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KS3 Mathematics
N8 Ratio and proportion
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Contents
N8 Ratio and proportion
N8.1 Ratio
N8.2 Dividing in a given ratio
N8.3 Direct proportion
N8.4 Using scale factors
N8.5 Ratio and proportion problems
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Stacking blocks
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Ratio
A ratio compares the sizes of parts or quantities to each
other.
For example,
What is the ratio of red
counters to blue counters?
red : blue
=9:3
=3:1
For every three red counters there is one blue counter.
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Ratio
A ratio compares the sizes of parts or quantities to each
other.
For example,
The
ratio
What
is of
theblue
ratiocounters
of blue to red counters is not the
same
as the
of red counters to blue counters.
counters
to ratio
red counters?
blue : red
=3:9
=1:3
For every blue counter there are three red counters.
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Ratio
What is the ratio of
red counters to yellow
counters to blue
counters?
red : yellow : blue
= 12 :
4
: 8
= 3 :
1
: 2
For every three red counters there is one yellow counter and
two blue counters.
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Simplifying ratios
Ratios can be simplified like fractions by dividing each part
by the highest common factor.
For example,
21 : 35
÷7
÷7
=3:5
For a three-part ratio all three parts must be divided by the
same number.
For example,
6 : 12 : 9
÷3
÷3
=2:4:3
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Equivalent ratio spider diagrams
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Simplifying ratios with units
When a ratio is expressed in different units, we must write
the ratio in the same units before simplifying.
Simplify the ratio 90p : £3
First, write the ratio using the same units.
90p : 300p
When the units are the same we don’t need to
write them in the ratio.
90 : 300
÷ 30
÷ 30
= 3 : 10
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Simplifying ratios with units
Simplify the ratio 0.6 m : 30 cm : 450 mm
First, write the ratio using the same units.
60 cm : 30 cm : 45 cm
60 : 30 : 45
÷ 15
÷ 15
= 4 :2 : 3
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Simplifying ratios containing decimals
When a ratio is expressed using fractions or decimals we
can simplify it by writing it in whole-number form.
Simplify the ratio 0.8 : 2
We can write this ratio in whole-number form by multiplying
both parts by 10.
0.8 : 2
× 10
× 10
= 8 : 20
÷4
÷4
=2:5
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Simplifying ratios containing fractions
Simplify the ratio
2
3
:4
We can write this ratio in whole-number form by multiplying
both parts by 3.
2
3
:4
×3
×3
= 2 : 12
÷2
÷2
=1:6
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Comparing ratios
We can compare ratios by writing them in the form 1 : m
or m : 1, where m is any number.
For example, the ratio 5 : 8 can be written in the form 1 : m
by dividing both parts of the ratio by 5.
5:8
÷5
÷5
= 1 : 1.6
The ratio 5 : 8 can be written in the form m : 1 by dividing
both parts of the ratio by 8.
5:8
÷8
÷8
= 0.625 : 1
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Comparing ratios
The ratio of boys to girls in class 9P is 4:5.
The ratio of boys to girls in class 9G is 5:7.
Which class has the higher proportion of girls?
The ratio of boys to girls in 9P is
÷4
4:5
÷4
= 1 : 1.25
The ratio of boys to girls in 9G is
÷5
5:7
÷5
= 1 : 1.4
9G has a higher proportion of girls.
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Contents
N8 Ratio and proportion
N8.1 Ratio
N8.2 Dividing in a given ratio
N8.3 Direct proportion
N8.4 Using scale factors
N8.5 Ratio and proportion problems
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Mixing paint
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Ratios and proportions on a metre rule
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Ratios and proportions on a metre rule
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Dividing in a given ratio
Divide £40 in the ratio 2 : 3.
A ratio is made up of parts.
We can write the ratio 2 : 3 as
2 parts : 3 parts
The total number of parts is
2 parts + 3 parts = 5 parts
We need to divide £40 by the total number of parts.
£40 ÷ 5 = £8
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Dividing in a given ratio
Divide £40 in the ratio 2 : 3.
Each part is worth £8 so
2 parts = 2 × £8 = £16
and
3 parts = 3 × £8 = £24
£40 divided in the ratio 2 : 3 is
£16 : £24
Always check that the parts add up to the original amount.
£16 + £24 = £40
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Dividing in a given ratio
A citrus twist cocktail contains orange juice, lemon
juice and lime juice in the ratio 6 : 3 : 1.
How much of each type of juice is contained in 750 ml
of the cocktail?
First, find the total number of parts in the ratio.
6 parts + 3 parts + 1 part = 10 parts
Next, divide 750 ml by the total number of parts.
750 ml ÷ 10 = 75 ml
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Dividing in a given ratio
A citrus twist cocktail contains orange juice, lemon
juice and lime juice in the ratio 6 : 3 : 1.
How much of each type of juice is contained in 750 ml
of the cocktail?
Each part is worth 75 ml so,
6 parts of orange juice = 6 × 75 ml = 450 ml
3 parts of lemon juice = 3 × 75 ml = 225 ml
1 part of lime juice = 75 ml
Check that the parts add up to 750 ml.
