Price ($)

b) 25

b)

4 For example:

1 a) b) c) d) 2 a)

Sir Vivian Richards

Sir Garfield Sobers

Frequency

3

Brian Lara

0-4

5-9

10-14

15-19

20-24

Distance (km)

12 10 8 6 4 2 0

b) i) $6.00 3 a) $15

2

4 6 8 10 12 Petrol (litres)

ii) $22.50 b) 152

iii) 10 litres

Answers 213

8 a) and b)

4 a) 14

7

12

6

10

Age

Frequency

16 8

5

8

4

6

3

4

2

2

1

0

0 15-17

18-20

21-23

24-26

Height (cm)

Time (sec)

b) 16

Page 126

2 a) 3 4 5 6 7 8 9

5 2 4 3 0 2 2

10 8 4

8 9 7 7 4 6 8 5

3

Miles per gallons

12

6

b) 67 48 44 40 36 32 28 24 20 16 12 8 4 0

14

7 8 9 7 9 6 4

Age

4 8 2 1 0

500 1000 1500 2000 2500 3000 3500 4000 Engine size (cc)

Engine size and miles per gallon are related. The greater the engine size, the less miles per gallon.


2 0

1

2

3

No. of siblings

c) Age and height are related The older the child, the taller they are. There is no relationship between age and no. of siblings. 5 a) i) Temperature in Wellington (°C)

1 15 16 17 18 19

80 90 100 110 120 130 140 150 160 170

27-29

18 16 14 12 10 8 6 4 2 0

2

4

6

8 10 12 14 16 18 20 22 24 26

Temperature in New York (°C)

b) Maths

ii) 26 Temperature in New York (°C)

24 22 20

Science

18 16 14 12

3 4 5 6 7 8

7 0 0 1 5 0

7 1 4 6 8 1 2 2 3

4 5 6 7 8

3 2 2 0 4

4 5 4 1

7 5 6 8 7 8 4 4

c) The Maths test was more difficult for the students. 2 a)

10 8

Key Maths Score Science Score

6 4 2 0

100

16 18 20 22 24 26 28 Temperature in Hong Kong (°C)

90 80 70

Test Scores

b) 0 0 1 2 5 9 1 1 5 6 2 0 2 4 5 c) As the temperature in Wellington falls the temperature in New York rises. As the temperature in Wellington falls the temperature in Hong Kong rises. As the temperature in New York falls the temperature in Hong Kong falls.

60 50 40 30 20 10 0 A B

C D E

Looking back (page 128) 1 a) 100

b) Maths

3 4 5 6 7 8

8 2 5 5 0 0 2 8 5 9

Science

3 4 5 6 7

0 0 6 0 5 5 5 5 6 4

80 70 50 40 30 20

0

G H

I

J

c)

10

100 90

10 20 30 40 50 60 70 80 90 100 Score in Science Test

80 Maths Score

Score in Maths Test

90

60

F

Student

70 60 50 40 30 20 10 0

10 20 30 40 50 60 70 80 90 100 Science Score

Answers 215

Maths: mean 56.4, mode 45, 50, range 51 Science: mean 53.6, mode 55, range 44 ii) Mathematics iii) The bar chart was the most effective in comparing scores. iv) On average the Maths scores are slightly higher, but the Science scores were more consistent. Generally the students who did well on the Maths test also did well on the Science test. i)

Revise and Consolidate 4 Page 133 1 2 3 4 5 6

A C C B D a)

Score

Chapter 12 Page 130

1

2

2

5

3

6

4

6

5

7

6

5

7

5

8

5

9

5

10

4

Looking back (page 131) 1 6 2 5 and 15 3 a) mean 5, median 4, mode 4, range 5 b) mean 6, median 5, mode 5, range 5 c) The range has stayed the same; the other values have increased by 1. d) i) mean 7, median 6, mode 6, range 5 ii) mean 4, median 3, mode 3, range 5


50

b) Test scores 7 6 5 4 3 2 1 0

1

2

3

4

5 6 Score

Page 131 1 a) Yes b) The mean 2 a) 13.1 b) 13 weeks 3 a) 17 b) 10 and 25 4 mean 12.4, mode 10, median 12

