GCSE Maths: Ratio, Proportion and Rates of Change Guide

Ratio

A ratio compares two or more quantities. To share an amount in a given ratio, add the parts of the ratio together to find the total number of shares, then divide the amount by this to find the value of one share.

Direct proportion

Two quantities are in direct proportion if one increases at the same rate as the other — doubling one doubles the other. This can be written as y = kx, where k is the constant of proportionality.

Inverse proportion

Two quantities are in inverse proportion if one increases as the other decreases at a matching rate — doubling one halves the other. This is written as y = k/x.

Rates of change

A rate of change compares how one quantity changes in relation to another, such as speed (distance ÷ time) or density (mass ÷ volume). On a distance-time graph, the gradient represents speed.

Common mistakes

• Sharing an amount by the ratio numbers themselves instead of finding the value of one share first.
• Mixing up direct and inverse proportion — check whether both quantities increase together or one increases as the other decreases.
• Forgetting to keep units consistent when calculating rates like speed.