Cambridge Lower Secondary

E
Mathematics
PL WORKBOOK 7
Lynn Byrd, Greg Byrd & Chris Pearce
M
SA

Second edition

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 Cambridge Lower Secondary

E
Mathematics
PL WORKBOOK 7
Lynn Byrd, Greg Byrd & Chris Pearce
M
SA

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
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PL
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 Contents

Contents
How to use this book 6
Acknowledgements7
5 Angles and constructions
5.1 A sum of 360° 66
1 Integers 5.2 Intersecting lines 68
5.3 Drawing lines and quadrilaterals 70

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1.1 Adding and subtracting integers 7
1.2 Multiplying and dividing integers 9
1.3 Lowest common multiples 12 6 Collecting data
1.4 Highest common factors 14 6.1 Conducting an investigation 73
1.5 Tests for divisibility 16 6.2 Taking a sample 76
1.6

2 E
 xpressions, formulae
and equations
2.1
2.2
2.3
2.4
Constructing expressions
PL
Square roots and cube roots

Using expressions and formulae

Collecting like terms
20
24
28
17
7 Fractions
7.1
7.2
7.3
7.4
7.5
Ordering fractions
Adding mixed numbers
Multiplying fractions
Dividing fractions
Making fraction calculations easier
80
83
88
93
97
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Expanding brackets 32
2.5 Constructing and solving equations 35 8 Shapes and symmetry
2.6 Inequalities39 8.1 Identifying the symmetry of 2D shapes 102
8.2 Circles and polygons 107
3 Place value and rounding 8.3 Recognising congruent shapes 111
8.4 3D shapes 115
SA

3.1 M ultiplying and dividing by powers of 10 43

3.2 Rounding 47
9 Sequences and functions
9.1  enerating sequences 1 
G 121
4 Decimals 9.2 Generating sequences 2  124
4.1 Ordering decimals 51 9.3 Using the nth term  129
4.2 Adding and subtracting decimals 54 9.4 Representing simple functions 134
4.3 Multiplying decimals 57
4.4 Dividing decimals 59 10 Percentages
4.5 Making decimal calculations easier 62
10.1 Fractions, decimals and percentages  137
10.2 Percentages large and small 139

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
3 to publication.
 1 Geometry
Contents

11 Graphs 14.4 Reflecting shapes 185

14.5 Rotating shapes 189
11.1 Functions  141 14.6 Enlarging shapes 193
11.2 Graphs of functions 144
11.3 Lines parallel to the axes 146
11.4 Interpreting graphs 148
15 Shapes, area and
volume
12 Ratio and proportion 15.1 Converting between units for area 199
15.2 Using hectares 202

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12.1 Simplifying ratios 153
15.3 The area of a triangle 204
12.2 Sharing in a ratio 157
15.4 Calculating the volume of cubes
12.3 Using direct proportion 161 and cuboids 209
15.5 Calculating the surface area of cubes
13 Probability and cuboids 214
13.1 The probability scale

13.3 Experimental probabilities

14 Position and
transformation
14.1 Maps and plans
PL
13.2 Mutually exclusive outcomes

14.2 Distance between two points

164
166
168

172
176
16 Interpreting and
discussing results
16.1 Two-way tables
16.2 Dual and compound bar charts
16.3 Pie charts and waffle diagrams
16.4 Infographics
220
227
234
239
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16.5 Representing data 245
14.3 Translating 2D shapes 179 16.6 Using statistics 247
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4 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 How to use this book

How to use this book

This workbook provides questions for you to practise what you have
learned in class. There is a unit to match each unit in your Learner’s Book.

Integers
Each exercise is divided into three parts:

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• Focus: these questions help you to master the basics
• Practice: these questions help you to become more confident in using

1 Adding and subtracting integers

what you have learned
• Challenge: these questions will make you think very hard.

:
9
example 1.1
You will also find these features:
Words you need to know.

b 2 − −5

w a number line if you need to.

PL 1
Key words
integers
inverse
number line
positive integers
negative integers Integers
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t at −5. Move 9 places to the right. 1.2 Multiplying and dividing integers

finish at 4. −5 + 9 = 4 1.1 Adding and subtracting integers

11 Copy and complete this multiplication grid.

Step-by-step
ubtract −5, add the inverse, 5. examples showing how ×Worked
6 4example 1.1 Key words
−30
5=2+5=7 to solve a problem. Work out: integers
−32 inverse
a −5 + 9 b 2 − −5
number line
SA

Challenge
Answer positive integers

se 1.1 Draw a number

a diagrams,
12 In these
circles next to it.
a –3 5
lineinifa square
the integer you need to.
is the product of the integers in the
Start at −5. Move 9 places to the right.
b 10 –2
negative integers

You finish at 4. −5 + 9 = 4
b To subtract −5, add the inverse, 5.
2 − −5 = 2 + 5 = 7
6 –2 –3 5

hese positive and Questions marked with this

negative integers. i Copy each diagram and fill in the squares.
Exercise 1.1
ii Add the numbers in the squares in each diagram.
3 + −4 symbol
b 6 + help
−5 you to practise
c −7 + 2 d13 a −5This+diagram
10 is similar to the diagrams in Question 10. –12

Focus
act these positivethinking and working Copy and complete the diagram. All the numbers are integers.
and negative integers. b
1
Is there more than one solution? Have you found all of the solutions? –15
Add these positive and negative integers.
–8

mathematically. Use the integers 3, 4 and −5 to complete this calculation.

−6 b −6 − 3 c 1 − −8 d14 a
( −5 − −6
a
+
−3 + −4
)× = −8
b 6 + −5 c −7 + 2 –10 −5 + 10
d
2 Subtract these positive and negative integers.
b What is the largest answer you can get when you put the integers 2,
and complete this addition table. − 6this calculation?b −6 − 3
−4aand 74 in c 1 − −8 d −5 − −6
+ )×
3 ( Copy and complete this addition table.
Give evidence to explain your answer.
4 −5 15 a +
The product −5integers is −20. Find the largest possible value of the sum of the
4of two
two integers.
2
b The product of two integers is −30. Find the largest possible sum of the two integers.
c Can−6you generalise the result of part a and part b?

4 Work out:
a 20 + −5 b −10 + −15 c −2 + −13 d −3 + 20

out:
Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
5 to publication.
0 + −5 b −10 + −15 c −2 + −13 d −3 + 20
7

11
 Acknowledgements
Thanks to the following for permission to reproduce images:
Cover image: ori-artiste/Getty Images
Inside: GettyImages/GI; Yoshiyoshi Hirokawa/GI; Lew Robertson/GI; Fajrul Islam/GI;
Norberto Leal/GI; Dave Greenwood/GI; Roman Milert/GI.
Key: GI= Getty Images.

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PL
M
SA

Original material
6 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1 Integers
1.1 Adding and subtracting integers
Worked example 1.1 Key words

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Work out: integers
inverse
a −5 + 9 b 2 − −5
number line
Answer
a

b
PL
Draw a number line if you need to.
Start at −5. Move 9 places to the right.
You finish at 4. −5 + 9 = 4
To subtract −5, add the inverse, 5.
2 − −5 = 2 + 5 = 7
positive integers
negative integers
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Exercise 1.1
Focus
1 Add these positive and negative integers.
SA

a −3 + −4 b 6 + −5 c −7 + 2 d −5 + 10
2 Subtract these positive and negative integers.
a 4−6 b −6 − 3 c 1 − −8 d −5 − −6
3 Copy and complete this addition table.

