gcse maths - 2tier-foundation for edexcel a (collins, 2006)

BRIAN SPEED
KEITH GORDON
KEVIN EVANS
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This does not mean that the contents will be used verbatim when setting examinations nor is it to be
read as being the official specification – a copy of which is available at www.edexcel.org.uk
This book provides indicators of the equivalent grade level of maths questions throughout. The publishers
wish to make clear that these grade indicators have been provided by Collins Education, and are not the
responsibility of Edexcel Ltd. Whilst every effort has been made to assure their accuracy, they should be
regarded as indicators, and are not binding or definitive.
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Edex_Found Math_00.qxd 17/03/06 10:25 Page ii
iii
CONTENTS
Chapter 1 Basic number 1
Chapter 2 Fractions 25
Chapter 3 Negative numbers 49
Chapter 4 More about number 67
Chapter 5 Perimeter and area 95
Chapter 6 Statistical representation 119
Chapter 7 Basic algebra 143
Chapter 8 Further number skills 167

Chapter 9 Ratios, fractions, speed and proportion 195
Chapter 10 Symmetry 213
Chapter 11 Averages 225
Chapter 12 Percentages 257
Chapter 13 Equations and inequalities 275
Chapter 14 Graphs 299
Chapter 15 Angles 327
Chapter 16 Circles 355
Chapter 17 Scale and drawing 373
Chapter 18 Probability 397
Chapter 19 Transformations 427
Chapter 20 Constructions 453
Chapter 21 Units 473
Chapter 22 Pie charts, scatter diagrams and surveys 483
Chapter 23 Pattern 505
Chapter 24 Surface area and volume of 3-D shapes 523
Chapter 25 Quadratic graphs 539
Chapter 26 Pythagoras’ theorem 549
Answers 563
Index 599
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Welcome to Collins GCSE Maths, the easiest way to learn and succeed in
Mathematics. This textbook uses a stimulating approach that really appeals to
students. Here are some of the key features of the textbook, to explain why.
Each chapter of the textbook begins with an Overview. The
Overview lists the Sections you will encounter in the chapter,
the key ideas you will learn, and shows how these ideas relate
to, and build upon, each other. The Overview also highlights
what you should already know, and if you’re not sure, there is
a short Quick Check activity to test yourself and recap.

Maths can be useful to us every day of
our lives, so look out for these
Really
Useful Maths!
pages. These double page
spreads use big, bright illustrations to
depict real-life situations, and present a
short series of real-world problems for
you to practice your latest mathematical
skills on.
Each
Section begins first by explaining
what mathematical ideas you are
aiming to learn, and then lists the key
words you will meet and use. The ideas
are clearly explained, and this is
followed by several examples showing
how they can be applied to real
problems. Then it’s your turn to work
through the exercises and improve your
skills. Notice the different coloured
panels along the outside of the exercise
pages. These show the equivalent exam
grade of the questions you are working
on, so you can always tell how well
you are doing.
iv
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Working through these sections in the right way should mean you achieve your very best
in GCSE Maths. Remember though, if you get stuck, answers to all the questions are at

the back of the book (except the exam question answers which your teacher has).
We do hope you enjoy using Collins GCSE Maths, and wish you every good luck in
your studies!
Brian Speed, Keith Gordon, Kevin Evans
Review the Grade Yourself pages at the
very end of the chapter. This will show
what exam grade you are currently
working at. Doublecheck
What you
should now know
to confirm that you
have the knowledge you need to
progress.
Every chapter in this textbook contains
lots of
Exam Questions. These provide
ideal preparation for your examinations.
Each exam question section also
concludes with a fully worked example.
Compare this with your own work, and
pay special attention to the examiner’s
comments, which will ensure you
understand how to score maximum
marks.
Throughout the textbook you will find
Puzzles and Activities – highlighted in
the green panels – designed to
challenge your thinking and improve
your understanding.
v

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You may use your calculator for this question
You should not use your calculator for this question
Indicates a Using and Applying Mathematics question
Indicates a Proof question
ICONS
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1
This chapter will show you …
● how to use basic number skills without a calculator
Visual overview
What you should already know
● Times tables up to 10 × 10
● Addition and subtraction of numbers less than 20
● Simple multiplication and division
● How to multiply numbers by 10 and 100
Quick check
How quickly can you complete these?
1 4 × 6 2 3 × 7 3 5 × 8 4 9 × 2
5 6 × 7 6 13 + 14 7 15 + 15 8 18 – 12
9 19 – 7 10 11 – 6 11 50 ÷ 5 12 48 ÷ 6
13 35 ÷ 7 14 42 ÷ 6 15 36 ÷ 9 16 8 × 10
17 9 × 100 18 3 × 10 19 14 × 100 20 17 × 10
1 Adding with
grids
2 Times table
check
3 Order of
operations and
BODMAS

