1. KS3 Mathematics N8 Ratio and proportion
The aim of this unit is to teach pupils to:
Understand the relationship between ratio and proportion, and use ratio and proportion to solve simple problems.
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
N8 Ratio and proportion

2. N8 Ratio and proportion Contents A1 N8.1 Ratio A1
N8.2 Dividing in a given ratio A1
N8.3 Direct proportion A1
N8.4 Using scale factors A1
N8.5 Ratio and proportion problems

3. 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?
Talk through the points on the slide showing, with reference to the diagram, that the ratio 9 : 3 is equivalent to the ratio 3 : 1. State that this is the ratio in its simplest form. Compare this to simplifying fractions.
Ask pupils what statements they can make about the number of red counters compared with the number of blue counters. For example, ‘the number of blue counters is a third of the number of red counters’ or ‘the number of red counters is three times the number of blue counters’.
To distinguish between ratio and proportion you may wish to ask,
What proportion of the counters are red? (three quarters)
Stress the ratio compares the sizes of parts to each other while proportion compares the sizes of parts to the whole.
red : blue
= 9 : 3
= 3 : 1
For every three red counters there is one blue counter.

4. Ratio A ratio compares the sizes of parts or quantities to each other.
For example,
What is the ratio of blue counters to red counters?
The ratio of blue counters to red counters is not the same as the ratio of red counters to blue counters.
Note the difference between comparing the number of red counters to the number of blue counters and comparing the number of blue counters to the number of red counters.
You may wish to introduce the idea of a multiplicative inverse stating that × 3 is the inverse of × 1/3. This idea is examined later on when multiplicative reasoning is developed through the scaling of any given number to another and back again.
blue : red
= 3 : 9
= 1 : 3
For every blue counter there are three red counters.

5. Ratio
What is the ratio of red counters to yellow counters to blue counters?
red : yellow : blue
= 12 : : 8
Show that ratios can compare more than two parts or quantities.
Explain with reference to the diagram that 12 : 4 : 8 simplifies to 3 : 1 : 2.
For every three red counters there is one yellow counter and two blue counters.
= 3 : : 2
For every three red counters there is one yellow counter and two blue counters.

6. Simplifying ratios
Ratios can be simplified like fractions by dividing each part by the highest common factor.
For example,
21 : 35
÷ 7
= 3 : 5
For a three-part ratio all three parts must be divided by the same number.
Discuss the simplification of ratios.
For example,
6 : 12 : 9
÷ 3
= 2 : 4 : 3

7. 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 can simplify them in the ratio.
Stress that ratios should always be expressed using the same units for each part.
90 : 300
÷ 30
= 3 : 10

8. 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
This example shows the simplification of a three part ratio expressed in m, cm and mm.
State that we could also simplify this ratio by converting the parts to metres (although in this example that would involve working with decimals) or millimetres (although in this example that would involve working with larger numbers).
÷ 15
= 4 :
2 : 3

9. 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
State that multiplying any number that has one number after the decimal point by 10, will give us a whole number. We must then multiply the other part by 10 to preserve equality.
Similarly, if a ratio contains parts with two digits after the decimal point we can multiply the parts by 100 to find an equivalent whole-number ratio.
The whole-number ratio must then be simplified if possible.
= 8 : 20
÷ 4
= 2 : 5

10. Simplifying ratios containing fractions
Simplify the ratio : 4. 2 3
We can write this ratio in whole-number form by multiplying both parts by 3. 2 3
: 4
× 3
Explain that if we multiply a fraction by its denominator then we will always have a whole number. In this example then, we can multiply both parts of the ratio by 3.
The whole-number ratio must then be simplified if possible.
= 2 : 12
÷ 2
= 1 : 6

11. Comparing ratios
We can compare ratios by writing them in the form 1 : m, 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
= 1 : 1.6
The ratio 5 : 8 can also be written in the form m : 1 by dividing both parts of the ratio by 8.
We could also write these using fraction as 1 : 8/5 and 5/8 : 1. However, when comparing the sizes of two different ratios it is usually easier to compare the sizes of decimals than the sizes of fractions.
Remind pupils that, unless they are told how many decimal places they should round to, they must decide whether to show their answers to one, two or three decimal places as appropriate.
5 : 8
÷ 8
= : 1

12. 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 : 5
÷ 4
= 1 : 1.25
The ratio of boys to girls in 9G is
5 : 7
Before solving this problem we must decide whether we should show the ratios in the form 1 : m or m : 1.
Discuss the fact that in this example the ratio 1 : m would tell us the number of girls for every boy and the ratio m : 1 would tell us the number of boys for every one girl. Conclude that it would be more appropriate to find the number of girls for every one boy. The ratio that has more girls for every one boy has the higher proportion of girls.
Talk through the example on the board and ask,
Is it possible for 9P to have more girls than 9G?
Stress that a higher proportion of girls does not mean more girls. 9P could have 27 pupils, 15 of which are girls, while 9G could have 24 pupils, 14 of which are girls.
÷ 5
= 1 : 1.4
9G has a higher proportion of girls.

