Mathematics

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Mathematics Higher Tier
Number
GCSE Revision
2
Higher Tier - Number revision
Contents Calculator questions Long
multiplication division Best buy
questions Estimation Units Speed, Distance
and Time Density, Mass and Volume Percentages
Products of primes HCF and LCM Indices Sta
ndard Form Ratio Fractions with the four
rules Upper and lower bounds Percentage
error Surds Rational and irrational
numbers Recurring decimals as
fractions Direct proportion Inverse
proportion Graphical solutions to equations
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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
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 !!
5
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?
6
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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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 !!
Estimation
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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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
10
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)
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Simple
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
Simple
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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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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Expressing 2 numbers as a product of primes can
help you calculate their Highest common factor
HCF Consider the numbers 20 and 30. Their factors
are 1, 2, 4, 5, 10, 20 and 1, 2, 3, 5, 6, 10,
15, 30 Their highest common factor is 10
HCF
e.g. Find the highest common factor of 84 and 120.
84 2 x 2 x 3 x 7
Pick out all the bits that are common to both.
120 2 x 2 x 2 x 3 x 5
Highest common factor 2 x 2 x 3 12
Expressing 2 numbers as a product of primes can
also help you calculate their Lowest common
multiple
LCM
LCM Consider the numbers 16 and 20. Their
multiples are 16, 32, 48, 64, 80, 96 and 20,
40, 60, 80, 100 Their lowest common multiple is
80
e.g. Find the lowest common multiple of 300 and
504.
Pick out the highest valued index for each prime
factor .
300 22 x 3 x 52
504 23 x 32 x 7
Lowest common multiple 23 x 32 x 52 x 7 12600
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10-4
Indices
115
Evaluate
91/2
3-2
75 ? 73
163/2
190
2-3
2-1
36-1/2
811/4
25 x 21
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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
56 x 103
0.2
8.6 x 10-1
7 x 10-2
8 900 000 000
9.2 x 103
3.5 x 10-3
Calculate 3 x 104 x 7 x 10 -1 without a calculator
Calculate 4.6 x 104 2.5 x 108 with a calculator
Calculate 1.5 x 106 3 x 10 -2 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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A journey of 37 km measured to the
nearest km could actually be as long as
37.49999999. km or as short as 36.5 km. It could
not be 37.5 as this would round up to 38 but the
lower and upper bounds for this measurement are
36.5 and 37.5 defined by 36.5 lt Actual
distance lt 37.5
Upper and lower bounds
e.g. Write down the Upper and lower bounds of
each of these values given to the accuracy stated
9m (1s.f.)
2.40m (2d.p.)
8.5 to 9.5
2.395 to 2.405
85g (2s.f.)
4000L (2s.f.)
84.5 to 85.5
3950 to 4050
180 weeks (2s.f.)
60g (nearest g)
175 to 185
59.5 to 60.5
e.g. A sector of a circle of radius 7cm makes an
angle of 320 at the centre. Find its minimum
possible area if all measurements are given to
the nearest unit. (? 3.14)
Area (?/360) x ? x r x
r Minimum area (31.5/360) x 3.14 x 6.5 x
6.5 Minimum area 11.61cm2
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If a measurement has been rounded off then it is
not accurate. There is a an error between the
measurement stated and the actual measurement.
error
The exam question that occurs most often is
Calculate the maximum percentage error between
the rounded off measurement and the actual
measurement.
Upper and lower bounds of 9.6 cm (1d.p.) ?
9.55 to 9.65
Maximum potential difference (MPD) between actual
and rounded off measurements ?
0.05
Max. pot. error (MPD/lower bound) x 100
(0.05/9.55) x 100 0.52
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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
?20
?4 x ?5
2 ?5
?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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Surds