4 Place value
and ordering
numbers
5 Rounding
6 Adding and
subtracting
numbers with
up to four digits
7 Multiplying and
dividing by
single-digit
numbers
BODMAS
In grids
Up to four digits
Rounding off
Single-digit numbers
Addition
Subtraction
Place value
Multiplication
Division
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2
Adding with grids
1.1
Key words
add
column

grid
row
In this section you will learn how to:
● add and subtract single-digit numbers in a grid
● use row and column totals to find missing numbers
in a grid
Adding with grids
You need a set of cards marked 0 to 9.
Shuffle the cards and lay them out in a 3 by 3 grid.
You will have one card left over.
Copy your grid onto a piece of paper. Then add up
each row and each column and write down their totals.
Finally, find the grand total and write it in the box at the
bottom right.
Look out for things that help. For example:
•
in the first column, 3 + 7 make 10 and 10 + 8 = 18
•
in the last column, 9 + 4 = 9 + 1 + 3 = 10 + 3 = 13
Reshuffle the cards, lay them out again and copy the new grid. Copy the new grid
again on a fresh sheet of paper, leaving out some of the numbers.
Pass this last grid to a friend to work out the missing numbers. You can make it
quite hard because you are using only the numbers from 0 to 9. Remember:
once a number has been used, it cannot be used again in that grid.
Example Find the numbers missing from this grid.
8
19
2
3
9

17
17
11
719
620
4
0
9
13
5
2
1
8
8
6
7
21
17
8
17
42
4
9
2
8
8
7
21
17
8

42
854
3
7
8
18
5
6
2
13
0
4
9
13
8
17
19
44
928
467
053
9876543210
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Find the row and column totals for each of these grids.
Find the numbers missing from each of these grids. Remember: the numbers missing from each grid
must be chosen from 0 to 9 without any repeats.
ab c
de f
gh i

1
5
6
7
3
14
6
2
16
16
9
11
36
1
7
14
5
8
15
3
4
16
6
15
24
35
9
4
14
3

2
5
5
8
19
18
9
11
38
2
8
19
5
13
4
0
10
16
13
13
42
1
12
2
9
4
3
15
16
12

0
9
17
2
45
13
3
17
42
8
15
2
4
10
15
36
9
18
7
6
3
4
12
4
20
2
5
11
1
6

8
17
17
13
38
abc1
9
6
3
2
5
7
8
4
0
8
9
6
1
5
7
4
3
0
1
9
8
6
3
7

2
4
de f2
3
8
4
5
9
6
7
1
5
6
2
9
1
7
3
8
4
0
7
1
8
2
6
3
4
5
gh i9

7
1
4
0
6
8
5
3
0
7
5
8
1
9
6
4
2
1
6
0
8
2
9
7
5
3
CHAPTER 1: BASIC NUMBER
3
Clues The two numbers missing from the second column must add up to 1, so they
must be 0 and 1. The two numbers missing from the first column add to 11, so they

could be 7 and 4 or 6 and 5. Now, 6 or 5 won’t work with 0 or 1 to give 17 across
the top row. That means it has to be:
You can use your cards to try out your ideas.
7
4
8
19
1
2
0
3
9
17
17
11
7
4
8
19
1
2
0
3
9
5
3
17
17
11
11

39
giving as the answer.
EXERCISE 1A
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4
Times table check
1.2
In this section you will:
● recall and use your knowledge of times tables
Special table facts
You need a sheet of squared paper.
Start by writing in the easy tables. These are the 1 ×, 2 ×, 5 ×, 10 × and 9 × tables.
Now draw up a 10 by 10 tables square
before you go any further. (Time yourself
doing this and see if you can get faster.)
Once you have filled it in, shade in all
the easy tables. You should be left with
something like the square on the right.
Now cross out one of each pair that
have the same answer, such as
3 × 4 and 4 × 3. This leaves you with:
Now there are just 15 table facts. Do learn them.
The rest are easy tables, so you should know all of them. But keep practising!
×
1
2
3
4
5

