When a problem is concerned with two variables, and the ratio of one of the variables to another is always the same, we say that the variables are in direct proportion.

An example might be the number of kilograms of potatoes that you buy and what you pay for them. For example, if potatoes cost 50p per kilogram and you buy 6 kg, you pay £3. If you buy 20 kg, you pay £10. The ratio 6 : 3 is the same as the ratio 20 : 10, since they both simplify to 2 : 1.

Direct proportion problems can also be solved by scaling.

In a box of sweets there are 5 toffees for every 2 chocolates. There are 15 toffees in the box. How many chocolates are there?

In this problem, the ratio of the number of toffees to the number of chocolates is constant. The ratio is 5:2. The solution can be modelled on a number line.

5 is multiplied by 3 to make 15 (i.e. scaled up by a factor of 3), so 2 must also be multiplied by 3 to get the answer of 6 chocolates.

We could also think of this by looking at the ratio:

Is equivalent to

I have multiplied the 5 to get 15
So I need to multiply the 2 by 3 as well.
So the solution is

5 : 2

15 : ?

15:6

Another approach to solving ratio problems is called the unitary method. In this method one of the variables is reduced to 1. Then it is possible to find out what the other variable would be for any number. For example:

A cake recipe for 6 people needs 120 g flour. How much flour will a cake for 7 people need?

6 people need 120 g flour.
1 person needs 120 ÷ 6 = 20 g flour.
7 people need 20 × 7 = 140 g flour.

Additional User Example

George and Linda are brother and sisters. George is 10yrs old and Linda is 15yrs old. For pocket money George gets £10 and Linda gets £15. The ratio for this is 10:15, which when simplified is 2:3

George gets 25 of the pocket money and Linda gets 35 of the pocket money given by their parents.

If Linda's pocket money goes up to £24 pounds it is possible to work out what George will get by thinking of the ratio: 2:3

2 : 3

16 24

×8

I have multiplied the 3 by 8 to get 24

So I also need to multiply the 2 by 8. Which gives 16

What this might look like in the classroom

Question 1:

If the ratio of boys : girls in a class is 3 : 1 could there be exactly 30 children in the class? Why? Could there be 20 boys? Which numbers of pupils are possible and why?

Answer 1:

There can’t be 30 pupils in the class because the number of pupils must be a multiple of four in order to be shared into 4 parts for a ratio of 3 : 1. If you do not have a multiple of 4 as the number of pupils in the class you will not have a ratio of 3:1. By the same reasoning you can’t have 20 boys because the number of boys must be a multiple of 3.

Question 2:

Find the missing numbers for the ratio 4:1:

16
a
12
40
d

:
:
:
:
:

4
8
b
c
3

Answer 2:

a = 32, b = 3, c = 10 and d = 12

Taking this mathematics further

Much work in science concerns the relationship between variables and rates, which are an application of direct proportion.

Proportion is a common theme in other areas of life. You might like to find out about body proportions, for example about Leonardo Da Vinci’s Vetruvian man.

Proportions, and the relationships between certain lengths, are also important in architecture and in art. Again, you might like to find out more about this.

Making connections

Proportionality and multiplicative relationships are one of the ‘big ideas’ in mathematics.

The graph of two variables that are in direct proportion is a straight line through (0, 0).

There are usually a number of methods that can be used to solve problems involving proportion.

This topic therefore links to others across the mathematics curriculum. There are direct links to fractions, decimals and percentage as well as to work in calculation but also links to algebra, enlargement, trigonometry, rates and scale.

Whilst two variables may be in direct proportion, they may also be related in a different way, for example by being inversely proportional.