+ 4 −5
2
−6
4 Work out:
a 20 + −5 b −10 + −15 c −2 + −13 d −3 + 20

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
7 to publication.
 1 Integers

5 Work out:
a 20 − −5 b −10 − −15 c −2 − −13 d −3 − 20

Practice
6 Fill in the missing numbers.
a 8 +   = 1 b −3 +   = 3
c −10 +   = −6 d 5 +   = −5
7 Fill in the missing numbers.

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a  − 3 = 6 b  − 3 = 2
c  − 3 = −1 d  − 3 = −6
8 Estimate the answers to these questions by rounding the numbers to the nearest
integer.

9
a −6.15 + 9.93
c −11.3 + −8.81

PL b 7.88 − −9.13
d 12.19 − 5.62
Estimate the answers to these questions by rounding the numbers.
a −28 − 53
c −888 − −111
b 514 + −321
d −61.1 + −29.3
10 Two integers add up to 2. One of the integers is 8. What is the other integer?
11 When you subtract one integer from another integer, the answer is 3.
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One integer is 1. Find the other integer.
12 Here are six integers: −5, −3, −2, 3, 4, 5.
Use each integer once to complete these additions.
a  +   = 1 b  +   = −2 c  +   = 3
SA

Challenge
13 Copy and complete this addition table.

+ 3
5 −2
1 −6
14 This subtraction table shows that 3 − 6 = −3. Copy and complete the table.

− −4 6
3 −3 1
−3

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8 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1.2 Multiplying and dividing integers

15 Copy and complete these addition pyramids.

a b

1 –2 –4

–3 4 6 –1
–6
16 This addition pyramid is more difficult than the pyramids
in Question 13.
Copy and complete the pyramid. Explain how you worked

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out the missing numbers. 5 3

1.2 Multiplying and dividing integers

Worked example 1.2
Work out:
a 4 × −8

Answer
a 4 × 8 = 32, so 4 × −8 = −32.
PL
b −20 ÷ (3 − −2)
Key word
product
M
b First, do the subtraction in the bracket.
3 − −2 = 3 + 2 = 5
So, −20 ÷ (3 − −2) = −20 ÷ 5 = −4.

Exercise 1.2
SA

Focus
1 Work out:
a 10 × −3 b 4 × −9 c 5 × −11 d 7 × −7
2 Work out:
a −24 ÷ 2 b −24 ÷ 6 c −50 ÷ 10 d −63 ÷ 9

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
9 to publication.
 1 Integers

3 Copy and complete this multiplication table.

× 4 7
−2
−6
4 Work out:
a (−5 + 2) × 4 b (−6 + −4) × 3
c (1 − −3) × −7 d (−2 − −5) × −10

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5 Work out:
a (−5 + −7) ÷ 4 b (−10 + −4) ÷ 2
c (6 − 14) ÷ 4 d (−5 − 13) ÷ 3

Practice
6

7
c −2 ×   = −26 PL
Work out the missing numbers.
a 3 ×   = −24

Work out the missing numbers.

a −27 ÷
c
 = −3
÷ 6 = −6
b
d

b
d
6 ×   = −18
−12 ×   = −60

−36 ÷  = −9
÷ 4 = −8
M
8 Estimate the answers to these questions by rounding the numbers.
a −4.1 × 2.8 b −7.1 × −3.2
c −1.1 × −7.9 d −9.1 ÷ 3.2
9 Estimate the answers to these questions by rounding the numbers.
a 423 × −2.9 b −32 × −28
SA

c −6.1 × 219 d −612 ÷ 2.92

10 The product of 2 and −9 is −18.
a Find three more pairs of integers with a product of −18.
b Are there more pairs of integers with a product of −18?
How can you be sure?

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10 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1.2 Multiplying and dividing integers

11 Copy and complete this multiplication grid.

× 6 4
−30
−32

Challenge
12 In these diagrams, the integer in a square is the product of the integers in the

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circles next to it.
a –3 5 b 10 –2

13 a 

i
ii
6 –2

PL
Copy each diagram and fill in the squares.
Add the numbers in the squares in each diagram.
This diagram is similar to the diagrams in Question 10.
Copy and complete the diagram. All the numbers are integers.
–3

b Is there more than one solution? Have you found all of the solutions? –15
5

–12

–8
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14 a Use the integers 3, 4 and −5 to complete this calculation.
–10
(   +   ) ×   = −8
b What is the largest answer you can get when you put the integers 2,
−4 and 7 in this calculation?
(   +   ) × 
SA

Give evidence to explain your answer.

 he product of two integers is −20. Find the largest possible value of the sum of the
15 a T
two integers.
b The product of two integers is −30. Find the largest possible sum of the two integers.
c Can you generalise the result of part a and part b?

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
11to publication.
 1 Integers

1.3 Lowest common multiples

Worked example 1.3 Key words
Find the lowest common multiple (LCM) of 6 and 9. common multiple
lowest common
Answer multiple (LCM)
The multiples of 6 are 6, 12, 18, 24, 30, 36, You may not multiple

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42, 48, 54, … need to list so
The multiples of 9 are 9, 18, 27, 36, 45, 54, … many multiples
each time.
The common multiples are 18, 36, 54, …
The lowest common multiple is 18.

Exercise 1.3
Focus
1
a 4 b 7
PL
Write down the first four multiples of:
c 12 d 30
M
2 How many multiples of 10 are less than 100?
3 a Work out the multiples of 8 that are less than 50.
b Work out the multiples of 5 that are less than 50.
c Write down the lowest common multiple of 8 and 5.
4 a Find the first five common multiples of 2 and 3.
SA

b Copy and complete this sentence:

The common multiples of 2 and 3 are multiples of
c Find the lowest common multiple of 2 and 3.
5 a Write down the first three common multiples of 6 and 4.
b Copy and complete this sentence:
The common multiples of 6 and 4 are multiples of
c Write down the lowest common multiple of 6 and 4.
6 Find the lowest common multiple of:
a 3 and 10 b 4 and 10 c 5 and 10

Original material
12 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1.3 Lowest common multiples

Practice
7 Show that the multiples of 3 and 5 are multiples of 15.
8 Find the lowest common multiple of 6 and 14.
9 a Find the lowest common multiple of:
i 7 and 2 ii 7 and 4 iii 7 and 6
b Is there an easy method to find the lowest common multiple
of 7 and a number less than 7?
c Does the method in part b work for 7 and a number more

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than 7?
10 Look at these numbers: 90 92 94 96 98  100
a i Which number is a multiple of 9 and 10?
ii Is this number the lowest common multiple?

ii

PL
b i Which number is a multiple of 2 and 7?
Is this number the lowest common multiple?
c i Which number is a multiple of 12 and 8?
ii Is this number the lowest common multiple?
11 Work out the lowest common multiple (LCM) of 2, 5 and 6.

Challenge
M
12 Find the LCM of 3, 8 and 9.
13 24 × 4 = 96
a Explain why 96 is a common multiple of 4 and 24.
b Is 96 the lowest common multiple of 4 and 24? Give evidence
to justify your answer.
SA

14 Two numbers have a LCM of 45. The two numbers add up to 14.
Find the two numbers.
15 The LCM of two numbers is 63. Work out the two numbers.

