1. Numbers

2. Contents : Calculator questions
Best buy questions
Long multiplication
Long division
Simple fractions questions
Negative numbers
Rounding
Estimating
Percentages
Types of number
Products of primes
HCF and LCM
Indices
Simplifying roots
Standard Form
Ratio
Fractions with the four rules
Recurring decimals as fractions
Units
Distance, Speed, Time questions
Density, Mass, Volume questions

3. Calculator questions
Which buttons would you press
to do these on a calculator ? –
3.5
2.3 – 0.2
8.5 x 103
3.4 x 10-1
2.5 3
1.56 –
6.31
1000 4

4. 400g 150g 78p 2.1L Best buy questions 87p 34p BeansOR
34p
87p
400g
150g
Always divide by the price to see
how much 1 pence will buy you
Beans
Large 40087 = 4.598g/p
Small 15034 = 4.412g/p
Large is better value
(more grams for every
penny spent)
0.95L
2.1L
78p
32p OR
Milk
Large 2.178 = L/p
Small 0.9532 = L/p
Small is better value
OR (looking at it differently)
{Large 782.1 = 37.14p/L
Small 320.95 = 33.68p/L}

5. Use the method that gives you the correct answer !!
Long multiplication
Use the method that gives you the correct answer !!
Question : 78 x 59 70 8 50
3500
400 9
630 72
Total =
Answer : 4602
Now try 84 x 46 and 137 x 23 and check on your calculator !!

6. Again use the method that gives you the correct answer !!
Long division
Again use the method that gives you the correct answer !!
Question :  23
23 times table 1 2 9 23 6 22 20 29 68
Answer : 129 r 20
Now try 1254  17 and check on your calculator – Why is the remainder different?
227

7. Simple fractions questions Equivalent fractions3 ? = 12 20 1. 5 6 = ? 24 2.
Fractions into decimals
9 . 12 = ? 3. ? =
5 . 25 4.
Divide top by bottom
9 . 12 =
9  12 = 0.75
Fractions of amounts 1 2 of
£30 1.
Divide by the bottom
then times by the top 4 5 1. 6 7 2. 5 9 3. 1 6 of
£30 2.
Put the following fractions
in order of size, smallest
to largest: 4. 4 7 2 3 5 8 5 6 of
£30 3. 5 9 of
$72 4.

8. Signs different  -ve answer
Negative numbers
Put these in order - smallest first
- 4 , , , , - 1 , 0 , 2 , - 1.4
Up and down the scale
Two signs next to each other 1. = 1. = = 2. = 2. = = 3. = 3. = 4. = 4. = 5. =
Multiplication and Division
Signs same  +ve answer
Signs different  -ve answer 1.
- 4  3 = 2.
- 5  - 2 = 3.
- 8  - 4 = 4.
20  - 5 = 5.
- 4  4 = 6.
(- 7)2 =

9. But there is always a trickier onecm
Rounding
Round this number off to :
1 decimal place
(b) 1 significant figure
2 decimal places
(d) 2 significant figures
the nearest centimetre
(f) the nearest metre
3 decimal places
(h) 3 significant figures cm
(b) 5000 cm cm
(d) 4700 cm
4716 cm
(f) 47 m cm
(h) 4720 cm
But there is always a trickier one
Round this number off to :
the nearest whole number
(b) 3 significant figures
2 decimal places 15
(b) 15.0
15.00

10. Always remember to write down the numbers you have rounded off
If you are asked to estimate an answer to a calculation – Round all the numbers off to 1 s.f. and do the calculation in your head.
DO NOT USE A CALCULATOR !!
Estimating
e.g. Estimate the answer to 4.12 x 5.98
 4 x 6 = 24
Always remember to write down the numbers you have rounded off
Estimate the answer to these calculations 1.
58 x 21 5.
7.12 x 39.2
0.87 8.
4.89 x 6.01
1.92 2.
399 x 31 3.
47 x 22 6.
377  19 9.
360 x 87 4.
4899  46 7.
1906  44
10.
58 x 21

