Contents
Exam board specification map
Introduction
Topic checker
Topic checker answers
iv
vi
x
xv
Number
The decimal number system
Order of operations
Negative numbers
Factors and multiples
Working with fractions
Fractions, decimals and percentages
Powers and roots
Standard index form
Ratio and proportion
Percentage calculations
2
4
6
8
10
12
14
16
18
20
Algebra
Algebraic expressions
Formulae and substitution
Solving equations
Trial and improvement
Rearranging formulae
Using brackets in algebra
Multiplying bracketed expressions
Inequalities
Number patterns and sequences
Sequences and formulae
Co-ordinates
Lines and equations
Quadratic graphs
22
24
26
28
30
32
34
36
38
40
42
44
46
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Shape, space and measures
Measures and accuracy
Dimensions
Time
Compound measures
Angle facts
Properties of shapes
Pythagoras’ Rule
Calculating areas
Circle calculations
Solid shapes
Volume calculations
Constructions
Loci
Transformations
48
50
52
54
56
58
60
62
64
66
68
70
72
74
Handling data
Statistical tables and diagrams
Pie charts
Finding averages
Comparing sets of data
Probability
Scatter diagrams and correlation
Exam questions and model answers
Complete the facts
Complete the facts – answers
Answers to practice questions
Glossary
Web Links
Last-minute learner
76
78
80
82
84
86
88
98
102
104
110
*
113
* Only available in the CD-ROM version of the book
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The decimal number system
Each place in a decimal number represents a power of ten:
,
, 1, 10, 100, etc.
Rounded numbers are used when less accuracy is required, or to
estimate the result of a calculation.
A
Place value
The system of numbers we use is called the
decimal system because it is based on the
number ten. Numbers in the decimal system
are composed of digits. The position of a digit
in a number controls its place value.
A decimal point separates the whole and
fractional parts of a number.
>>
key fact
Digits to the left of
the decimal point represent whole
numbers; those to the right, fractions.
B
Number
Position of ‘2’ Value of ‘2’
125
tens
2 130 559 millions
20
2 000 000
3.28
tenths
0.0026
thousandths
Decimal calculations
key fact
To multiply a number
by 10, 100 or 1000, simply move its
digits 1, 2 or 3 places to the left:
45.67  10  456.7
45.67  100  4567
45.67  1000  45 670.
>>
The digit 2 means something different in each
of these numbers:
key fact
To divide by 10, 100 or
1000, move digits 1, 2 or 3 places to
the right:
45.67  10  4.567
45.67  100  0.456 7
45.67  1000  0.045 67.
Multiplying by 0.1 is the same as dividing by 10.
You can use the result of a whole number
multiplication to ﬁnd the answer to many
decimal multiplications:
12  2 
so 12  0.2 
12  0.02 
and 1.2  2 
1.2  0.2 
1.2  0.02 
24,
2.4
0.24
2.4
0.24
0.024, etc.
If one of the numbers is made ten times
smaller, the answer will be ten times smaller.
A similar rule works with division, but if the
number you are dividing by gets smaller, the
answer gets bigger:
12  2 
so 12  0.2 
12  0.02 
and 1.2  2 
1.2  0.2 
1.2  0.02 
6,
60
600
0.6
6
60, etc.
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C
Rounded numbers
Sometimes an answer to a question will
contain a lot of decimal digits. Try 10  7
on your calculator – it should ﬁll the display
completely. In practice, you don’t always need
an answer this accurate, so you may need to
round it. This changes the value slightly, but
makes it much easier to read.
Rounding to a given number of decimal
places (dp)
Suppose you needed to round 1.428 571 4
to 2 dp.
Rounding to the nearest 10, 100, etc.
•
Split the number after the required number
of decimal places: 1.42 | 85714
•
Look at the ﬁrst digit after the split. It’s 8.
This is over 5, so you round the number
up to 1.43.
The number 24 is closer to 20 than it is to 30,
so 24 rounded to the nearest ten is 20. 25 is
exactly halfway. In a ‘halfway’ situation, you
always round up. So 25 rounded to the nearest
ten is 30.
>>
Rounding to a given number of signiﬁcant
ﬁgures (sf)
Here, the type of rounding depends on the
size of the number. Count digits from the ﬁrst
non-zero digit in the number, and split after
this. The table shows some different cases.
key fact
If the ‘next’ digit
is 5 or over, round up.
Number
Round to
Split number
Type of rounding
Rounded number
6 415
2 sf
64 | 15
nearest hundred
6400
0.066 666 66
3 sf
0.066 6 | 6666
4 dp
0.066 7
84.9
1 sf
8 | 4.9
nearest ten
80
D
Estimating answers
Sometimes it is useful to check a calculation
by making an estimate of the answer.