A surd is the name given to a number which has
been left in the form of a root. So ?5 has been
left in surd form.
SIMPLIFYING EXPRESSIONS WITH SURDS IN
A surd or a combination of surds can be
simplified using the rules ?M x ?N ?MN and
visa versa ?M ?N ?M/N and visa versa
Tips Deal with a surd as you would an algebraic
term and always look for square numbers
?12
?4 x ?3
2 ?3
?135 ?3
?9 x ?5
3 ?5
?45
?5(?5 ?20)
5 ?100
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(?3 1)2
(?3 1) (?3 1)
3 ?3 ?3 1
4 2?3
LEAVING ANSWERS IN SURD FORM
Pythagoras ? (?14)2 (?6)2 x2 14 6
x2 x ?8
Answer x 2?2
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Rational and irrational numbers
Rational numbers can be expressed in the form
a/b. Terminating decimals (3.17 or 0.022) and
recurring decimals (0.3333..or 4.7676..) are
rational.
Irrational numbers cannot be made into fractions.
Non-terminating and non-recurring decimals
(3.4526473 or ? or ?2) are irrational.
State whether the following are rational or
irrational numbers
1/5
2.3/5.7
2.7
What do you need to do to make the following
irrational numbers into rational numbers
?2
6?
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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 0.413213213.. as a fraction.
Let n 0.4132132132..
so 10000n 4132.132132132.. and 10n
4.132132132 so 9990n 4128
so n 4128/9990 n 688/1665
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Direct proportion
If one variable is in direct proportion to
another (sometimes called direct variation)
their relationship is described by
p ? t
p kt
Where the Alpha can be replaced by an Equals
and a constant k to give
e.g. y is directly proportional to the square of
r. If r is 4 when y is 80, find the value of r
when y is 2.45 .
Write out the variation
y ? r2
Possible direct variation questions
Change into a formula
y kr2
Sub. to work out k
80 k x 42
g ? u3
g ku3
k 5
c ? ?i
c k?i
So
y 5r2
s ? 3?v
s k3?v
And
2.45 5r2
t ? h2
t kh2
r 0.7
x ? p
x kp
Working out r
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Inverse proportion
If one variable is inversely proportion to
another (sometimes called inverse variation)
their relationship is described by
p ? 1/t
p k/t
Again Alpha can be replaced by a constant k
to give
e.g. y is inversely proportional to the square
root of r. If r is 9 when y is 10, find the
value of r when y is 7.5 .
Write out the variation
y ? 1/?r
Possible inverse variation questions
Change into a formula
y k/?r
Sub. to work out k
10 k/?9
g ? 1/u3
g k/u3
k 30
c ? 1/?i
c k/?i
So
y 30/?r
s ? 1/3?v
s k/3?v
And
7.5 30/?r
t ? 1/h2
t k/h2
r 16 (not 2)
x ? 1/p
x k/p
Working out r
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Graphical solutions to equations
If an equation equals 0 then its solutions lie
at the points where the graph of the equation
crosses the x-axis.
e.g. Solve the following equation
graphically x2 x 6 0
All you do is plot the equation y x2 x 6
and find where it crosses the x-axis (the line
y0)
There are two solutions to x2 x 6 0 x -
3 and x 2
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Graphical solutions to equations
If the equation does not equal zero Draw the
graphs for both sides of the equation and where
they cross is where the solutions lie
e.g. Solve the following equation
graphically x2 2x 11 9 x
Plot the following equations and find where
they cross y x2 2x 20 y 9 x
There are 2 solutions to x2 2x 11 9 x x
- 4 and x 5
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If there is already a graph drawn and you are
being asked to solve an equation using it, you
must rearrange the equation until one side is
the same as the equation of the graph. Then plot
the other side of the equation to find the
crossing points and solutions.
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Rearranging the equation x3 4x 5 5x 5 to
get x3 8x 7
Add 2 to both sides
x3 4x 5 5x 5
x3 4x 7 5x 7
Take 4x from both sides
x3 8x 7 x 7
So we plot the equation y x 7 onto the graph
to find the solutions
Solutions lie at 3, 0 and 3
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• State the graphs you need to plot to solve the
• following equations describing how you will find
• your solutions
• 3x2 4x 2 0
• 7x 4 x2 4x
• x4 5 0
• 0 8x2 5x
• 2x 9
• 6x3 2x2 5

• If you have got the graph of y 4x2 5x 6 work
  
• out the other graph you need to draw to solve
  each
• of the following equations
• 4x2 4x 6 0
• 4x2 x - 2 7
• 4x2 3x 2x
• 3x2 5

Solve this equation graphically x3 8x2 3x
2x2 2x