Frequency

Total

Frequency

1 Girls: mean 28, median 28.5, range 18 Boys: mean 28.2, median 29, range 10 The mean and the median are both very similar for both boys and girls, but the large range for the girls compared to the boys suggests that the boys were more consistent than the girls. 2 Archer A: mode 0, mean 4.5, range 10 Archer B: mode 5, mean 4, range 5 Archer B should go through as they are more consistent. 3 a) Athlete A: mean 8.712, median 8.71, range 0.14 Athlete B: mean 8.65, median 8.625, range 0.38 b) Students’ own answers 4 2 5 5 6 a) 4 b) 3.13

Tally

7 a)

Mark

Tally

7

8

9

Frequency

1–5

2

6–10

5

11–15

5

16–20

9

21–25

6

26–30

5

31–35

10

36–40

8

Total

50

10

b)

4 a)

Frequency

Scores in Science Test 10 9 8 7 6 5 4 3 2 1 0 0.5

b) c) d) e) f) g)

5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.5 Score

Chapter 13

Page 138 1 a) p 4, q 1 c) d 0, e 2 e) x 4, y 0.5 2 Answers as page 136

b) d) f) b) d) f)

x 0.5, y 2 x 3, y 4 v 79, w 43 c 10, d 2 x 4, y 3 g 6, h 1

b) x 5, y 8 d) x 160, y 230 f) h 3, k 9

Page 141 1 1 a) 4, 8, 10, 42 1 b) 4, 8, 10, 42 1 1 c) 2, 0.1, 2, 4, 8, 10, 3, 4, 1, 1.25, 4, 42 1 1 d) 2, 0.1, 2, 4, 3, 4, 1, 1.25, 4, 42 1 e) 2, 0.1, 2, 3, 4, 1, 1.25, 4 f) 8, 10 1 1 g) 2, 0.1, 2, 4, 8, 3, 4, 1, 1.25, 4, 42 1 h) 4, 8, 10, 42 1 i) 4, 8, 10, 42 j) 2, 3, 4 1 1 k) 2, 0.1, 4, 8, 10, 1, 1.25, 4, 42 1 1 l) 2, 0.1, 4, 8, 10, 1, 1.25, 4, 42 2 a) i) 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ii) 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5 iii) 10 iv) 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 b) Students’ own answers 3 a) x 5 b) x 4 c) x 22 d) x 80 e) a 2 f) b 2 g) c 6/11 h) z 0.2

6 5 4 3 2 1 0

1

2

18

19

20

21

22

23

24

25

26

75

76

77

78

79

80

81

82

83

3 2 1

0

1

2

3

4

5

3 2 1

0

1

2

3

4

5

112 111 0

1 11

2 11

3 11

4 11

5 11

6 11

7 11

3

8 11

0.6 0.5 0.4 0.3 0.2 0.1 0 3 35

9 11

10 11

1 1111

0.1 0.2 0.3

1 w 342 2 h 413

b) p f 8.6 d) 4 c 27 b) x 4 2 x 2 d) x 211 Perimeter is 2w 2(4 w) 200 52 m m (m 200) 8 a) Total saving is m 5400 2 6 b) $3 200 in bonds, $1 610 in stocks and $570 in a savings account 9 $48 5 a) c) 6 a) c) 7 a) b)

c) mode 17 and 32 median 24 mean 23.62

Page 136 1 a) b 1, c 1 c) r 1, s 5 e) p 7, q 3 2 a) x 2, y 1 c) p 3, q 14 e) j 0.2, k 1.4

h)

654321 0 1 2 3 4 5 6 7 8

Looking back (page 143) x 34569, y 5283 x 3, y 5 x 0.5, y 4 x 7, y 4 x 49, y 115 3x 2y 26 4x y 28 b) $6 27 and 45 $9 a) 3c 4p 98 2c 5p 91 b) $18 c) $11 a) p 12 c) 11 m e) 11 w a) p 23 c) x 7 e) y 5

1 a) c) e) g) i) 2 a)

3 4 5

6

7

8 a) b) c) d) e) f)