6
7
8
9
10
12345678910
24 32
21 28
18 24
12 16
9
48 56
42 49
36
64
×
1
2
3
4
5
6
7
8
9
10
12345678910
24 32 48 56 64
21 28 42 49 56
18 24 36 42 48

12 16 24 28 32
912 182124
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Write down the answer to each of the following without looking at the multiplication square.
a 4 × 5 b 7 × 3 c 6 × 4 d 3 × 5 e 8 × 2
f 3 × 4 g 5 × 2 h 6 × 7 i 3 × 8 j 9 × 2
k 5 × 6 l 4 × 7 m 3 × 6 n 8 × 7 o 5 × 5
p 5 × 9 q 3 × 9 r 6 × 5 s 7 × 7 t 4 × 6
u 6 × 6 v 7 × 5 w 4 × 8 x 4 × 9 y 6 × 8
Write down the answer to each of the following without looking at the multiplication square.
a 10 ÷ 2 b 28 ÷ 7 c 36 ÷ 6 d 30 ÷ 5 e 15 ÷ 3
f 20 ÷ 5 g 21 ÷ 3 h 24 ÷ 4 i 16 ÷ 8 j 12 ÷ 4
k 42 ÷ 6 l 24 ÷ 3 m 18 ÷ 2 n 25 ÷ 5 o 48 ÷ 6
p 36 ÷ 4 q 32 ÷ 8 r 35 ÷ 5 s 49 ÷ 7 t 27 ÷ 3
u 45 ÷ 9 v 16 ÷ 4 w 40 ÷ 8 x 63 ÷ 9 y 54 ÷ 9
Write down the answer to each of the following. Look carefully at the signs, because they are
a mixture of ×, +, – and ÷ .
a 5 + 7 b 20 – 5 c 3 × 7 d 5 + 8 e 24 ÷ 3
f 15 – 8 g 6 + 8 h 27 ÷ 9 i 6 × 5 j 36 ÷ 6
k 7 × 5 l 15 ÷ 3 m 24 – 8 n 28 ÷ 4 o 7 + 9
p 9 + 6 q 36 – 9 r 30 ÷ 5 s 8 + 7 t 4 × 6
u 8 × 5 v 42 ÷ 7 w 8 + 9 x 9 × 8 y 54 – 8
Write down the answer to each of the following.
a 3 × 10 b 5 × 10 c 8 × 10 d 10 × 10 e 12 × 10
f 18 × 10 g 24 × 10 h 4 × 100 i 7 × 100 j 9 × 100
k 10 × 100 l 14 × 100 m 24 × 100 n 72 × 100 o 100 × 100
p 20 ÷ 10 q 70 ÷ 10 r 90 ÷ 10 s 170 ÷ 10 t 300 ÷ 10
u 300 ÷ 100 v 800 ÷ 100 w 1200 ÷ 100 x 2900 ÷ 100 y 5000 ÷ 100
CHAPTER 1: BASIC NUMBER

5
EXERCISE 1B
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Suppose you have to work out the answer to 4 + 5 × 2. You may say the answer is 18, but the correct
answer is 14.
There is an order of operations which you must follow when working out calculations like this. The × is
always done before the +.
In 4 + 5 × 2 this gives 4 + 10 = 14.
Now suppose you have to work out the answer to (3 + 2) × (9 – 5). The correct answer is 20.
You have probably realised that the parts in the brackets have to be done first, giving 5 × 4 = 20.
So, how do you work out a problem such as 9 ÷ 3 + 4 × 2?
To answer questions like this, you must follow the
BODMAS rule. This tells you the sequence in which you
must do the operations.
B Brackets
O Order (powers)
D Division
M Multiplication
A Addition
S Subtraction
For example, to work out 9 ÷ 3 + 4 × 2:
First divide: 9 ÷ 3 = 3 giving 3 + 4 × 2
Then multiply: 4 × 2 = 8 giving 3 + 8
Then add: 3 + 8 = 11
And to work out 60 – 5 × 3
2
+ (4 × 2):
First, work out the brackets: (4 × 2) = 8 giving 60 – 5 × 3
2