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
13to publication.
 1 Integers

1.4 Highest common factors

Worked example 1.4 Key words
a Find the factors of 24. common factor
conjecture
b Find the highest common factor (HCF) of 24 and 80.
consecutive
Answer factor

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a 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6 highest common
factor (HCF)
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
b You must find the highest factor of 24 that is also a factor of 80.




24 is not a factor of 80.
PL
8 is a factor of 80 because 80 ÷ 8 = 10.
12 is not a factor of 80 because 80 ÷ 12 = 6 remainder 4.

The highest common factor of 24 and 80 is 8.

Exercise 1.4
M
Focus
1 Find the factors of:
a 21 b 32 c 50 d 72 e 43
2 Find the factors of:
SA

a 51 b 52 c 53 d 54 e 55
3 a Find the common factors of 16 and 28.
b Find the highest common factor of 16 and 28.
4 a Find the common factors of 30 and 45.
b Find the highest common factor of 30 and 45.

Original material
14 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1.4 Highest common factors

Practice
5 Find the highest common factor of:
a 18 and 21 b 18 and 27 c 18 and 36
6 Find the highest common factor of:
a 27 and 45 b 50 and 75 c 40 and 72 d 24 and 35
7 Find the highest common factor of:
a 70 and 77 b 70 and 85 c 70 and 84

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8 a Find the highest common factor of 32 and 40.
b Use your answer to part a to simplify the fraction 32 .
40
9 a Find the highest common factor of 52 and 91.
b There are 91 rooms in a hotel. 52 rooms are reserved. What fraction of
the rooms are reserved?

Challenge
PL
10 Two numbers have a highest common factor of 5. The two numbers add up to
35. Show that there are three possible pairs of values for the two numbers.
11 The HCF of two numbers is 4. Both numbers are more than 4 and less
than 30.
a Show that the numbers could be 8 and 12.
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b Show that the numbers are not 8 and 16.
c Find all the other possible values of the two numbers.
12 The HCF of two numbers is 3. One of the numbers is 9. What are the possible
values of the other number?
SA

13 a Find the highest common factor of:

i 9 and 10 ii 20 and 21 iii 32 and 33
b 9 and 10 are consecutive numbers. 20 and 21 are consecutive numbers.
Use part a to make a conjecture about the highest common factor of two
consecutive numbers.
c What is the lowest common multiple of two consecutive numbers?

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
15to publication.
 1 Integers

1.5 Tests for divisibility

Worked example 1.5 Key words
Use tests for divisibility to show that 3948 is divisible by 3 and 6 but divisible
not by 9. tests for
divisibility
Answer

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The sum of the digits is 3 + 9 + 4 + 8 = 24.
24 is divisible by 3, so 3948 is also divisible by 3.
3948 is even and divisible by 3, so 3948 is also divisible by 6.
24 is not divisible by 9, so 3948 is also not divisible by 9.

Exercise 1.5
Focus
1
PL
Use tests for divisibility to show that 5328 is divisible by 4 and by 9.
2 a Show that 2739 is divisible by 11.
M
b When the digits are reversed, the number is 9372.
Is 9372 divisible by 11? Give a reason for your answer.
3 a Show that 67 108 is divisible by 4.
b Is 67 108 divisible by 8? Give a reason for your answer.
4 The number 3812* is divisible by 3. The final digit is missing. What can you say
SA

about the missing digit?

Practice
5 What integers less than 12 are factors of 7777?
6 a Use the digits 5, 4, 2 and 1 to make a number that is divisible by:
i 5 ii 3
b Can you arrange the digits 5, 4, 2 and 1 to make a number that is divisible by:
i 9? ii 11?
7 322 is divisible by 7. Use this fact to find a number that is divisible by 7, 2 and 3.
8 Find the smallest positive integer that is not a factor of 2520. Give reasons for
your answer.

Original material
16 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1.6 Square roots and cube roots

9 Here are some numbers where all the digits are 9:

9  99  999  9999  99999  ...
In numbers where all the digits are 9, which are multiples of 11?

Challenge
10 A number is divisible by 15 if it is divisible by 3 and 5.
a Show that 7905 is divisible by 15.
b The number 208** is divisible by 15. Find the possible values

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of the two missing digits.
11 Find three numbers less than 20 that are factors of 3729.
Give reasons for your answers.
12 Show that 8897 is divisible by only one number between 1 and 12.

a 2 b 3
PL
13 The numbers 4, 5, 6, … are consecutive numbers; for example, 4567
is a number with four consecutive digits.
Find all the numbers with four consecutive digits that are divisible by:

1.6 Square roots and cube roots

c 5 d 11
M
Worked example 1.6 Key words
Work out 3 125 − 49 . consecutive
cube number
Answer
cube root
SA

53 = 5 × 5 × 5 = 125, so 3 125 = 5. square number

72 = 7 × 7 = 49, so 49 = 7. square root
3
125 − 49 = 5 − 7 = −2

Exercise 1.6
Focus
1 Work out:
a 32 + 42 b 6² + 72 c 92 + 102

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
17to publication.
 1 Integers

2 Work out:
a 64 b 100 c 225 d 169
3 Work out:
a 13 + 23 b 33 + 53 c 43 − 23
4 Work out:
a 64 − 3 64 b 25 − 3 125 c 3
27 − 16
5 Work out:
a 3 216 b 3
512 c 3
1000 d 3
1728

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Practice
6 Copy and complete:
a = 20 b = 25 c = 30 d = 35
7

9

Copy and complete:
a 3 =6 b

PL 3

Find the integer that is closest to:

a 38
45 is between 6 and 7.
b 220

Write down a similar statement for:

= 10 c

c
3

3
70
= 11 d 3 = 15
M
a 90 b 135
10 Mustafa thinks of a number. The number is between 100 and 200. The square
root of the number is a multiple of 3.
What is Mustafa’s number?
11 a F
 ind all the numbers between 100 and 200 that have an integer
SA

square root.
b Find all the numbers between 100 and 200 that have an integer cube root.

Challenge
12 Find the highest common factor of 12 + 22 + 32 and 42 + 52 + 62.
13 Jiale thinks of a number. She works out the square root of the
number. Then she works out the cube root of the square root of the
number. The answer is 2.
a Find Jiale’s number.
b Show that Jiale gets the same answer if she finds the cube root first and then
the square root.

Original material
18 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 1.6 Square roots and cube roots

14 289 and 324 are two consecutive square numbers. Find the next
square number after 324.
15 1331 and 1728 are two consecutive cube numbers. Find the next
cube number after 1728.
16 a Show that 64 is a square number and a cube number.
b There is one number between 100 and 1000 that is a square
number and a cube number. What is this number?
c What method did you use to answer part b? Could you use the
same method to look for another number that is both a square

E
number and a cube number?

PL
M
SA

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19to publication.
 2 Expressions,
formulae and equations

E
2.1 Constructing expressions
Here are three bags, each with a different number of balls inside.
Key word

This bag has

two balls.
PL
This bag has
four balls.
This bag has b balls. You
cannot see how many
balls are in this bag, so
you can choose any letter
to represent the number.
expression

Tip
Remember that
the correct order
of operations is
used in algebra as
M
well as in number
calculations.
Exercise 2.1 Divisions and
multiplications
come before
Focus additions and
SA

1 Write down the missing number for each of the following. subtractions.

If you do not know the number, choose your own letter to
represent it.
Tip
a This box has  counter. In part c you do
not know how
many counters
b This box has  counters.
are in this box,
so choose your
c This box has counters. own letter.