1 0 4

5

1

2

3

6

7

8

b) d) f) h)

x 3, y 2 x 3, y 2 x 5, y 3 x 2.5, y 1

b) d) f) b) d) f)

d 15 2 t 3 k k 44 t 8 a 23 4

5

6

20 19 18 17 16 232425262728 0

1 3

2 3

1 1 13 1 23 2

3 3.4

4

Answers 217

4

Chapter 14

y 6x 7

Page 146 1

Straight lines

Quadratic curves

12

y 3x 3

y x 2

yx

y x2 4

y 7 3x

y x2 x 2

2 a) v d) iv 3 a)

b) iii e) vi

y 8x 3

y 13 11 10 9

c) ii f) i

8 7 6

x

5 4 3 2 1 0

1

2

y 7 3x

22 19 16 13 10

7

4

1 2 5

x

5 4 3 2 1 0

1

2

3

4

5 4

b)

3 yx 4 2

21 12

0 3 4 3 0

5

b) (0, 3) e) (0, 7) b) (0, 4)

4 Q1 a) (0, 0) d) (0, 4) Q3 a) (0, 7)

3 5

4 12

2

y 18 x 2

c) (0, 0) f) (0, 2)

5

4

3

1

2

1 0 1

Page 150 1 a), d), e) 2 a), c) and d), f) 3

y 7x 2

y 7x 4

y 7x

y 11

10

y 2 16 x

2

3

4

5

6

7

8

9

1

2

3

4 x

10

9 8

5 a)

7

y 5

6

4

5

3

y x 47

4

2

3

1

2

5

1

2

1

0 1

2

3

4

5

6

7


1

2 x

4

3

2

1 0 1

2

3

4

5

6

1

2

3

4

5 x

b)

y 10

y 4x 23

9

2 a) x number of T-shirts, y number of shorts, 600 100x 90y; x 1; y 1 b) x1

y 7

8 7

6

6

5

5

9y 60 10x

4

4

3

3

2

y1

2

1

1

5

4

3

2

1 0 1

1

2

3

4

5 x

6

5

4

3

2

1 0 1

1

2

3

4

6 x

5

2

2

3

3

4

4

5

c) 3 d) $30 3 a) x cost of pens, y cost of books 5x 2y 50, 6x y 45 b) y

6

7

8

45

9

40 35 30

Page 152 1 a) 80 m d)

Height (metres)

6x y 45

b) 40 m

25

c) 90 m

20

90

15

80

10

70

5

60

20 15 10 5 0 5

50 40

10

30

15

20

5x 2y 50

20

10 O

5 10 15 20 25 x

20 40 60 80 100 Distance (metres)

25

c) x $5.71, y $10.71 4 a) (0, 5) 5 man hours (all by Jack) (0, 12) 12 man hours (all by Jack) (10, 5) 15 man hours (10 by James, 5 by Jack) (10, 12) 22 man hours (10 by James, 12 by Jack) b) (10, 12) c) 22 hours d) James works 10 hours, Jack works 12 hours 5 a) x 3 b) A x2 3x c) (8, 40): length 8 cm, area 40 cm2 (10, 40): length 10 cm, area 40 cm2 (10, 70): length 10 cm, area 70 cm2 Omit points where area is negative. d) 8 cm e) 70 cm2, 10 cm

Answers 219

Looking back (page 154)

18 a)

1 All lines are parallel. 2 The slope 3 x 3, y 2 x 4, y 3 x 1, y 0 x 2, y 1.5 4 6x 42 67

y 4 3

y x2

2 1

x

3 2 1 0 1

25 6

1

2

3 x

y x 2 2 3

y

18 b) (1, 1) and (2, 4)

Chapter 15

x

Page 159 a) 5 m b) 25 cm c) 5 cm d) 9 cm 14.14 m 25 cm 12.5 cm c) 42 42 < 82 a) 12 22 < 32 d) hypotenuse not the longest side 6 b) 5.29 m 7 b) 5 m c) i) 5.83 m ii) 7.68 m 1 2 3 4 5

5 x 2; y 1, y 23x 2

Revise and Consolidate 5

e) 3.5 m

Page 155 1 a) a 19, b 33 c) x 6623, y 2293 46 , 21

3 7

b) x 1, y 1 d) x 2, y 1

f) x 2, y 5 e) x y g) x 2, y 12 2 a) 3x 2y 260 4x y 280 b) $60 3 39, 58 4 $12 5 a) 3x 4y 73.50 2x 2y 43 b) $12.50 c) $9 b) 5 y 6 a) x 12 c) x 3 d) 4 y e) x 5 f) x 1 g) x 3 h) x 13 7 x 0, 1, 2, 3, … 8 x 4, 3, 2, 1, 0, 1, … 9 c) and d) 10 a) and c) 11 b) 12 a) 13 d) 14 c) b) 6 c) 14 d) 152 e) 14 15 a) 14 16 a) $280, $380 b) i) $420 ii) 1 July 1996 iii) A 180 20N 17 (1, 4)