13. N8.2 Dividing in a given ratio
Contents
N8 Ratio and proportion A1
N8.1 Ratio A1
N8.2 Dividing in a given ratio A1
N8.3 Direct proportion A1
N8.4 Using scale factors A1
N8.5 Ratio and proportion problems

14. 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

15. 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

16. 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

17. 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 ml + 75 ml =
750 ml

18. N8 Ratio and proportion Contents A1 N8.1 Ratio A1
N8.2 Dividing in a given ratio A1
N8.3 Direct proportion A1
N8.4 Using scale factors A1
N8.5 Ratio and proportion problems

19. 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?
Discuss the various ways of expressing a proportion in words, as a fraction, as a decimal and as a percentage.
Ask pupils to tell you what proportion of the counters are blue.
Compare this slide to the ratio slide that shows the ratio of red counters to blue counters.
Stress that ratio compares the sizes of parts or quantities to each other while proportion compares the size of a part to the size of the whole.
Ratios can also be expressed as fractions, decimals and percentages and so pupils often confuse these two.
We can express this proportion as: 3 4
12 out of 16
3 in every 4
0.75 or
75%

20. 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
180 g.
Establish that the number of packets of crisps and the weight of the packet are in direct proportion as long as the weight of each packet is the same.
That means that if we double the number of packets, as in the example, we double the weight. If we half the number of packets, we half the weight, and so on.
The number of packets and the weight of the packets are in direct proportion if the ratio of the number of packets : weight of packets is always the same.
If we double the number of packets then we double the weight.
The number of packets and the weights are in direct proportion.

21. Direct proportion problems
3 packets of crisps weigh 90 g.
How much does 1 packet 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.
Establish that if we find out the weight of one packet, then we can work out the weight of any given number of packets by multiplying.
For example, 8 packets would weigh (8 × 30) g = 240 g.
Once we know the weight of one packet we can work out the weight of any number of packets.

22. 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
State that a unitary method is a method where we first find out the value of one.
7 packets weigh
210 g.
This is called using a unitary method.

23. N8 Ratio and proportion Contents A1 N8.1 Ratio A1
N8.2 Dividing in a given ratio A1
N8.3 Direct proportion A1
N8.4 Using scale factors A1
N8.5 Ratio and proportion problems

24. How can we get from 4 to 5 using only multiplication and division?
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.
State that when solving ratio and proportion problems we are usually required to scale from one number to another number.
Discuss the scaling from 4 to 5 in terms of multiplication and division.
(4 × 5) ÷ 4 = 5

25. How can we divide by 4 and multiply by 5 in a single step?
Using scale factors
How can we divide by 4 and multiply by 5 in a single step? × 5 4
Dividing by 4 and multiplying by 5 is equivalent to 4 × 5 =
We call the 5 4
a multiplier or a scale factor.
Introduce the idea of a scale factor as a number that scales one number to another number using multiplication only.
Is it possible to scale from any number to any other number? (yes) 5 4 =
We can write the scale factor as a decimal,
1.25. 5 4 =
We can also write it as a percentage,
125%.

26. Using a diagram to represent scale factors
We can represent the scaling from 4 to 5 using a diagram: × 5 4 5 4 1
× 5
This diagram gives a visual representation of the scaling from 4 to 5.
÷ 4 or × 1 4

27. How can we get from 5 to 4 using only multiplication and division?
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.
This slide show how scaling from 5 to 4 differs from scaling from 4 to 5.
(5 × 4) ÷ 5 = 4

28. How can we divide by 5 and multiply by 4 in a single step?
Using scale factors
How can we divide by 5 and multiply by 4 in a single step? × 4 5
Dividing by 5 and multiplying by 4 is equivalent to 5 × 4 =
We call the 4 5
a multiplier or a scale factor.
Introduce the idea of using a single number as a multiplier or scale factor. 4 5 =
We can write the scale factor as a decimal,
0.8. 4 5 =
We can also write it as a percentage,
80%.

29. Using a diagram to represent scale factors
We can represent the scaling from 5 to 4 using a diagram: 1 5 × 4 5 4
This diagram gives a visual representation of the scaling from 5 to 4, using an intermediate step and using a single step.
× 4
÷ 5 or × 1 5

30. 4 5 Inverse scale factors 5 To scale from 4 to 5 we multiply by . 4 4
When we scale from a smaller number to a larger number the scale factor must be more than 1. × 5 4 4 5
4/5 is the multiplicative inverse (or reciprocal) of 5/4.
You may wish to state that we can also find the inverse by dividing by the 5/4.
Pupils will be more familiar with finding the inverse operation, division, and keeping the same operator. However, in the context of scale factors we want to keep the same operation, multiplication, and find the inverse operator (the multiplicative inverse or reciprocal).
The concept of a multiplicative inverse may be a difficult concept for many pupils. It is not necessary to discuss this point with lower ability pupils.
Ask pupils how we can tell when a fraction is greater than 1 (the numerator is bigger than the denominator).
Ask pupils how we can tell when a fraction is less than 1 (the denominator is bigger than the numerator).
When we scale from a larger number to a number smaller the scale factor must be less than 1. × 4 5

31. 4 9 7 3 Using scale factors b a . To scale a to b we multiply by a b .
To scale b to a we multiply by
For example, × 9 4 × 3 7
Again, stress that when we scale from a smaller number to a larger number the scale factor must be more than 1 (that is, the numerator must be greater than the denominator) and when we scale from a larger number to a smaller number the scale factor must be less than 1 (that is, the numerator must be smaller than the denominator). 4 9 7 3 × 4 9 × 7 3

32. 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?
To scale from £8 to £2 we multiply × 1 4
or × 0.25. × 1 4
or × 0.25 × 1 4
or × 0.25
£8 is worth 13€
£2 is worth
(13 ÷ 4)€
= 3.25€

33. 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 8
or ×
Alternatively, to scale from 8 to 13 we multiply × 13 8
or × 1.625
£8 is worth 13€
£2 is worth
(2 × 1.625)€ =
3.25€ × 13 8
or × 1.625

34. 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 8
or × 1.625
pounds
euros × 8 13
or × (to 3 dp)