11. Percentages of amounts
Percentage increase and decrease
A woman’s wage increases by 13.7% from
£240 a week. What does she now earn ?
Increase:
New amount:
13.17% of £240 =
Her new wage is
£ a week
13.17
100 x
240 =
31.608
30% =
75% =
10% =
£600
1% =
Percentages of amounts
25% =
45% =
5% =
(Do these without
a calculator)
85% =
20% =
50% =
2% =

12. 50% 0.5 1 2 % Percentages Fractions, decimals and percentages Frac Dec
Copy and complete:
0.17
56% 1 2
0.5
0.04 9 50
83% 4 25
0.92 19 20
28%

13. Reverse %
e.g. A woman’s wage increases by 5% to £660 a week. What was her original wage to the nearest penny?
Original
amount
x 1.05
£660
Original
amount
£660
÷ 1.05
Original amount = 660 ÷ 1.05 = £628.57
e.g. A hippo loses 17% of its weight during a diet. She now weighs 6 tonnes. What was her former weight to 3 sig. figs. ?
Original
weight
x 0.83
6 ton.
Original
weight
6 ton.
÷ 0.83
Original weight = 6 ÷ 0.83 = 7.23 tonnes

14. This is not the correct method: This is not the correct method:
Repeated %
This is not the correct method:
12000 x = 780
780 x 5 = 3900
= £15900
e.g. A building society gives 6.5% interest p.a. on all money invested there. If John pays in £12000, how much will he have in his account at the end of 5 years.
£12000
x 1.065 ?
He will have = x (1.065)5 = £
This is not the correct method:
40000 x 0.23 = 9200
9200 x 4 = 36800
40000 – = $3200
e.g. A car loses value at a rate of approximately 23% each year. Estimate how much a $40000 car be worth in four years ?
£40000
x 0.77 ?
The car’s new value = x (0.77)4 = $14061 (nearest $)

15. 9 12 6 1 20
100 7 25 3 11 13 16 2 27
Types of number
From this set of numbers
list the:
Odd numbers
Even numbers
Multiples of 8
Factors of 12
Prime numbers
Square numbers
Cube numbers
Some useful words to know the meaning of:
Sum = add together
Product = multiply together
Difference = subtract one number from another
Reciprocal of a number = 1 divided by the number
(e.g. Reciprocal of 4 = ¼ or 0.25)

16. 40
Products of primes
Express 40 as a product of primes 2 20 2 10
40 = 2 x 2 x 2 x 5 (or 23 x 5) 2 5
630
Express 630 as a product of primes 2
315
Now do the same for
100 , 30 , 29 , 144 3
105 3 35
630 = 2 x 3 x 3 x 5 x 7 (or 2 x 32 x 5 x 7) 5 7

17. Finding the HCF and LCM of a pair of numbers
HCF stands for the Highest Common Factor (the biggest number that
will go into both numbers)
LCM stands for Lowest Common Multiple (the first number to appear
in both numbers times table)
e.g. Find the HCF and LCM of the two numbers 140 and 112
Write both numbers as a product of primes
140 = 2 x 2 x 5 x 7 and = 2 x 2 x 2 x 2 x 7
For the HCF write out all the primes that appear in both answers
HCF = 2 x 2 x 7 = 28
For the LCM write out the largest number of each prime that exists in either number
LCM = 2 x 2 x 2 x 2 x 5 x 7 = 560

18. 104
Indices 52 91 32 9
102
75  73 23
190 26 53
171 25 43

19. 12 4 x 3 2 3 8 4 x 2 2 2 45 9 x 5 3 5 72 36 x 2 6 2
Simplifying roots
Tip: Always look for square numbered
factors (4, 9, 16, 25, 36 etc)
e.g. Simplify the following into the form a  b
12
4 x 3
2 3 8
4 x 2
2 2
45
9 x 5
3 5
72
36 x 2
6 2
700
100 x 7
10 7