+ 8
Then the order (power): 3
2
= 9 giving 60 – 5 × 9 + 8
Then multiply: 5 × 9 = 45 giving 60 – 45 + 8
Then add: 60 + 8 = 68 giving 68 – 45
Finally, subtract: 68 – 45 = 23
6
Order of operations and
BODMAS
1.3
Key words
brackets
operation
sequence
In this section you will learn how to:
● work out the answers to a problem with a number
of different signs
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CHAPTER 1: BASIC NUMBER
7
Dice with BODMAS
You need a sheet of squared paper and three dice.
Draw a 5 by 5 grid and write the numbers from 1 to 25
in the spaces.
The numbers can be in
any order
.
Now throw three dice. Record the score on

each one.
Use these numbers to make up a number
problem.
You must use all three numbers, and you must not put them together to make a
number like 136. For example, with 1, 3 and 6 you could make:
1 + 3 + 6 = 10 3 × 6 + 1 = 19 (1 + 3) × 6 = 24
6 ÷ 3 + 1 = 3 6 + 3 – 1 = 8 6 ÷ (3 × 1) = 2
and so on. Remember to use BODMAS.
You have to make only one problem with each set of numbers.
When you have made a problem, cross the answer off on the grid and throw the dice
again. Make up a problem with the next three numbers and cross that answer off the
grid. Throw the dice again and so on.
The first person to make a line of five numbers across, down or diagonally is the
winner.
You must write down each problem and its answer so that they can be checked.
Just put a line through each number on the grid, as you use it. Do not cross it out so
that it cannot be read, otherwise your problem and its answer cannot be checked.
This might be a typical game.
First set (1, 3, 6) 6 × 3 × 1 = 18
Second set (2, 4, 4) 4 × 4 – 2 = 14
Third set (3, 5, 1) (3 – 1) × 5 = 10
Fourth set (3, 3, 4) (3 + 3) × 4 = 24
Fifth set (1, 2, 6) 6 × 2 – 1 = 11
Sixth set (5, 4, 6) (6 + 4) ÷ 5 = 2
Seventh set (4, 4, 2) 2 – (4 ÷ 4) = 1
14 13 18 7 24
15 1 16 17 6
23 8 2 12 5
3 22 4 10 19
25 21 9 20 11

14 13 18 7 24
15 1 16 17 6
23 8 2 12 5
3 22 4 10 19
25 21 9 20 11
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Work out each of these.
a 2 × 3 + 5 = b 6 ÷ 3 + 4 = c 5 + 7 – 2 =
d 4 × 6 ÷ 2 = e 2 × 8 – 5 = f 3 × 4 + 1 =
g 3 × 4 – 1 = h 3 × 4 ÷ 1 = i 12 ÷ 2 + 6 =
j 12 ÷ 6 + 2 = k 3 + 5 × 2 = l 12 – 3 × 3 =
Work out each of the following. Remember: first work out the bracket.
a 2 × (3 + 5) = b 6 ÷ (2 + 1) = c (5 + 7) – 2 =
d 5 + (7 – 2) = e 3 × (4 ÷ 2) = f 3 × (4 + 2) =
g 2 × (8 – 5) = h 3 × (4 + 1) = i 3 × (4 – 1) =
j 3 × (4 ÷ 1) = k 12 ÷ (2 + 2) = l (12 ÷ 2) + 2 =
Copy each of these and put a loop round the part that you do first. Then work out the answer.
The first one has been done for you.
a 3 × 3 – 2 = 7 b 3 + 2 × 4 = c 9 ÷ 3 – 2 =
d 9 – 4 ÷ 2 = e 5 × 2 + 3 = f 5 + 2 × 3 =
g 10 ÷ 5 – 2 = h 10 – 4 ÷ 2 = i 4 × 6 – 7 =
j 7 + 4 × 6 = k 6 ÷ 3 + 7 = l 7 + 6 ÷ 2 =
Work out each of these.
a 6 × 6 + 2 = b 6 × (6 + 2) = c 6 ÷ 6 + 2 =
d 12 ÷ (4 + 2) = e 12 ÷ 4 + 2 = f 2 × (3 + 4) =
g 2 × 3 + 4 = h 2 × (4 – 3) = i 2 × 4 – 3 =
j 17 + 5 – 3 = k 17 – 5 + 3 = l 17 – 5 × 3 =
m 3 × 5 + 5 = n 6 × 2 + 7 = o 6 × (2 + 7) =
p 12 ÷ 3 + 3 = q 12 ÷ (3 + 3) = r 14 – 7 × 1 =