Page 164 1 a) 0.344 b) 3.487 c) d) 1.732 e) 1.664 f) 2 a) 35.7° b) 89.1° c) d) 40.2° e) 0.4° 3 a) 0.203 b) 1.103 c) d) 0.479 e) 0.139 4 a 51.3°, b 79°, c 82.5° 5 n 8.34 m, p 18.81 m, q 68.69 m

0.466 1 54.5° 1.477

Page 166 1 a) 0.766 b) 0.707 c) 0.951 d) 0.866 e) 0.999 f) 1 2 a) 0.579 b) 0.462 c) 0.955 d) 0.876 e) 0.148 3 a) 0.956 b) 0.996 c) 0.515 d) 0 e) 0.866 f) 0.707 4 a) 0.340 b) 0.995 c) 0.540 d) 0.805 e) 0.698 5 a 48.2°, b 23.6°, c 23.6°, d 36.9°, e 48.6° 6 p 74.31 m, q 7.51 cm, r 6.18 m, s 6.18 cm, t 46.82 cm 7 a) 10.6 cm b) 21.2 cm 8 a) 3.38 m b) 7.25 m Page 170 1 2 3 4 5 6 7

1.93 m 25.73 m 4.663 m a) 10.72 m b) 41.41 m a) 2.5 km b) 41.1° a) 8 m b) 13.86 m Sissy 73.86 m, Joan 77.67 m

8 AB AD 7.62 cm, DC BC 4.24 cm, ABC ADC 11.7°, DCB 90°, DAB 46.4° Looking back (page 172) 1 a 7.7 m, b 103.3 mm, c 111.2 m, d 17.8 cm, e 14.7 cm 2 48.1 cm b b b c c c b) c) d) e) f) 3 a) a c a a a b 4 a) 8.7 cm b) 5 cm 5 a) 30° b) 60° 6 84.46 cm (2 d.p.) 7 b) 49.81 m

3 a) 16.62 cm b) 21.2 cm 4 149.92 m 5 a) 0.469 b) 0.875 c) 0.532 d) 0.875 e) 0.469 f) 1.881 6 a 2.1 cm, b 15 cm, c 54.6°, d 111.9 cm, e 1 766.9 cm 7 6.6 m 8 a) 45.68 m b) 64.13 m c) diagonal route d) yes 9 Students’ own diagrams

Chapter 17 Page 185 1 a) 0.51 b) 0.66 c) 0.84 2 a) Machine Correct weight

Chapter 16 Page 173

A

Students’ own diagrams Enlargement centre (0, 0), scale factor 2 Enlargement centre (0, 0), scale factor 4 a) Students’ own diagrams b) They are in a straight line. 5 a) Centre A, scale factor 2 b) Centre J, scale factor 3 6 Coordinates 3, coordinates 4

B

52%

C

19 30

1 2 3 4

Page 176 1 a) AB 8 cm, BC 10 cm, CD 20 cm, DA 10 cm 2 i) a) 1 : 2 b) A 12B c) scale factor 2 d) 12 cm ii) a) 1 : 5 b) A 15B c) scale factor 5 d) 25 cm iii) a) 1 : 3 b) A 13B c) scale factor 3 d) 60 cm 3 i) 32 cm ii) 24 cm iii) 10 cm 4 No, because one side is enlarged by a scale factor of 4, the other side is enlarged by a scale factor of 3.

d) 0.4

0.73

b) 300 39% 117 c) 90 bags 3 0.3 4 a) Square b) P(triangle) 133600, P(square) 133650, P(circle) 39650 c) Triangle: 43 times, square: 45 times, circle: 32 times. 5 Ella is not correct. The angles at the centre of the spinner determine the probability, and they are not equal. Page 186 1 a) Pupil B should be most reliable as the number of throws is the greatest. b) Number of heads

0

1

2

3

Total frequency

22

80

74

24

Page 177

c)

1 a) 36 cm b) Picture A 64 cm, picture B 192 cm, ratio 1 : 3 c) The perimeter is based on the length of the sides and so the ratio of the perimeter changes with the ratio of the lengths. 2 b) A 63°, B 90°, C 27° d) The angles are the same. e) The sides are in proportion so the angles will be the same.