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20 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.1 Constructing expressions

d This bag has  apples.

e This bag has  apples.

2 Two balls are added to each of these bags. Write down the
missing numbers. Tip

E
a There is one ball in the bag. In part c choose
Add two balls, so there are now 1 + 2 = balls your own letter
in the bag. for the number
b There are balls in the bag. of balls; for

3
c
balls in the bag.
There are

the bag.
PL
Add two balls, so there are now

balls in the bag.
Add two balls, so there are now

A bag has n counters in it.

+2=

+2 balls in

Match the statement in each rectangle to its correct expression in

example, b.
You cannot work
out b + 2,
so just leave
the expression
as it is.
M
the ovals. The first one has been done for you: A and iv. Tip
I add three I add one I add six I add nine I add seven An expression is
counters to counter to counters to counters to counters to
a statement that
the bag. the bag. the bag. the bag. the bag.
contains letters
A B C D E
and sometimes
SA

n+7 n+6 n+1 n+3 n+9 numbers; for

example, n + 7.
i ii iii iv v

4 Sofia has a box that contains t toys. Write an expression for the
total number of toys she has in the box when: Tip
a she puts in four more toys In each part of
b she takes out two toys the question Sofia
c she adds five toys starts with t toys.
d she takes out half of the toys.

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
21to publication.
 2 Expressions, formulae and equations

5 Cheng has s strawberries. Write an expression for someone

who has:
a two more strawberries than Cheng
b three times as many strawberries as Cheng
c six fewer strawberries than Cheng
d half as many strawberries as Cheng.

Practice

E
6 Write down an expression for the answer to each of these questions.
a Ali has x paintings. He buys two more paintings.
How many paintings does he now have?
b Hamza has t free text messages on his phone each month.

PL
So far this month he has used 15 text messages.
How many free text messages does he have left?
c Ibrahim is i years old and Tareq is t years old. What is the
total of their ages?
d Aya can store v video clips on one memory card. How many
video clips can he store on two memory cards?
Nesreen thinks of a number, n. Write an expression for the number
Nesreen gets each time.
a She multiplies the number by 6.
M
b She divides the number by 5.
c She multiplies the number by 5, then adds 1.
d She multiplies the number by 7, then subtracts 2.
e She divides the number by 10, then adds 3.
f She multiplies the number by 3, then subtracts the result
SA

from 25.
8 The cost of an adult meal in a fast-food restaurant is $a. The cost
of a child’s meal in the same restaurant is $c.
Write an expression for the total cost of the meals for these groups.
a one adult and one child
b one adult and three children
c four adults and one child
d four adults and five children

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22 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.1 Constructing expressions

9 Match each description (a to f ) to the correct expression (i to vii).

Description Expression
a Multiply x by 5 and subtract from 4. i 4x + 5
b Multiply x by 4 and add 5. ii 4x − 5
c Multiply x by 5 and subtract 4. iii 4 − 5x
d Multiply x by 5 and add 4. iv 5 − 4x

E
e Multiply x by 4 and subtract 5. v 5x − 4
f Multiply x by 4 and subtract from 5. vi 5 − 5x
vii 5x + 4

a match:

PL
Marcus writes this description for the expression that did not have

My x y 5 n t 5.

Is Marcus correct? Explain your answer.

M
10 Write an expression for each of these situations.
You can choose your own letters, but make sure you write what
your letters represent.
a The total cost of seven drinks and six bags of potato chips.
b The total value of three rings is doubled.
SA

Challenge
11 Write an expression for each of these descriptions.
a k more than g b h less than t
c y more than eight times x d three times a multiplied by b
12 Write a description for each of these expressions. Tip
a v + 7u b 8w − d
Remember the correct
c 5x + 3y d 7pq
order of operations.
Multiplication comes
before addition and
subtraction.

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
23to publication.
 2 Expressions, formulae and equations

13 Kai has two pieces of wood. The lengths of the pieces of wood are shown
in the diagram.

8a + 15 2b – 3c

Write an expression for:
a the total length of the pieces of wood
b the difference in the lengths of the pieces of wood.
14 p and q are whole numbers, such that p + q = −2 and pq = −8. Also p > q.

E
a Sadie thinks that p = −12 and q = 10. Explain why Sadie is wrong.
b Work out the values of p and q.

numbers but has no = sign.

For example: p
PL
2.2 Using expressions and formulae
An expression is a statement that contains letters and sometimes

x + 2 n − 3
You can substitute numbers into an expression to work out the value of
the expression.
4m
Key words
formula
substitute
M
For example, the value of x + 2 when x = 5 is 5 + 2 = 7. Tip
A formula is a statement that contains letters and sometimes numbers Substitute the x
and has an = sign. for the number
For example: A = 2p y = x − 2 v = 5n − 3 R = m + k 5 and then work
out 5 + 2.
SA

Exercise 2.2
Focus
1 For each of these statements, write the letter E if it is an expression
or write the letter F if it is a formula. The first two have been done
for you.
a 8h E b v = 9u F c 9u + 3
d m = 4 + n e G = 2x + y f b − c Tip
In part a,
2 Work out the value of x + 6 when:
x + 6 = 1 + 6 = 
a x = 1 b x = 2 c x = 3 d x = 4

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24 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.2 Using expressions and formulae

3 Work out the value of n − 1 when:

Tip
a n = 5 b n = 6 c n = 7 d n = 8
In part a,
4 Work out the value of the expression in each rectangle and match
it to its correct answer in the oval. The first one has been done for n − 1 = 5 − 1 = 
you: A and iii.

y+2 y–2 4+y 10 – y y+3

when y = 3 when y = 10 when y = 2 when y = 1 when y = 0
A B C D E

E
8 6 5 3 9
i ii iii iv v

5 Write down ‘True’ or ‘False’ for each of these statements.

If a statement is false, work out the correct value of the expression.

e
The value of

The value of
2
x
3
PL
The first one has been done for you.
a The value of 2m when m = 5 is 7. False; when m = 5, 2m = 10.
b The value of 3m when m = 8 is 24.
c The value of 9p when p = 2 is 11.
d
w
when w = 6 is 3.

when x = 12 is 6.

Tips
2m means 2 × m,
so 2 × 5 = 10.
w
2
means w ÷ 2.
M
Practice
6 Work out the value of each expression.
a a + 10 when a = 6 b b − 3 when b = 120
c c + z when c = 3 and z = 17 d d − y when d = 40 and y = 15
SA

f
e 3e when e = 20 f when f = 35
5
g g + 2x when g = 1 and x = 6 h h − 4w when h = 17 and w = 2
j
i 2i + 3v when i = 3 and v = 2 j + u when j = 30 and u = 3
2
24 p+q
k − 3 when k = 8 l when p = 11 and q = 22
k 3

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
25to publication.
 2 Expressions, formulae and equations

7 Jana uses this formula to work out how much money her friends
will collect from their sponsored walk.

My d =  d (m) × d  ($ r m)

How much money do the following friends collect?

a Miriam walks 5 kilometres at a sponsored rate of
$16 per kilometre.