>>
Then calculate with these numbers:
s key fact
Round the numbers
in the calculation to 1 sf first.
To estimate the answer to
s ,
The actual answer using the original numbers
is
 47.88 to 2 dp. This is close to the
estimate, so you can be conﬁdent it’s correct.
round the
numbers so the calculation becomes
  40.
s .
>> practice questions
1
Do a whole number calculation first, then use the result to answer the question.
(a) 2.5  3
2
(b) 0.3  1.2
(d) 1.44  0.03
Round each number in the three different ways given.
(a) 61.25;
(i) nearest unit
(b) 588.621;
3
(c) 6.4  8
(i) 1 sf
(ii) 1 dp
(ii) 2 sf
(iii) 2 dp
(iii) 4 sf
Estimate the answer to each calculation, then find the exact answer, rounded to 3 sf.
(a) (31.42  15.7)  2.25
(b)
13 .7 35 .1
14
(c) 5.89 2
(d)
3.57
1.81
2.26
4.009
3
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Order of operations
There is a standard order in which you carry out operations.
Brackets can be used to change this order.
Use BIDMAS to remember the order.
A
‘Stronger’ and ‘weaker’ operations
Mathematical operations allow numbers to be
combined in a calculation.
The usual operations are:
• addition and subtraction
2  4  5 means 2  20  22,
not 6  5  30.
>>
• multiplication and division
• powers and roots.
When you see more than one operation in a
calculation, there is a standard order you must
use to work out the answer.
>>
key fact
Multiplication and
division should be done before
addition and subtraction as they
are ‘stronger’ operations.
B
key fact
Powers and roots
are the ‘strongest’ and should
always be done first.
3  22 means 3  4  12, not 62  36.
When operations of the same ‘strength’ are
mixed, you simply work through them in
order.
So 10  3  2  5  4.
Brackets
The only way to change the order in a calculation is to use brackets.
The part of the calculation that occurs in the bracket must be evaluated ﬁrst.
(2  4)  5 means 6  5  30.
(3  2) 2 means 6 2  36.
Sometimes, you may see 8  (9  3) written as 8(9  3), as in algebra.
Some calculations contain ‘hidden’ brackets.
In this one, fraction-style notation has been used to show division.
To work out
, you actually calculate (23.85  46.2)  30.
A square root sign can also ‘hide’ a bracket.
3 2  4 2 means the square root of (3 2  4 2), not
3 2  4 2.
Occasionally, you may ﬁnd brackets inside brackets. This is called nesting.
Always start with the innermost bracket in the calculation.
((23  37)  4) 2  (60  4) 2  15 2  225
4
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C
Calculators
A scientiﬁc calculator obeys the correct rules
for operations. The display may show answers
to some of the steps in a calculation as you
work. Don’t be put off by this! A standard
calculator will probably do calculations in the
order they are entered, and will give incorrect
results for some calculations.
You can also use your calculator’s memory to
store part of a long calculation. Make sure you
know how these functions work – read your
calculator’s instruction manual!
Your calculator should have a pair of bracket
Or these:
keys
[(
)]
The memory keys should look like these:
Min
M-
M+
STO
MC
RCL
CLR
. These alter the order of
operations in a calculation in exactly the way
you would expect.
D
BIDMAS
B brackets
key fact
You can use the
made-up word BIDMAS to help you
remember the order of operations.
Each letter stands for an operation.
I indices (this covers powers and roots)
D division
M multiplication
A addition
>> practice questions
S subtraction
1
2
3
4
}
}
Use the correct order of operations to calculate these.
(a) 10  3  5
(b) 12  2 2
(c) 24  3  7
(d) 5  2 2
(e) 4  10  10  9
(f) 3.5  15  2
Carry out these calculations involving brackets.
(a) (6  2)  4
(b) 10  (23  21)
(c) 2  (2  2)  2
(d) 10(3.6  2.1)
(e) (3  (2  1)) 2
(f) (12  3) 3
Some of these calculations need brackets to make them correct.
Add brackets where required.
(a) 4  2  3  18
(b) 12  6  3  4
(c) 5  6  6  5  46
(d) 5  6  6  5  121
(e) 4  3 2  13
(f) 1  2  3  4  5  1
Evaluate these using your calculator.
Round your answer to 3 significant figures if needed.
1000
(a) (2.15  4.22)  55
(b)
(d) (5 2  40) 2
(e) (14.1  (6.25  8.15) 2 ) 2
16 .3 41 .45
(c)
(f)
42  22
(10) 3
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