Number of heads

0

1

Total frequency

22

80


0.11 0.4

2

3

74

24

0.37 0.12

d) Sum of estimated probabilities 1 2 Answers will vary. 3 Answers will vary. 4 a) 600 b) 215 c) 261050 0.36 (2 d.p.) Page 189 1 a)

Looking back (page 178) 1 Students’ own diagrams

Faces uppermost Total

1,2,3

1,2,4

1,3,4

2,3,4

6

7

8

9

Revise and Consolidate 6 Page 182 1 252 602 652 2 a) 11.4

b) 9.8

c) 10.6

Answers 221

e)

b) Total

Frequency


4

4 80

12

1 2 80

0.15

14

17

17 80

0.2125

15

19

1 9 80

0.2375

16

14

1 4 80

0.175

17

9

9 80

0.1125

18

5

5 80

0.0625

12 13

c)

6

6

12 13 14 15

7

13 14 15 16

8

14 15 16 17

9

15 16 17 18

8

0.05

girl girl boy

girl boy boy

9

Total

Frequency


Theoretical probability

12

4

4 80

0.05

1 16

0.0625

13

12

1 2 80

0.15

2 16

0.125

14

17

1 7 80

0.2125

3 16

0.1875

15

19

19 80

0.2375

4 16

0.25

16

14

1 4 80

0.175

3 16

0.1875

17

9

9 80

0.1125

2 16

0.125

5

5 80

0.0625

1 16

0.0625

girl

2 girls

boy

1 girl, 1boy

girl

1 boy, 1 girl

boy

2 boys

girl

boy

boy

2 boys, 1 girl

girl

2 boys, 1 girl

boy

3 boys

1

2

3

4

5

6

0

0

1

2

3

4

5

6

1

1

2

3

4

5

6

7

2

2

3

4

5

6

7

8

3

3

4

5

6

7

8

9

4

4

5

6

7

8

9

10

5

5

6

7

8

9

10

11

6

6

7

8

9

10

11

12

Total

P(2 girls) 0.25 P(1 girl) 0.5 P(0 girls) 0.25

Frequency

Theoretical probability

0

1

0.02

1

2

0.04

2

3

0.06

3

4

0.08

4

5

0.1

5

6

0.12

6

7

0.14

7

6

0.12

8

5

0.1

9

4

0.08

10

3

0.06

11

2

0.04

12

1

0.02

Note: the theoretical probabilities do not all up to 1 due to rounding. b) Answers will vary. c) Answers will vary. Looking back (page 190) 1 a) 0.45 2 a) 50% d) 10% 3 a) 115

b) 0.55 c) 4 b) 15% e) 0% b) 1145

4 a)

d) 2 c) 85% f) 9813%

b) Y

Y G G


1 boy, 2 girls

girl

0

There are differences between the 2 types of probability but we do not expect them to be exactly the same. e) By increasing the number of times the experiment is carried out. 2 a) Answers will vary. b) Answers will vary. c) Child 1 Child 2 Outcomes

d) Answers will vary.

girl

3 girls

boy

Number of spots

d)

18

boy

P(3 girls) 0.125 2 girls, 1 boy P(2 girls) 0.375 2 girls, 1 boy P(1 girl) 0.375 P(0 girls) 0.125 1 girl, 2 boys

girl

3 a)

7

Child 1 Child 2 Child 3 Outcomes

Y B

Y R Y B G G

5 a) 0.35 (H), 0.65 (P) b) 1 c) Tom’s was more accurate because he did more trials. d) No, if he did more trials his result would be more accurate.