s (14 – 7) × 1 = t 2 + 6 × 6 = u (2 + 5) × 6 =
v 12 – 6 ÷ 3 = w (12 – 6) ÷ 3 = x 15 – (5 × 1) =
y (15 – 5) × 1 = z 8 × 9 ÷ 3 =
Copy each of these and then put in brackets where necessary to make each answer true.
a 3 × 4 + 1 = 15 b 6 ÷ 2 + 1 = 4 c 6 ÷ 2 + 1 = 2
d 4 + 4 ÷ 4 = 5 e 4 + 4 ÷ 4 = 2 f 16 – 4 ÷ 3 = 4
g 3 × 4 + 1 = 13 h 16 – 6 ÷ 3 = 14 i 20 – 10 ÷ 2 = 5
j 20 – 10 ÷ 2 = 15 k 3 × 5 + 5 = 30 l 6 × 4 + 2 = 36
m 15 – 5 × 2 = 20 n 4 × 7 – 2 = 20 o 12 ÷ 3 + 3 = 2
p 12 ÷ 3 + 3 = 7 q 24 ÷ 8 – 2 = 1 r 24 ÷ 8 – 2 = 4
CHAPTER 1: BASIC NUMBER
8
EXERCISE 1C
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Three dice are thrown. They give scores of 3, 1 and 4.
A class makes the following questions with the numbers.
Work them out.
a 3 + 4 + 1 = b 3 + 4 – 1 = c 4 + 3 – 1 =
d 4 × 3 + 1 = e 4 × 3 – 1 = f (4 – 1) × 3 =
g 4 × 3 × 1 = h (3 – 1) × 4 = i (4 + 1) × 3 =
j 4 × (3 + 1) = k 1 × (4 – 3) = l 4 + 1 × 3 =
Three different dice give scores of 2, 3, 5. Put ×, +, ÷, – or ( ) in each sentence to make it true.
a 2 3 5 = 11 b 2 3 5 = 16 c 2 3 5 = 17
d 5 3 2 = 4 e 5 3 2 = 13 f 5 3 2 = 30
The ordinary counting system uses place value, which means that the value of a digit depends upon its
place in the number.
In the number 5348
the 5 stands for 5 thousands or 5000
the 3 stands for 3 hundreds or 300

the 4 stands for 4 tens or 40
the 8 stands for 8 units or 8
And in the number 4 073 520
the 4 stands for 4 millions or 4 000 000
the 73 stands for 73 thousands or 73 000
the 5 stands for 5 hundreds or 500
the 2 stands for 2 tens or 20
You write and say this number as:
four million, seventy-three thousand, five hundred and twenty
Note the use of narrow spaces between groups of three digits, starting from the right. All whole and mixed
numbers with five or more digits are spaced in this way.
CHAPTER 1: BASIC NUMBER
9
Place value and ordering
numbers
1.4
Key words
digit
place value
In this section you will learn how to:
● identify the value of any digit in a number
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Write the value of each underlined digit.
a 341 b 475 c 186 d 298 e 83
f 839 g 238
0
h 15
07
i 6530 j 25

436
k 29 054 l 18 254 m 4308 n 52 994 o 83 205
Copy each of these sentences, writing the numbers in words.
a The last Olympic games in Greece had only 43 events and 200 competitors.
b The last Olympic games in Britain had 136 events and 4099 competitors.
c The last Olympic games in the USA had 271 events and 10 744 competitors.
Write each of the following numbers in words.
a 5 600 000 b 4 075 200 c 3 007 950 d 2 000 782
Write each of the following numbers in numerals or digits.
a Eight million, two hundred thousand and fifty-eight
b Nine million, four hundred and six thousand, one hundred and seven
c One million, five hundred and two
d Two million, seventy-six thousand and forty
Write these numbers in order, putting the smallest first.
a 21, 48, 23, 9, 15, 56, 85, 54
b 310, 86, 219, 25, 501, 62, 400, 151
c 357, 740, 2053, 888, 4366, 97, 368
Write these numbers in order, putting the largest first.
a 52, 23, 95, 34, 73, 7, 25, 89
b 65, 2, 174, 401, 80, 700, 18, 117
c 762, 2034, 395, 6227, 89, 3928, 59, 480
CHAPTER 1: BASIC NUMBER
10
EXAMPLE 1
Put these numbers in order with the smallest first.
7031 3071 3701 7103 7130 1730
Look at the thousands column first and then each of the other columns in turn.
The correct order is:
1730 3071 3701 7031 7103 7130
EXERCISE 1D