E
b Yara walks 8 kilometres at a sponsored rate of
$18 per kilometre.
8 a Write a formula for the number of hours in any number of
days, using:
i words ii letters

10 a
b

b
in four days.

i PL
Use your formula in part a ii to work out the number of hours

Use the formula A = bh to work out A when:

b = 4 and h = 5 ii b = 3 and h = 12
Work out the value of b when A = 52 and h = 4.
Write a formula for the number of hours for any number of
minutes, in:
Tip
bh means b × h.
M
i words ii letters
b Use your formula in part a ii to work out the number of hours
in 360 minutes.
11 Hiroto uses this formula to work out the times it should take him
to travel from his house to any of his friends’ houses. Tip
D
SA

T= , where: T is the time, in hours D means D ÷ S.

S S
D is the distance, in kilometres
S is the average speed, in kilometres per hour.
How long does it take Hiroto to travel from his house to:
a Souta’s house, which is 60 kilometres away, at an average
speed of 20 kilometres per hour?
b Hina’s house, which is 140 kilometres away, at an average
speed of 40 kilometres per hour?

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26 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.2 Using expressions and formulae

Challenge
12 The weight of an object is calculated using the formula W = mg, where:
W is the weight, in newtons (N)
m is the mass, in kg
g is the acceleration due to gravity, in m/s2
On Earth g = 10 m/s2, and on the Moon g = 2 m/s2.
The mass of a man is 75 kg and the mass of a lunar landing module is
10 344 kg.

E
a Work out the weight of the:
i man on Earth ii lunar landing module on Earth.
b Work out the weight of the:
i man on the Moon ii lunar landing module on the Moon.

to give you the same answer?

x + 12 4x 6x − 8

PL
13 What value of x can you substitute into each of these expressions

14 Kwame uses the formula F = ma, where:

F is the force, in newtons (N)
m is the mass, in kg
a is the acceleration, in m/s2
M
He works out that F = 75 N when m = 25 kg.
What value of a did he use? Explain how you worked out your answer.
15 This is part of Kali’s homework. She has spilt tea on some of her work.

Qn:
SA

U  a M =

o k t   f M n:
i P = 51 d h = 17 i P = 65 d h = 13
Sn:
i M= =3 i M = =5

a What is the formula that Kali uses?

b Work out the value of M when P = 56 and h = 4.

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
27to publication.
 2 Expressions, formulae and equations

16 A cookery book shows how long it takes to cook a piece of meat.

Electric oven T = 70W Where:

T is the time, in minutes
Microwave oven T = 28W W is the weight of the
piece of meat, in kg.
a How much longer will it take to cook a 2 kg piece of meat in
an electric oven than in a microwave oven?
b A piece of meat takes 1 hour and 52 minutes to cook in a
microwave oven. How long would this same piece of meat take
to cook in an electric oven?

E
2.3 Collecting like terms Key words

Like terms are terms that

contain the same letter.

PL
The letter in an expression represents an unknown number but, if you
find it easier, you can think of the letter as an object.
You simplify an expression by collecting like terms.
Example:
2b + 3b = 5b
a + a = 2a
collecting like
terms
like terms
simplify
term

Tip
M
2a and 3a are like
terms. 2a and 3b
are not like terms.
Exercise 2.3 Think of a as an
apple, so
Focus
SA

1 Match the expression in each rectangle to its correct simplified Think of b as a

expression given in the oval. banana, so
The first one has been done for you: A and iii.

a + 4a a+a 3a + a a+a+a 5a + 4a 2a + 2a + 2a

A B C D E F
9a 3a 5a 6a 2a 4a Tip
i ii iii iv v vi Remember that a
means 1a.

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28 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.3 Collecting like terms

2 Look at these statements. Three of the statements are true and

three of the statements are false.
Write ‘true’ or ‘false’ for each statement. If the statement is false,
write the correct statement.
a b + 2b = 3b b 4d + 2d = 5d
c 6f + 4f = 11f d c + c + c = 3c
e h + 3h + 5h = 8h f 7v + 2v + v = 10v
3 Each of these expressions simplifies to 3s or 5s.
For each rectangle A to E, write down whether it simplifies to:

E
Tip
i 3s or ii  5s.
The first one has been done for you: A and ii. Think of s as a
strawberry. For
9s – 4s 5s – 2s 8s – 5s 6s – s 10s – 5s example, for A,
you start with

A B

PL C D

Look at these statements. Three of the statements are true and

three of the statements are false.
Write ‘true’ or ‘false’ for each statement. If the statement is false,
write the correct statement.
a 7b − 2b = 3b
c 6f − f = 6
b 4d − 2d = 2d
d 8c − 2c − 3c = 3c
E nine, you eat four,
so you have five
strawberries left.
M
e 9h − h − h = 7h f 10v − 6v − v = 4v
5 Look at these expressions. Some of the expressions can be
simplified and some cannot be simplified. Copy the expressions. Tip
If the expression can be simplified, write a tick and work out the You can simplify
simplified answer. only when
SA

If the expression cannot be simplified, write a cross. the letters are

For example: 4m + 2m ✓ 6m the same.
2a + 3b ✗
a 5p + p b 6p + 2 c 5n − 2w d 8u − u

Practice
6 Pedro has striped, checked and spotted bricks.
The length of a striped brick is x.
The length of a checked brick is y.
x y z
The length of a spotted brick is z.

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
29to publication.
 2 Expressions, formulae and equations

Work out the total length of each arrangement of bricks. Give each
answer in its simplest form.
a b
? ?

c d
? ?

e f
? ?

7 Simplify each expression.

E
a a + a + a + a b 4b + 3b c 4c + 7c
d 2d + 3d + 4d e 6e + 6e + e f 10f + f + 4f
g 9g − 3g h 4h2 −3h2 i 9i − i
j 8j + 2j − 4j k k + 6k − 3k l 12y3 − 4y3 − 7y3
8

x 7x
10x
PL
In an algebraic pyramid, you find the expression in each block by adding
the expressions in the two blocks below it.
Copy and complete these pyramids.
a

3x
b

4x
7x
15x
M
9 Simplify these expressions by collecting like terms.
a 3x + 4x + 5y b 5z + 5z + 5a + a
c 3a + 4b + 4a + 5b d 4x + 5 + 3x + 2
e d + 1 + d + 1 f 5f − 3f + 12g − 3g
g 45 − 15 + 12w − w h 7x + 5y − 3x + y
SA

i 8a + 6b − 4a − 5b j 4w + 3x + 7y − 2w − 3x + 13y
k 200a + 20g + 100 − 15g − 70
10 Write each expression in its simplest form.
a 4ab + 2ab + 3xy + 5xy b 3rd + 3rd + 5th + 6th
c 5tv + 6tv + 9jk − 5kj d 8ej + 7yh − 3je − 4hy
e 5v + 15rv − 2v + vr f 7un − 4nu + 11ef − 11fe

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30 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.3 Collecting like terms

Challenge
11 This is part of Maddi’s homework.

Qn
W  e n r t m.
a 2x + 8 + 7x − 4 b 5g + 4t − t + 2gr
Sn

E
a 2x + 8 = 10x, 7x − 4 = 3x, 10x + 3x = 13x
b 5g + 4t − t + 2gr = 5g + 4 + 2gr

PL
Maddi has made several mistakes. Explain the mistakes Maddi has made.
12 Copy and complete this algebraic pyramid.
Remember, you find the expression in each block by adding the
expressions in the two blocks below it.