17 Time (h) Distance (km)

1 2

1

12

1

30

60

90

1

22

2

3

120 150 180

180

Test your knowledge Test 1 1 c 2 b 3 c 4 d 5 c 6 b 7 a) i) 20 x ii) 16 x b) (20 x) (16 x) x 36 8 play both games b) 33 c) 21 32 8 a) 25 1 6 d) 5 e) 7 9 a) 4.36 102 b) 3 106 c) 2.4 104 3 1 d) 5.3 10 e) 7.08 10 10 a) i) 110112 ii) 42335 iii) 62758 b) i) 13128 ii) 11225 iii) 11100112

Distance (km)

150

12 13 14 15 16

4 5

3 a 7 d

Length (cm)

24 22 20 18 16 14 12 10 8 6 4 2 0

270 260 250

0

1 2

18 No. of hours

4 d 8 d

The Georgetown shirt is cheaper. 3 hours a) average 48 b) 64 marks minimum a)

60

0

Length (cm)

a 2 b d 6 a c 10 b ii) 4 a) i) 172 b) i) 48.28 ii) 4.7

90

30

Test 2 1 5 9 11

120

19 a) 46 e) 46

1

0

1 12 2 Time (h)

1

2

3

4

2 12

5

6

1

7

8

9

24 22 20 18 16 14 12 10 8

6

0 1 2 3 4 5 6 7 8 9 10 11 12 No. of hours

b) 49 f) 43

c) 32 g) 60

d) 60 h) 29

Weight in grams

240 230 220 210 200

Test 3 1 a 2 c 4 2.5 hours 5 a 6 b 9 d 10 c 12 $1 000, $12

3 c 7 d 11 b

8 b

190 180 170 150 170 190 210 230 250 270 290 310 330 350 Number of pages

x y 95 2

Answers 223

13

13 a) e) 14 a) 15 a) b)

90 80 70

km/h

60

13% b) 36% c) 44% 67% f) 81% g) 91% n 4 b) n 6 Students’ own constructions triangles XYM and ZYM

d) 60% h) 96% c) n 3

Test 5 1 b 2 c 3 d 5 b 6 b 7 d ii) 334 9 b) i) 51230 b) i) 1.554 ii) 8 10 a) i) 27 ii) 16 iii) 17 b) 12, 52 11 a) i) $192 ii) $320 b)

50 40 30 20

4 c 8 a

10 8 5

10 15 m/s

20

a) 25 km/h b) 17 m/s c) 72 km/h d) 9 m/s e) 47 km/h 33 cm2 a) 117 m b) 14.8 m2 a) 13.0 b) 60.8 c) 8.4 d) 9.7 a) $213 b) $195.25 a) i) BD$39.20 ii) TT$149.60 iii) Guy$246.60 b) i) US$200 ii) US$48.66

7

25

5 4 3 2 1 0

0

Test 4 1 a 2 c 3 c 4 d 5 c 6 a 7 b 8 a 9 a) a 5, b 3 b) i) 2 ii) 1 iii) y 2x 1 10 a) y 5 b) x2y 5xy 2xy2 x2y2 11 a) i) EC$106.80 ii) Guy$5 656 b) i) US$187.27 ii) US$42.43 12 a) Weights of Form 3 Students 40 30 20 10 0 19.5 29.5 39.5 49.5 59.5 69.5 79.5 Weight (kg)

b) median 53.3 in the interval 45–54, mean 53.75


x

JA Dollars

19 a) ZXY 60°, XZY 76°, XYZ 44° b) NM 10.3 cm, LN M 50°, MLN 80°

Frequency

6

50 10 0 15 0 20 0 25 0 30 0 35 0 40 0 45 0 50 0 55 0 60 0

14 15 16 17 18

0

US Dollars

0

c) i) $416

12 13 14 15

ii) US $4.60

iii) US $8.59 2a a) x 2 b) x b a)–c) Students’ own diagrams d) 4.9 cm, 7.75 cm a) Students’ own diagrams b) 109° c) 0.30, 0.66 a) i) p : x → 2x 1 q : x → x2 4 r : x → factors of x ii) f and q b) i) 1 ii) 2 iii) 3 iv) 23

Answers 225


Index A addition 1 bases 21 decimals 8–9 algebra expressions 79–80 factorisation 79–80 fractions 74–5 graphs 144–54 notation 78 alternate angles 110 angles alternate angles 110 bearings 169 bisector 106, 112 construction 112–14 corresponding angles 110 cosine ratio 165 degrees 161 depression 169 elevation 169 minutes 161 sine ratio 165 tangent ratio 161–4 arcs 103, 108 area 38–42 surface area 43–4 ascending sequences 66 averages 129

B Babylon 19 balance (hire purchase) 86 bar graphs 121 bases 19–22 addition 21 conversion 20 multiplication 22 subtraction 22 bearings (angles) 169 binary operations 78 binary system 19 biodiversity 132 bisectors 106–7 angles 106, 112 lines 106 BOMDAS 1 brackets 1, 72, 79

