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 sometimes 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 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 as finding a pair of equivalent fractions such that:
Another approach is the unitary method, in which the value of one of the variables is reduced to 1.
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.