8a + 6b
17a + 11b

5a + 2b
2a + b
M
13 Simplify these expressions. Write your answers in their simplest form.
3a a b 3b c
a − b + c 3c +
4 2 5 10 7
14 In this diagram, the expressions in each rectangle simplify to the
expression in the circle.
SA

Tip
1 Remember:
x × y = xy, so
4 3
2x × y = 2xy and
4x × 3y = 12xy.
2 8ab 2

3 4

a Copy and complete the diagram in two different ways.

b Is it possible to say how many ways there are to complete the
diagram? Explain your answer.

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
31to publication.
 2 Expressions, formulae and equations

15 In a magic square, the rows, columns and diagonals all add up to

the same number. Copy the magic squares in parts a and b.
a Write the numbers 1 to 9 in this magic square, so that all the
rows, columns and diagonals add up to 15.
You can use each number only once.
Three numbers have been written in the magic square for you.

E
9

b Write the algebraic expressions from the cloud in the magic
square, so that all the rows, columns and diagonals add up
to 3b.

a+b – c
b–c

b–a
b+c
PL
b–a – c

b+c – a
a+b

a+b+c
M

2.4 Expanding brackets Key words

brackets
SA

You can use a box method to multiply numbers together, like this: expand
4 × 16 = 4 × (10 + 6)

× 10 6 Tips
4 40 24 You can write
4 × (10 + 6) as
4 × 16 = 40 + 24 = 64 4(10 + 6).
You use the
table to expand
the brackets
4(10 + 6) to get
(4 × 10) + (4 × 6).

Original material
32 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.4 Expanding brackets

Exercise 2.4
Focus
1 Copy and complete the boxes to work out the answers.
a 5 × 13 b 2 × 38
× 10 3 × 30 8
5 2

E
5 × 13 =   +   =  2 × 38 =   +   = 
c 7 × 21 d 4 × 17
× 20 1 × 10 7

a 3 × x
7
7 × 21 =   + 
Simplify these expressions.
b 4 × p
 = 

PL

c
4
4 × 17 = 

9 × f
Copy and complete the boxes to simplify these expressions.
The first one has been done for you.
a 2(x + 3) b 3(x + 4)
 + 

d
 = 

5 × m
Tip
3 × x can be
written simply
as 3x.
M
× x 3 × x 4
2 2x 6 3
2(x + 3) = 2x + 6 3(x + 4) =   + 
c 5(m + 1) d 4(n + 2)
SA

× m 1 × n 2
5 4
5(m + 1) =   +  4(n + 2) =   + 
4 Copy and complete the boxes to simplify these expressions. The
first one has been done for you.
a 3(x − 2) b 5(x − 6)
× x −2 × x −6
3 3x −6 5
3(x − 2) = 3x − 6 5(x − 6) =   − 

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
33to publication.
 2 Expressions, formulae and equations

c 2(y − 4) d 6(k − 3)
× y −4 × k −3
2 6
2(y − 4) = − 6 (k − 3) = −

Practice
5 Expand the brackets.

E
a 3(a + 2) b 5(b + 3) c 3(c + 2) d 5(d − 1)
e 4(e − 9) f 3( f − 8) g 4(2 + f ) h 8(7 + z)
i 9(3 + y) j 4(4 − x) k 7(1 − w) l 7(2 − v)
6 Multiply out the brackets.

7
a

j
5(2p + 1)
d 11(3s − 4a + 7)
g 6(1 + 2v)
5(3 − 5x)
b
e
h
k
PL 7(3q + 2)
2(2t − 5)
8(6 + 4w − 3g)
5(4 − 3x)
This is part of Paul’s homework. Paul has made a mistake in
every solution.

Qn
c
f
i
l
9(2r + 3)
4(5u − 1)
10(6 + 7x)
5(5k − 8x − 6h)
M
My t  .
a 5(a + 3) b 3(4b − 5) c 4(3 − c)
Sn
a 5(a + 3) = 5a + 3
SA

b 3(4b − 5) = 12b − 8
c 4(3 − c) = 12 − 4c = 8c

Explain the mistakes Paul has made.

Tip
8 Which one of these expressions is the odd one out?
Explain your answer. The ‘odd one
out’ means the
2(9x + 12) 2(10x + 8) 6(4 + 3x) 3(8 + 6x) 1(18x + 24) expression that is
different from all
of the others.

Original material
34 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.5 Constructing and solving equations

Challenge
9 The diagram shows a rectangle.
2x – 3y cm
The width of the rectangle is 2x − 3y cm.
The length of the rectangle is 12 cm. 12 cm
Write an expression, in its simplest form, for the:
a area of the rectangle b perimeter of the rectangle.
10 Expand and simplify these expressions.
a 3(x + 2) + 4x b 4(9 + x) − 24

E
c 5(2x − 2) + x + 17 d 6(3x − 4) −8x + 4
e 4(x + 4) + 7(x + 1) f 8(5 + 2x) + 3(x − 6) Tip
11 Show that 4(2x + 7) + 3(6x − 5) ≡ 13(2x + 1).
≡ means ‘is
12 Work out the missing numbers in these expansions. equivalent to’ or
a 9(3x + 2 ) = 3

a
c
x+

2x +
(

(6 y − 10 ) =
= 8x + PL
y − 80
)
b 5(8 − 6 z ) = 10
13 Work out the missing numbers in these expansions.
All the numbers are in the cloud. Only use each number once.
( ) b
d
5(
7(
x−
y+
)
(

)=
=
−

x − 35
y + 42
z ) ‘is the same as’.

7
2

15
8
3
9
36
4 6
14
48
M
2.5 Constructing and solving equations
When you solve an equation, you find the value of the unknown letter.
Key words
You can use a flow chart like this to solve an equation using
inverse operations. inverse operation
SA

Solve: x + 5 = 12 x +55 12 solve

So x=7 7 −55 12

Tip
Exercise 2.5 Reverse the flow
chart to work out
Focus the value of x.

1 Write down the missing numbers.

a +4=6 b +1=6 c + 2 = 10
d 8− =5 e 9− =2 f 17 − = 10

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
35to publication.
 2 Expressions, formulae and equations

2 Copy and complete these flow charts to work out the value of x.

a x+3=7 x +3 7 b x +1 = 9 x +1 9

x= −3 7 x= −1 9

c x + 6 = 11 x +6 11 d x + 2 = 13 x +2 13

x= − 11 x= 13

e x + 4 = 12 x +4 12 f x + 9 = 15 x

E
x= x=
3 Copy and complete these flow charts to work out the value of x.

e
x−1=5

x=

x−2=9

x=

x − 10 = 8
x

x
PL
−1

+1

−2

− 10
5

8
b

f
x−3=8

x=

x − 5 = 12

x=

x−4=5
x

x
−3

+3

−5
8

12

12
M
x= x= 12
4 Copy and complete these flow charts to work out the value of y.

a 2y = 6 y ×2 6 b 4y = 8 y ×4 8
SA

y= ÷2 6 y= ÷4 8

Practice
5 Solve each of these equations. Check your answers by substituting your answer back
into the original equation.
a x+2=6 b x+6=9 c 4 + x = 11 d 15 + x = 21
e x − 5 = 10 f x−4=6 g x − 15 = 12 h 5x = 20
i 3x = 30 j 4x = 28
6 Solve each of these equations. Check your answers by substituting your answer back
into the original equation.
a 14 = x + 3 b 9=x+5 c 12 = x − 6
d 20 = x − 5 e 14 = 2x f 50 = 10x

Original material
36 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.5 Constructing and solving equations

7 Write an equation for each of these statements. Solve each equation

to find the value of the unknown number.
a b

I think of a number I think of a number

and then add 5. My and then subtract
answer is 21. What 5. My answer is 21.
is the number I first What is the number I

E
thought of? first thought of?