C calculators 162–3 centres of enlargement 173 charts (statistics) compound bar charts 121 pie charts 121–2 circles arcs 103, 108 area 40 circumference 35–7 circumference 35–7 classes (data) 119 compasses 106 complements (sets) 24, 26, 30 compound bar charts 121 consecutive terms (sequences) 66 constants 56, 148 construction 101–15 angles 112–14 bisectors 106–7 parallel lines 110–11 perpendiculars 108–9 rectangles 110 triangles 101–5 continuous data 119 continuous variables 151 conversion (bases) 20 corresponding angles 110 cosine ratio 165 credit 86 cube roots 7 cubes surface area 43 volume 89 cuboids surface area 43 volume 89

D data (see also statistics) 119, 129–32 continuous data 119 discrete data 119 decimals addition 8–9 division 12–13 multiplication 10–11 recurring decimals 14–15 subtraction 8–9

degrees 161 denary system 19 denominators 74 deposits 86 depression (angles) 169 descending sequences 66 diagonals 93, 94 diameter 35 differences 1 direct proportion 56 discrete data 119 discrete variables 151 disjoint sets 24 distance–time graphs 53 division 1 decimals 12–13 divisors 12

E Elements 24 elevation (angles) 169 elevation (solids) 97 elimination (simultaneous equations) 135–6 empty set 24 enlargements 173–8 scale factors 173, 175 equal sets 24 equations 72–3 simultaneous equations 135–8 equidistant points 106 equilateral triangles 112 equivalent sets 24 estimates 8, 10, 12 even numbers 1 event 184 exchange rates 84 experimental probability 186–9 exponents 3 expressions (algebra) 79–80

F factorisation 79–80 factors 1, 79 HCFs 79 foreign exchange 84–5 formulae 32 subjects 76–7 fractional indices 63–4

Index 227

fractions (algebra) 74–5 frequency axis (bar graphs) 121 frequency polygons 122–3 frequency tables 119–20 grouped frequency tables 119 functions 78 graphs 144–7 quadratic functions 145–6

G general terms (sequences) 68–9 gradients 148, 151 graphs bar graphs 121 distance–time graphs 53 functions 144–7 gradient 148–9 line graphs 121 linear graphs 148 scatter graphs 125–6 speed–time graphs 53 grouped frequency tables 119

H HCFs (highest common factors) 79 hire purchase 86–7 histograms 122 hypotenuses 104, 158, 165

I indices 3–4, 61–2 fractional indices 63–4 negative indices 63–4 indirect proportion 57 inequalities 139–42 solution set 139, 141 instalments (hire purchase) 86 integers 1 intercepts 151 interest 86 intersecting arcs 103 intersections (sets) 24, 28–30 inverse operations 12, 76 inverse proportion 57

L line graphs 121 linear graphs 148 lines bisectors 106 gradient 148–9 graphs 148–50 parallel lines 110–11 perpendiculars 108–9 lines of symmetry 93–4


M mappings (see functions) means (statistics) 129 medians 129 mid-points 106 minutes (angles) 161 modes 129 money 84–8 foreign exchange 84–5 hire purchase 86–7 multiples 1 multiplication 1 bases 22 decimals 10–11 multiplicative inverses 74

N natural numbers 1 negative indices (powers) 63–4 notation (algebra) 78 number lines 140 numerators 74

O odd numbers 1 operations binary operation 78 inverse operations 76 outcomes 184

P parabolas 145–6 parallel lines 110–1 parallelograms 38 area 38, 40 perimeters 35–7 perpendicular bisectors 106 perpendiculars 108–9 pi () 35 pictographs (statistics) 121 pie charts 121–2 plan (solids) 97 polygons frequency polygons 122–3 regular polygons 35 powers 3, 61 prime number 1 prisms surface area 43 volume 91–2 probability comparing 188–9 definitions 184–5 estimating 186–7 experimental 186–9 theoretical 184–5, 186–9

products 1 proportion 56–9, 175, 177 pyramids 43 Pythagoras’ theorem 6, 158–60

Q quadratic curves 145–6 quadratic sequences 70–1 quadrilaterals 93–4 quilt maker (case study) 181 quilts 179–81 quotients 1