20. Standard form
Write in Standard Form
Write as an ordinary number
9.6
0.0001
4.7 x 109
8 x 10-3
3 600
0.041
1 x 102
5.1 x 104
23 600
0.2
7 x 10-2
8.6 x 10-1
9.2 x 103
3.5 x 10-3
0.003
46.7
6 x 106
2 x 100
Do 3 x 104 x 7 x 105 with and without a calculator

21. Splitting in a given ratio
1 : ?
0.5 : ?
? : 1
2100 : ?
? : 12
14 : ?
? : 6
? : 10
21 : ?
49 : ?
7:2
Ratio
Equivalent
Ratios
Splitting in a given ratio
Total parts = 12
Anne gets 2 of 600 = £100 12
£600 is split between
Anne, Bill and Claire in
the ratio 2:7:3. How
much does each receive?
Basil gets 7 of 600 = £350 12
Claire gets 3 of 600 = £150 12

22. + – × ÷ Fractions with the four rules
Learn these steps to complete all fractions questions:
Always convert mixed fractions into top heavy fractions before you start
When adding or subtracting the “bottoms” need to be made the same
When multiplying two fractions, multiply the “tops” together and the “bottoms” together to get your final fraction
When dividing one fraction by another, turn the second fraction on its head and then treat it as a multiplication

23. Fractions with the four rules
4⅔ + 1½
4⅔  1½ 14 3 2 + = 14 3 2  = 9 6 28 + = 14 3 2  = 37 6 = 28 9 = = 6 1 = 3 1 9

24. Recurring decimals as fractions
Learn this technique which changes recurring decimals into
fractions:
Express
…..
as a fraction.
Let n = …..
so 10n = …..
so n = 7
so n = 7/9
Express
…..
as a fraction.
Let n = …..
so 100n = …..
so 99n = 232
so n = 232/99
Express
…..
as a fraction.
Let n = …..
so 1000n = …..
so n = 132
so n = 132/999
n = 44/333

25. kg g cg mg km m cm mm kl l cl ml Units 1 inch  2.5 cm 1 yard  0.9 m
5 miles  8 km
2.2 lbs  1 kg
1 gallon  4.5 litres
Learn these rough conversions
between imperial and metric units
Learn this pattern for converting between the various metric units
Metric capacity conversions
Metric weight conversions
Metric length conversions kg g cg mg
x 1000
x 10
x 100
÷ 1000
÷ 10
÷ 100 km m cm mm
x 1000
x 10
x 100
÷ 1000
÷ 10
÷ 100 kl l cl ml
x 1000
x 10
x 100
÷ 1000
÷ 10
÷ 100

26. D S = T Speed, Distance, Time questions
Speed, Distance and Time are linked by this formula
To complete questions check that all units are
compatible, substitute your values in and rearrange if necessary.
Speed = 45 m/s
Time = 2 minutes
Distance = ?
2. Distance = 17 miles
Time = 25 minutes
Speed = ?
3. Speed = 65 km/h
Distance = 600km
Time = ?
45 m/s and 120 secs
17 miles and hours
S = D T
S = D T
S = D T
65 =
600 . T
45 =
D .
120
S =
17 .
0.417
T =
600 . 65
45 x 120 = D
S = 40.8 mph
T = 9.23 hours
D = 5400 m

27. M D = V Density, Mass, Volume questions
Density, Mass and Volume are linked by this formula
To complete questions check that all units are
compatible, substitute your values in and rearrange if necessary.
Density = 8 g/cm3
Volume = 6 litres
Mass = ?
2. Mass = 5 tonnes
Volume = 800 m3
Density = ?
3. Density = 12 kg/m3
Mass = 564 kg
Volume = ?
8 g/cm3 and 6000 cm3
800 m3 and 5000 kg
D = M V
D = M V
D = M V
12 =
564 . V
8 =
M .
6000
D =
5000 .
800
V =
564 . 12
8 x 6000 = M
D = 6.25 kg/m3
V = 47 m3
M = g
( or M = 48 kg)