8 Write an equation for each of these statements. Solve each equation

to find the value of the unknown number.
a I think of a number and add 14. The answer is 20.

a
c

Challenge
2a + 4 = 18
30 = 7b + 9
PL
b I think of a number and subtract 17. The answer is 20.
c I think of a number and multiply it by 5. The answer is 20.
Solve each of these equations. Check your answers by substituting
your answer back into the original equation.
b 5a − 2 = 18
d 18 = 6b −12
M
10 This is part of Sofia’s homework.

Qn:
T gm  a gt g d o
SA

o r g, 3x d 2x.

3x
Wk t  ize f  g n  gm. 2x

Sn:
En 3x + 2x = 90°
Sy 5x = 90°
Se x = 90 = 18°
5
A 3x g  3 × 18 = 54°.
2x g  2 × 18 = 36°.
Ck 54 + 36 = 90° ✓

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
37to publication.
 2 Expressions, formulae and equations

For each of these diagrams:

i Write an equation involving the angles.
ii Simplify your equation by collecting like terms.
iii Solve your equation to find the value of x.
iv Work out the sizes of the angles in the diagram.
v Check that your answers are correct by substituting your
answer back into the original equation.
Tip
a b
What do the

E
6x 4x angles on a
5x
4x straight line and
in a triangle add
c d up to?
5x

a
c 3z + 6 = −30
3x 4x 2x

12 Zara solves these equations:

2a − 11 = −77
PL
11 Solve each of these equations and check your answers.
x + 12 = −6

4 = b + 16
b y − 9 = −4
d 4w − 3 = −27
3x

I think that
4x
M
a×b < c+d
6c + 7 = 25 −12 = 3d + 9
Is Zara correct? Show all your working and
explain your answer.
13 Use the formula w = 2x + y − 3z to work out:
a w when x = 8, y = −3 and z = 7
SA

b x when w = 15, y = 9 and z = −4

c y when w = −10, x = 6 and z = 2
d z when w = 20, x = 15 and y = 8

Original material
38 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.6 Inequalities

2.6 Inequalities
Remember: < means ‘is less than’
Key words
> means ‘is greater than’.
inequality
integer
Exercise 2.6

E
Focus
1 Write ‘true’ or ‘false’ for each of these statements. The first one has
been done for you.
a x < 9 means ‘x is less than 9’. True

A
B
x > 4
x < 4
PL
b x > 3 means ‘x is less than 3’.
c x < 2 means ‘x is greater than 2’.
d x > 7 means ‘x is greater than 7’.
Match each inequality (A to D) with its correct meaning (i to iv).
The first one has been done for you: A and ii.
i
ii
x is less than 12.
x is greater than 4.
M
C x > 12 iii x is less than 4.
D x < 12 iv x is greater than 12.

3 Write these statements as inequalities. The first one has been done for you.
a x is greater than 2.   x > 2 b y is greater than 5.
SA

c m is less than 15. d b is less than 7.

4 Match each inequality (A to D) with its correct number line
Tip
(i to iv). The first one has been done for you: A and iv.
Remember: You
A x > −5 i
use an open circle
3 4 5 6 7 (o) for the < and
B x < 5 ii > inequalities.
–9 –8 –7 –6 –5 –4
C x < −5 iii
3 4 5 6 7
D x > 5 iv
–6 –5 –4 –3 –2 –1

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
39to publication.
 2 Expressions, formulae and equations

5 a Copy this number line.

12 13 14 15 16 17
Show the inequality y < 15 on the number line.

b Copy this number line. 12 13 14 15 16 17

Show the inequality y > 15 on the number line.

c Write down the inequality shown on

–4 –3 –2 –1 0 1

E
this number line. Use the letter x.

d Write down the inequality shown on

–5 –4 –3 –2 –1 0
this number line. Use the letter x.

Practice
6
a

c
x > 8

x < −3
7

–5
PL
Copy each number line and show each inequality on the number line.
8

–4
9

–3
10

–2
11

–1
b

d
x < 6

x > 0
3

–1
4

0
5

1
6

2
7

3
M
7 Write down the inequality shown on these number lines. Use the letter x.

a b
12 13 14 15 16 30 31 32 33 34
SA

c d
–9 –8 –7 –6 –5 –4 –3 –2 –1 0
8 Write down if A, B or C is the correct answer to each of the following.
a For the inequality h > 5, the smallest integer that h could be is:
A 4 B 5 C 6
b For the inequality j > −7, the smallest integer that j could be is:
A −8 B −7 C −6
c For the inequality k < 12, the largest integer that k could be is:
A 11 B 12 C 13
d For the inequality m < −1, the largest integer that m could be is:
A −3 B −2 C −1

Original material
40 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 2.6 Inequalities

9 Zara looks at the inequality z < −5.

A list of the integer

values that z could be is:
-4, -3, -2, -1, 0, 1, …

E
Is Zara correct? Explain your answer.
10 For each of these inequalities, write down:
i the smallest integer that p could be
ii a list of the integer values that p could be.
a p > 8

Challenge
PL
b p > −3
11 For each of these inequalities, write down:
i the largest integer that q could be
ii a list of the integer values that q could be.
a q < −1 b q < 16
c

c
p > 4.7

q < 3.9
M
12 Copy each number line and show each inequality on the number line.
a x > 1.5 b x < 3.75
0 1 2 3 4 1 2 3 4 5

c y > 4.6 d y < 8.25

3 4 5 6 5 6 7 8 9
SA

13 Write down the inequality shown on these number lines. Use the letter y.

a b
–2 –1 0 1 2 10 11 12 13

c d
2 3 4 5 25 26 27 28 29

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
41to publication.
 2 Expressions, formulae and equations

14 Arun looks at this number line: 

–5 –4 –3 –2 –1

The number line

shows the inequality
x > -4.5.

E

Explain the mistake that Arun has made.
15 Write down the inequality shown on these number lines. Use the letter w.

a b

a
–8 –7 –6

–3
PL
–5

–2
–4

–1
–3
16 Copy each number line and show each inequality on the number line.
y < −1.5 
0 1
b
–13

y > −4.4 
–12 –11

–5
–10

–4 –3 –2
M
SA

Original material
42 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 3 Place value
and rounding

E
3.1 Multiplying and dividing by
powers of 10
Exercise 3.1
Focus
1
PL
Each oval card has the same value as a square card.

A 1000 B 100 C 100 000 D 10 000 000

Key words
power
powers of 10

E 10 000
M
i 105 ii 107 iii 103 iv 104 v 102

Copy and complete the list of pairs of cards with the same value.
The first one has been done for you.
Tip
SA

A and iii because 1000 = 103.

B and because 100 = The number of zeros after the 1 is
C and because 100 000 = the same as the power of the 10.
D and because 10 000 000 = 3 zeros after the 1 = power of 3
E and because 10 000 =
2 Copy and complete the following.
Tip
a 6 × 104 = 6 × 10 000 = 60 000
b 9 × 104 = 9 × = Remember:
104 = 10 000
c 3 × 104 = 3 × =

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
43to publication.
 3 Place value and rounding

3 Copy and complete the following.

Tip
a 2 × 105 = 2 × 100 000 = 200 000
b 7 × 105 = 7 × = Remember:
105 = 100 000
c 5 × 105 = 5 × =
4 Write whether A, B or C is the correct answer for each of these.
a 8 × 103 = A 800 B 8000 C 80 000
b 4 × 10 =6
A 4 000 000 B 400 000 C 40 000 000
c 3 × 10 =
8
A 3 000 000 B 30 000 000 C 300 000 000

E
5 Rafa uses this method to work out 6 000 000 ÷ 105.