R radii 35 random event 188–9 ranges (statistics) 129 rates 51 rational numbers 1 ratios 173, 175, 177 simplest form 175 real-life problems graphs 151–3 trigonometry 169–71 reciprocals (see multiplicative inverses) rectangles area 40 construction 110 recurring decimals 14–15 regular polygons 35 right-angled triangles 158–72 cosine ratios 165 hypotenuse 104, 158, 165 Pythagoras’ theorem 6, 158–60 sine ratios 165 tangent ratios 161–4 roots 5–7, 63 cube roots 7 square roots 5 rounding 8

S scale factors 173, 175 scatter graphs 125–6 scientific notation 16–17 sequences 66–7 general term 68–9 quadratic sequences 70–1 sets 24–34 complements 24, 26, 30 intersections 24, 28–9, 30 unions 24, 26, 30 Venn diagram 30–3 signwriter (case study) 100 similar shapes 175, 177

simplest form (ratios) 175 simultaneous equations 135–8 elimination 135–6 substitution 137–8 sine ratios 165 sketches 103 solids 97–8 solution sets (inequalities) 139, 141 solutions 151 speed 51, 53–5 speed–time graphs 53 square numbers 3 Pythagoras’ theorem 6 square roots 5, 63 standard form 16 standard method (multiplication) 10 statistics 119–32 averages 129 bar graphs 121 compound bar charts 121 frequency polygons 122–3 frequency tables 119–20 histograms 122 line graphs 121 means 129 medians 129 modes 129 pictographs 121

pie charts 121, 122 ranges 129 scatter graphs 125, 126 stem-and-leaf diagrams 125 stem-and-leaf diagrams 125 straight-line graphs 148–50 subject (formulae) 76–7 simultaneous equations 137 subsets 24 substitution (simultaneous equations) 137–8 subtraction 1 base systems 22 decimals 8–9 sums 1 surface areas 43–4 symmetry 93, 94

T tables (trigonometry) 161 tangent ratios 161–4 teacher (case study) 48 term-to-term rule (sequences) 66 terms (algebra) 79 terms (sequences) 66 general terms 68–9 tessellation 179 theoretical probability 184–9 transformations 173–83

enlargements 173–8 trapezia 38 area 38, 40, 41, 79 triangles area 40 construction 101–5 equilateral triangles 112 right-angled triangles 104, 158–72 trigonometry (see right-angled triangles)

U unions (sets) 24, 26, 30 universal sets 24 unknowns 72

V variables 72, 151 simultaneous equations 135 Venn diagrams 30–3 volume 89–92 cubes 89 cuboids 89 prisms 91–2

Y y-intercepts 151

Index 229


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Spectrum Phonics, Grade 3

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Building Fluency, Grade 3

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Daily Academic Vocabulary, Grade 3

Decodable Book 7 Grade 2

Grammar Practice Book: Grade 6

Decodable Book 2 Grade 2

Decodable Book 4 Grade 2

Building Spelling Skills: Grade 3

Grammar Practice Book-Grade 2

Spectrum Language Arts, Grade 3

Grammar Practice Book: Grade 5

Grammar Practice Book-Grade 1

Decodable Book 18 Grade 2

Decodable Book 11 Grade 2

Decodable Book 15 Grade 2

Maths Connect for Jamaica Grade 3 Pupil Book

Maths for AQA Modular GCSE: Foundation Pupil Book

Decodable Book 3 Grade 2

Fairyland 4 : Pupil' Book

Grammar Practice Book - Grade 3 (Story Town)

De Pupil

Spectrum Phonics, Grade 3

Jamaica

Jamaica

Building Fluency, Grade 3

De Pupil

Maths for chemists

Maths for chemists

De pupil

De Pupil

De pupil

Jamaica

Daily Academic Vocabulary, Grade 3

Decodable Book 7 Grade 2

Grammar Practice Book: Grade 6

Decodable Book 2 Grade 2

Decodable Book 4 Grade 2

Building Spelling Skills: Grade 3

Grammar Practice Book-Grade 2

Spectrum Language Arts, Grade 3

Grammar Practice Book: Grade 5

Grammar Practice Book-Grade 1

Decodable Book 18 Grade 2

Decodable Book 11 Grade 2

Decodable Book 15 Grade 2

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