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Copy each sentence and fill in the missing word, smaller or larger.
a 7 is …… than 5 b 34 is …… than 29
c 89 is …… than 98 d 97 is …… than 79
e 308 is …… than 299 f 561 is …… than 605
g 870 is …… than 807 h 4275 is …… than 4527
i 782 is …… than 827
a Write as many three-digit numbers as you can using the digits 3, 6 and 8. (Only use each digit
once in each number).
b Which of your numbers is the smallest?
c Which of your numbers is the largest?
Using each of the digits 0, 4 and 8 only once in each number, write as many different three-digit
numbers as you can. (Do not start any number with 0.) Write your numbers down in order, smallest
first.
Write down in order of size, smallest first, all the two-digit numbers that can be made using 3, 5 and
8. (Each digit can be repeated.)
You use rounded information all the time. Look at
these examples. All of these statements use
rounded information. Each actual figure is either
above or below the approximation shown here.
But if the rounding is done correctly, you can find
out what the maximum and the minimum figures
really are. For example, if you know that the
number of matches in the packet is rounded to the
nearest 10,
•
the smallest figure to be rounded up to 30 is
25, and
•

the largest figure to be rounded down to 30 is
34 (because 35 would be rounded up to 40).
So there could actually be from 25 to 34 matches
in the packet.
CHAPTER 1: BASIC NUMBER
Rounding
1.5
Key words
approximation
rounded down
rounded up
In this section you will learn how to:
● round a number
11
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What about the number of runners in the marathon? If you know that the number 23 000 is rounded to
the nearest 1000,
•
The smallest figure to be rounded up to 23 000 is 22 500.
•
The largest figure to be rounded down to 23 000 is 23 499.
So there could actually be from 22 500 to 23 499 people in the marathon.
Round each of these numbers to the nearest 10.
a 24 b 57 c 78 d 54 e 96
f 21 g 88 h 66 i 14 j 26
k 29 l 51 m 77 n 49 o 94
p 35 q 65 r 15 s 102 t 107
Round each of these numbers to the nearest 100.
a 240 b 570 c 780 d 504 e 967

f 112 g 645 h 358 i 998 j 1050
k 299 l 511 m 777 n 512 o 940
p 350 q 650 r 750 s 1020 t 1070
On the shelf of a sweetshop there are three jars like the ones below.
Look at each of the numbers below and write down which jar it could be describing.
(For example, 76 sweets could be in jar 1.)
a 78 sweets b 119 sweets c 84 sweets d 75 sweets
e 186 sweets f 122 sweets g 194 sweets h 115 sweets
i 81 sweets j 79 sweets k 192 sweets l 124 sweets
m Which of these numbers of sweets could not be in jar 1: 74, 84, 81, 76?
n Which of these numbers of sweets could not be in jar 2: 124, 126, 120, 115?
o Which of these numbers of sweets could not be in jar 3: 194, 184, 191, 189?
80
sweets
(to the
nearest
10)
120
sweets
(to the
nearest
10)
190
sweets
(to the
nearest
10)
Jar 1 Jar 2 Jar 3
CHAPTER 1: BASIC NUMBER
12

EXERCISE 1E
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Round each of these numbers to the nearest 1000.
a 2400 b 5700 c 7806 d 5040 e 9670
f 1120 g 6450 h 3499 i 9098 j 1500
k 2990 l 5110 m 7777 n 5020 o 9400
p 3500 q 6500 r 7500 s 1020 t 1770
Round each of these numbers to the nearest 10.
a 234 b 567 c 718 d 524 e 906
f 231 g 878 h 626 i 114 j 296
k 279 l 541 m 767 n 501 o 942
p 375 q 625 r 345 s 1012 t 1074
Which of these sentences could be true and which must be false?
a There are 789 people living in Elsecar. b There are 1278 people living in Hoyland.
c There are 550 people living in Jump. d There are 843 people living in Elsecar.
e There are 1205 people living in Hoyland. f There are 650 people living in Jump.
These were the numbers of spectators in the crowds at nine Premier Division games on a weekend
in May 2005.
a Which match had the largest crowd?
b Which had the smallest crowd?
c Round all the numbers to the nearest 1000.
d Round all the numbers to the nearest 100.
Give these cooking times to the nearest 5 minutes.
a 34 min b 57 min c 14 min d 51 min e 8 min
f 13 min g 44 min h 32.5 min i 3 min j 50 s
Aston Villa v Man City 39 645
Blackburn v Fulham 18 991
Chelsea v Charlton 42 065
C. Palace v Southampton 26 066

