Mathematics

1
Mathematics Intermediate Tier
Number
GCSE Revision
2
Intermediate Tier - Number revision
Contents Calculator questions Best buy
questions Long multiplication Long
division Simple fractions questions Negative
numbers Rounding Estimating Percentages Ty
pes 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
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Calculator questions
Which buttons would you press to do these on a
calculator ?
2.5 4.1 3.5
1.7 2.8 2.3 0.2
1.56
4
Best buy questions
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)
Milk Large? 2.1?78 0.0269L/p Small? 0.95?32
0.0297L/p Small is better value OR (looking at it
differently) Large? 78?2.1 37.14p/L Small?
32?0.95 33.68p/L
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Long multiplication
Use the method that gives you the correct answer
!!
Question 78 x 59
70
8
50
3500
400
9
630
72
Total 3500 400 630 72
Answer 4602
Now try 84 x 46 and 137 x 23 and check on your
calculator !!
6
Long division
Again use the method that gives you the correct
answer !!
Question 2987 ? 23
1
2
9
6
22
20
Answer 129 r 20
Now try 1254 ? 17 and check on your calculator
Why is the remainder different?
7
Simple fractions questions
Equivalent fractions
Fractions into decimals
Divide top by bottom
Fractions of amounts
Divide by the bottom then times by the top
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Negative numbers
Put these in order - smallest first
- 4 , - 3.6 , - 0.4 , - 0.36 , - 1 , 0 , 2 , - 1.4
Up and down the scale
Two signs next to each other
- 3 - 1
- 4 2
Multiplication and Division
Signs same ? ve answer Signs different ? -ve
answer
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4715.692803 cm
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

• 4715.7 cm
• (b) 5000 cm
• 4715.69 cm
• (d) 4700 cm
• 4716 cm
• (f) 47 m
• 4715.692 cm
• (h) 4720 cm

But there is always a trickier one
14.9999

• Round this number off to
• the nearest whole number
• (b) 3 significant figures
• 2 decimal places

• 15
• (b) 15.0
• 15.00

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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
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Percentages
Percentage increase and decrease
A womans wage increases by 13.7 from 240 a
week. What does she now earn ?
Increase
New amount
13.17 of 240
240 31.608
271.608
Her new wage is 271.61 a week
13.17 100
x
240
31.608
30
75
10
1
Percentages of amounts
25
45
5
(Do these without a calculator)
85
20
50
2
12
Percentages
Fractions, decimals and percentages
50
Copy and complete
0.5
13
Reverse
e.g. A womans wage increases by 5 to 660 a
week. What was her original wage to the nearest
penny?
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 6 0.83 7.23 tonnes
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Repeated
This is not the correct method 12000 x 0.065
780 780 x 5 3900 12000 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.
He will have 12000 x (1.065)5 16441.04
This is not the correct method 40000 x 0.23
9200 9200 x 4 36800 40000 36800 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 ?
The cars new value 40000 x (0.77)4 14061
(nearest )
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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)

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40
Products of primes
Express 40 as a product of primes
40 2 x 2 x 2 x 5 (or 23 x 5)
630
Express 630 as a product of primes
Now do the same for 100 , 30 , 29 , 144
630 2 x 3 x 3 x 5 x 7 (or 2 x 32 x 5 x 7)
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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 112 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
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104
Indices
52
91
32
102
75 ? 73
23
190
26
53
171
25
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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
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Standard form
Write in Standard Form
Write as an ordinary number
0.0001
9.6
4.7 x 109
8 x 10-3
3 600
0.041
1 x 102
5.1 x 104
23 600
0.2
8.6 x 10-1
7 x 10-2
8 900 000 000
9.2 x 103
3.5 x 10-3
46.7
0.003
2 x 100
6 x 106
Do 3 x 104 x 7 x 105 with and without a calculator
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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 273. How much does each receive?
Basil gets 7 of 600 350 12
Claire gets 3 of 600 150 12
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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

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Fractions with the four rules
4? 1½
4? ? 1½
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Recurring decimals as fractions
Learn this technique which changes recurring
decimals into fractions
Express 0.77777777.. as a fraction.
Let n 0.77777777.. so 10n
7.77777777.. so 9n 7
so n 7/9
Express 2.34343434.. as a fraction.
Let n 2.34343434.. so 100n
234.34343434.. so 99n 232
so n 232/99
Express 0.13213213.. as a fraction.
Let n 0.132132132.. so
1000n 132.132132132.. so 999n 132
so n 132/999 n 44/333
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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 length conversions
Metric weight conversions
Metric capacity conversions
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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 0.417 hours
45 x 120 D
S 40.8 mph
T 9.23 hours
D 5400 m
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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
8 x 6000 M
D 6.25 kg/m3
V 47 m3
M 48000 g
( or M 48 kg)