105  5 z, o I l  5 z f  r

6 000 000.

6
a 800 000 ÷ 105
d 90 000 ÷ 104 e
PL
60 0 0 0 0 0, h g  n r f 60.

Use Rafa’s method to work out:

b 2 000 000 ÷ 105
3 000 000 ÷ 106
Write whether A, B or C is the correct answer.
c
f
400 000 ÷ 104
500 000 000 ÷ 106
M
a 50 000 ÷ 103 = A 50 B 5 C 500
b 710 000 ÷ 10 = 4
A 7100 B 710 C 71
c 89 000 000 ÷ 10 =
6
A 890 B 89 C 8900
d 470 000 000 ÷ 105 = A 47 B 470 C 4700
SA

Practice
7 Work out:
a 56 × 102 b 877 × 104 c 13 × 106
d 6.5 × 104 e 33.2 × 103 f 0.65 × 106
8 Copy and complete these calculations.
a 3.7 × 104 = b 34.6 × = 34 600
c × 10 = 8 900 000
6
d 78.34 × 10 = 783 400 000
9 Work out:
a 9000 ÷ 103 b 520 000 ÷ 104 c 8 000 000 ÷ 105

Original material
44 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 3.1 Multiplying and dividing by powers of 10

10 Copy and complete this table.

÷ 102 ÷ 103 ÷ 104 ÷ 105 ÷ 106

400 000 4000
56 000 5.6
3000 0.03
720 0.72

11 Write down whether A, B or C is the correct answer.

E
a 240 000 ÷ 105 A 24 B 2.4 C 0.24
b 7020 ÷ 10 6
A 0.00702 B 0.0702 C 0.702
c 8 700 000 ÷ 108
A 87 B 0.87 C 0.087 Tip
12 Arun thinks of a starting number. He multiplies his number by 103,

PL
then divides the answer by 105. He multiplies this answer by 106 and
finally divides this answer by 102.
Arun thinks that a quicker method would be
to just multiply his starting number by 10.
Is Arun correct?
Show your working and explain your answer.

Challenge
Try different
starting numbers
and work out
the answers.
M
13 These formulae show how to convert
between different metric units of mass. Tip
Number of milligrams = number of grams × 103 The units are:
SA

milligram (mg)
Number of milligrams = number of kilograms × 106
gram (g)
Number of milligrams = number of tonnes × 109 kilogram (kg)
tonne (t)
Use the formulae to work out the missing numbers in
these conversions.
a mg = 28 g b mg = 0.75 g
c mg = 2 kg d mg = 0.083 kg
e mg = 53 t f mg = 0.0025 t

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
45to publication.
 3 Place value and rounding

14 a Use the formulae in Question 13 to complete the

following formulae.

Number of grams = number of milligrams ÷ 103

…
Number of kilograms = number of milligrams ÷ 10

…
Number of tonnes = number of milligrams ÷ 10

E
b Use your formulae in part a to work out the missing numbers
in these conversions.

i g = 45 000 mg
ii kg = 7 600 000 mg
iii

Object
Moon
space station
PL
t = 65 700 000 mg
15 The table shows the distances from Earth to different objects in
space. The distances are shown as a decimal number multiplied by
a power of 10.

Distance from Earth (km)

3.844 × 105
4.08 × 102
M
Venus 4.14 × 107
Neptune 4.35 × 109
weather satellite 3.6 × 104
Jupiter 6.287 × 108
SA

Aki wants to write the objects in order, from the closest to Earth to
the farthest from Earth.
a Without doing any calculations, write down what you think
Aki’s list will be. Explain how you made your decisions.
b Work out the multiplications and then compare the distances.
Was your list in part a correct? If it was incorrect, write down
the correct list.

Original material
46 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.
 3.2  Rounding

16 In this diagram, the calculations in

0.078 3 103
each rectangle simplify to the
answer given in the circle.
4 10 7800 4 102
Copy and complete the
diagram with a different
3 10 78 3 10
calculation in each box.
Two have been done
for you. 4 10 4 10

3 10

E
17 Work out the following.
a (3.6 ×10 4 ) + 15 000 × (2 310 000 ÷ 106)
b 4  Tip
3
c (0.005 × 103)2 d (256 000 000 ÷ 107) − (0.000049 × 105) Remember to use

3
 .2  Rounding
Exercise 3.2
Focus
PL the correct order
of operations.

Key words
degree of
accuracy
M
round
1 Round each of these numbers to one decimal place. The first two
have been done for you.
a 4.53 = 4.5 (1 d.p.) b 3.69  = 3.7 (1 d.p.) c 8.82
d 7.24 e 2.37 f 4.09
SA

2 Round each of these numbers to one decimal place. Write whether

A or B is the correct answer.
a 4.671 = A 4.6 B 4.7
b 9.055 = A 9.0 B 9.1
c 3.733 = A 3.7 B 3.8
d 6.915 = A 6.9 B 7.0
e 0.858 = A 0.8 B 0.9
3 Round each of these numbers to two decimal places. The first two
have been done for you.
a 2.473 = 2.47 (2 d.p.) b 8.659  = 8.66 (2 d.p.) c 3.314
d 8.065 e 1.938 f 2.422

Original material © Cambridge University Press 2019. This material is not final and is subject to further changes prior
47to publication.
 3 Place value and rounding

4 Round each of these numbers to two decimal places. Write whether

A or B is the correct answer.
a 0.6651 = A 0.66 B 0.67
b 2.3015 = A 2.30 B 2.31
c 8.5544 = A 8.55 B 8.56
d 0.0593 = A 0.05 B 0.06
e 5.6058 = A 5.60 B 5.61

Practice

E
5 Round each of these numbers to two decimal places (2 d.p.). The first one
has been done for you.
a 4.983 = 4.98 (2 d.p.) b 9.037 c 24.332
d 128.641 e 0.66582 f 0.03174
6

7
a 7.2845 = 7.285 (3 d.p.)
d 0.67893

has been done for you.

PL
Round each of these numbers to three decimal places (3 d.p.). The first one
has been done for you.

a 3.882615 = 3.8826 (4 d.p.)
b 65.8823
e 300.00442

b 61.89022
c 134.9028
f 0.0085411
Round each of these numbers to four decimal places (4 d.p.). The first one

c 143.56228
M
d 200.006789 e 300.000555 f 18.25252525
8 Write whether A or B is the correct answer to each of the following.
a 34.9892 to 1 d.p. = A 35 B 35.0
b 7.4955 to 2 d.p. = A 7.50 B 7.5
c 0.009666 to 3 d.p. = A 0.010 B 0.01
SA

9 A red blood cell has a length of 0.0065982 millimetres. Write down the length,
correct to five decimal places.
10 Use a calculator to work out the answers to the following. Round each of your
answers to the given degree of accuracy.
11
a 19 ÷ 11 (2 d.p.) b 12 − ( 4 d.p.) c 89 + 26 (3 d.p.)
13

Original material
48 © Cambridge University Press 2019. This material is not final and is subject to further changes prior to publication.