Everton v Newcastle 40 438
Man.Utd v West Brom 67 827
Middlesbrough v Tottenham 34 766
Norwich v Birmingham 25 477
Portsmouth v Bolton 20 188
Welcome to
Elsecar
Population 800
(to the nearest 100)
Welcome to
Hoyland
Population 1200
(to the nearest 100)
Welcome to
Jump
Population 600
(to the nearest 100)
CHAPTER 1: BASIC NUMBER
13
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Addition
There are three things to remember when you are adding two whole numbers.
•
The answer will always be larger than the bigger number.
•
Always add the units column first.
•
When the total of the digits in a column is more than 9, you have to carry a digit into the next column
on the left, as shown in Example 2. It is important to write down the carried digit, otherwise you may

forget to include it in the addition.
Subtraction
These are four things to remember when you are subtracting two whole numbers.
•
The bigger number must always be written down first.
•
The answer will always be smaller than the bigger number.
•
Always subtract the units column first.
•
When you have to take a bigger digit from a smaller digit in a column, you must first remove 10 from
the next column on the left and put it with the smaller digit, as shown in Example 3.
14
Adding and subtracting
numbers with up to four digits
1.6
Key words
addition
column
digit
subtract
In this section you will learn how to:
● add and subtract numbers with more than one
digit
EXAMPLE 2
Add: a 167 + 25 b 2296 + 1173
ab
2296
+1173
3469

1
1 67
+25
1 92
1
EXAMPLE 3
Subtract: a 874 – 215 b 300 – 163
ab
2
3
9
0
1
0
– 163
137
8
6
7
1
4
– 215
6 5 9
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Copy and work out each of these additions.
ab c d e
fghi j
Copy and complete each of these additions.
a 128 + 518 b 563 + 85 + 178 c 3086 + 58 + 674

d 347 + 408 e 85 + 1852 + 659 f 759 + 43 + 89
g 257 + 93 h 605 + 26 + 2135 i 56 + 8407 + 395
j 89 + 752 k 6143 + 557 + 131 l 2593 + 45 + 4378
m 719 + 284 n 545 + 3838 + 67 o 5213 + 658 + 4073
Copy and complete each of these subtractions.
ab c d e
fghi j
kl m n o
Copy and complete each of these subtractions.
a 354 – 226 b 285 – 256 c 663 – 329
d 506 – 328 e 654 – 377 f 733 – 448
g 592 – 257 h 753 – 354 i 6705 – 2673
j 8021 – 3256 k 7002 – 3207 l 8700 – 3263
Copy each of these additions and fill in the missing digits.
ab c d
4
□
7
+
□
5
□
936
4 5
+
□□
9 3
□
7
+3

□
8 4
5 3
+2
□
□
9
5375
– 3547
8034
– 3947
8432
– 4665
8043
– 3626
6254
– 3362
580
– 364
650
– 317
638
– 354
602
– 358
673
– 187
732
– 447
572

– 158
954
– 472
908
– 345
637
– 187
562
93
+ 197
175
+ 276
438
147
+ 233
4676
+ 3584
483
+ 832
287
+ 335
317
416
+ 235
4872
+ 1509
95
+56
365
+ 348

CHAPTER 1: BASIC NUMBER
15
EXERCISE 1F
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ef g h
ij
Copy each of these subtractions and fill in the missing digits.
ab c d
ef g h
ij
Multiplication
There are two things to remember when you are multiplying two whole numbers.
•
The bigger number must always be written down first.
•
The answer will always be larger than the bigger number.
8 0 7 6
–
□□□□
6 1 8 7
□
4
□
– 5 5 8
2
□
5
□□□
– 2 4 7

3 0 9
4 6 2
–
□□□
1 8 5
5 4
□
–
□□
6
3 2 5
□
1 4
–2 5
□
3
□
7
6 7
□
–
□□
3
1 3 5
8 5
–
□□
2 7
□
7

–3
□
5 4
7 4
–2
□
□
1
3 5 7 8
+
□□□□
8 0 7 6
□
4
□
+ 3 3 7
7
□
5
□□□
+ 3 4 8
8 0 7
4 6 9
+
□□□
7 3 5
54
□
+
□□

6
822
□
1 8
+2 5
□
8
□
7
CHAPTER 1: BASIC NUMBER
16
Multiplying and dividing by
single-digit numbers
1.7
Key words
division
multiplication
In this section you will learn how to:
● multiply and divide by a single-digit number
EXAMPLE 4
Multiply 231 by 4.
213
× 4
852
1
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Note that the first multiplication, 3 × 4, gives 12. So, you need to carry a digit into the next column on the
left, as in the case of addition.
Division