450 ml + 225 ml + 75 ml = 750 ml
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Dividing in a given ratio spider diagram
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Contents
N8 Ratio and proportion
N8.1 Ratio
N8.2 Dividing in a given ratio
N8.3 Direct proportion
N8.4 Using scale factors
N8.5 Ratio and proportion problems
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Proportion
Proportion compares the size of a part to the size of a whole.
There are many ways to express a proportion. For example,
What proportion of these
counters are red?
We can express this proportion as:
12 out of 16
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3 in every 4
3
4
0.75
or
75%
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Proportional sets
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Direct proportion problems
3 packets of crisps weigh 90 g.
How much do 6 packets weigh?
3 packets weigh 90 g.
×2
×2
6 packets weigh 120 g.
If we double the number of packets then we double the weight.
The number of packets and the weights are in
direct proportion.
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Direct proportion problems
3 packets of crisps weigh 90 g.
How much do 6 packets weigh?
3 packets weigh 90 g.
÷3
÷3
1 packet weighs 30 g.
If we divide the number of packets by three then divide the
weight by three.
Once we know the weight of one packet we can work out the
weight of any number of packets.
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Direct proportion problems
3 packets of crisps weigh 90 g.
How much do 7 packets weigh?
3 packets weigh 90 g.
÷3
÷3
1 packet weighs 30 g.
×7
×7
7 packets weigh 210 g.
This is called using a unitary method.
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Contents
N8 Ratio and proportion
N8.1 Ratio
N8.2 Dividing in a given ratio
N8.3 Direct proportion
N8.4 Using scale factors
N8.5 Ratio and proportion problems
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Using scale factors
How can we get from 4 to 5 using only
multiplication and division?
We could divide 4 by 4 to get 1 and then multiply by 5.
(4 ÷ 4) × 5 = 5
We could also multiply 4 by 5 to get 20 and then divide by 4.
(4 × 5) ÷ 4 = 5
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Using scale factors
How can we divide by 4 and multiply by
5 in a single step?
5
Dividing by 4 and multiplying by 5 is equivalent to ×
4
5
4×4 =5
5
We call the
a multiplier or scale factor.
4
We can write the scale factor as a decimal,
We can also write it as a percentage,
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5
= 1.25
4
5
= 125%
4
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Using a diagram to represent scale factors
We can represent the scaling from 4 to 5 using a diagram:
×
5
4
5
4
×5
÷ 4 or
1
×
4
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1
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Using scale factors
How can we get from 5 to 4 using only
multiplication and division?
We could divide 5 by 5 to get 1 and then multiply by 4.
(5 ÷ 5) × 4 = 4
We could also multiply 5 by 4 to get 20 and then divide by 5.
(5 × 4) ÷ 5 = 4
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Using scale factors
How can we divide by 5 and multiply by
4 in a single step?
4
Dividing by 5 and multiplying by 4 is equivalent to ×
5
4
5×5 =4
4
We call the
a multiplier or scale factor.
5
We can write the scale factor as a decimal,
We can also write it as a percentage,
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4
= 0.8
5
4
= 80%
5
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Using a diagram to represent scale factors
We can represent the scaling from 5 to 4 using a diagram:
5
4
×
5
4
÷ 5 or
1
×
5
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×4
1
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Inverse scale factors
5
To scale from 4 to 5 we multiply by
4
To scale from 5 to 4 we multiply by
When we scale from a smaller
number to a larger number the
scale factor must be more than 1.
5
×
4
4
5
×
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4
5
4
5
When we scale from a larger
number to a number smaller the
scale factor must be less than 1.
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Scale factor diagrams
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Scale factor diagrams
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Using scale factors
b
To scale a to b we multiply by
a
To scale b to a we multiply by
a
b
For example,
×
9
4
4
9
×
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×
4
9
3
7
7
3
×
7
3
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Using scale factors and direct proportion
We can use scale factors to solve problems involving direct
proportion.
£8 is worth 13 euros.
How much is £2 worth?
1
To scale from £8 to £2 we ×
or × 0.25
4
1
×
4
or × 0.25
£8 is worth 13€
1
×
4
£2 is worth (13 ÷ 4)€
or × 0.25
= 3.25€
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Using scale factors and direct proportion
We can use scale factors to solve problems involving direct
proportion.
£8 is worth 13 euros.
How much is £2 worth?
13
Alternatively, to scale from 8 to 13 we ×
or × 1.625
8
13
or × 1.625
×
8
£8 is worth 13€
£2 is worth (2 × 1.625)€ = 3.25€
×
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13
or × 1.625
8
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Using scale factors and direct proportion
We can use scale factors to solve problems involving direct
proportion.
£8 is worth 13 euros.
How much is £2 worth?
We can convert between any number of pounds or euros using
13
or × 1.625
×
8
pounds
euros
8
or × 0.615 (to 3 dp)
×
13
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Using scale factors and direct proportion
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Contents
N8 Ratio and proportion
N8.1 Ratio
N8.2 Dividing in a given ratio
N8.3 Direct proportion
N8.4 Using scale factors
N8.5 Ratio and proportion problems
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Direct proportion spider diagrams
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Cog wheels
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