There are two things to remember when you are dividing one whole number by another whole number:
•
The answer will always be smaller than the bigger number.
•
Division starts at the left-hand side.
This is how the division was done:
•
First, divide 3 into 4 to get 1 and remainder 1. Note where to put the 1 and the remainder 1.
•
Then, divide 3 into 11 to get 3 and remainder 2. Note where to put the 3 and the remainder 2.
•
Finally, divide 3 into 27 to get 9 with no remainder, giving the answer 139.
Copy and work out each of the following multiplications.
ab c d e
fghi j
kl m n o
Calculate each of the following multiplications by setting the work out in columns.
a 42 × 7 b 74 × 5 c 48 × 6
d 208 × 4 e 309 × 7 f 630 × 4
g 548 × 3 h 643 × 5 i 8 × 375
j 6 × 442 k 7 × 528 l 235 × 8
m 6043 × 9 n 5 × 4387 o 9 × 5432
253
× 6
340
× 4
320
× 3
200
× 4

50
× 3
85
× 5
53
× 4
42
× 7
34
× 6
23
× 5
18
× 6
19
× 2
17
× 3
13
× 5
14
× 4
CHAPTER 1: BASIC NUMBER
17
EXAMPLE 5
Divide 417 by 3.
417 ÷ 3 is set out as:
139
3
|

4
1
1
2
7
EXERCISE 1G
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Calculate each of the following divisions.
a 438 ÷ 2 b 634 ÷ 2 c 945 ÷ 3
d 636 ÷ 6 e 297 ÷ 3 f 847 ÷ 7
g 756 ÷ 3 h 846 ÷ 6 i 576 ÷ 4
j 344 ÷ 4 k 441 ÷ 7 l 5818 ÷ 2
m 3744 ÷ 9 n 2008 ÷ 8 o 7704 ÷ 6
By doing a suitable multiplication, answer each of these questions.
a How many days are there in 17 weeks?
b How many hours are there in 4 days?
c Eggs are packed in boxes of 6. How many eggs are there in 24 boxes?
d Joe bought 5 boxes of matches. Each box contained 42 matches. How many matches did Joe buy
altogether?
e A box of Tulip Sweets holds 35 sweets. How many sweets are there in 6 boxes?
By doing a suitable division, answer each of these questions.
a How many weeks are there in 91 days?
b How long will it take me to save £111, if I save £3 a week?
c A rope, 215 metres long, is cut into 5 equal pieces. How long is each piece?
d Granny has a bottle of 144 tablets. How many days will they last if she takes 4 each day?
e I share a box of 360 sweets between 8 children. How many sweets will each child get?
CHAPTER 1: BASIC NUMBER
18
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CHAPTER 1: BASIC NUMBER
19
Letter sets
Find the next letters in these sequences.
a
O, T, T, F, F, …
b
T, F, S, E, T, …
Valued letters
In the three additions below, each letter stands for a single numeral. But a letter may
not necessarily stand for the same numeral when it is used in more than one sum.
abc
Write down each addition in numbers.
Four fours
Write number sentences to give answers from 1 to 10, using only four 4s and any
number of the operations +, –, × and ÷ . For example:
1 = (4 + 4) ÷ (4 + 4) 2 = (4 × 4) ÷ (4 + 4)
Heinz 57
Pick any number in the grid on the right. Circle the
number and cross out all the other numbers in the row
and column containing the number you have chosen.
Now circle another number that is not crossed out and
cross out all the other numbers in the row and column
containing this number. Repeat until you have five
numbers circled. Add these numbers together. What do
you get? Now do it again but start with a different
number.
Magic squares

9
16
14
11
6
3
1
8
Now try to complete
this magic square
using every number
from 1 to 16.
6
7
2
1
5
9
8
3
4
This is a magic square.
Add the numbers in any
row, column or diagonal.
The answer is always 15.
19 8 11 25 7
12 1 4 18 0
16 5 8 22 4
21 10 13 27 9
14 3 6 20 2

FOUR
+FIVE
NINE
TWO
+TWO
FOUR
ONE
+ O N E
TWO
Hints
Letter sets Think about numbers.
Valued letters a Try E = 3, N = 2
b Try O = 7, U = 3
c Try N = 5, O = 9
There are other answers to each sum.
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