Ratio and ProportionAQA GCSE Maths: Revision notes

Ratio notation (e.g. $3:2$) compares two or more quantities and should always be reduced to its simplest form by dividing by the highest common factor.

A ratio can be expressed as a fraction: expressing one quantity as a fraction of another may give a fraction less than 1 (a part of a whole) or greater than 1 (a comparison of unequal parts). A multiplicative relationship between two quantities (e.g. "one is 3 times the other") can also be written as a ratio or fraction, and ratios relate directly to linear functions ($y = kx$).

Ratio notation and scale factors are used to compare lengths, areas and volumes, linking to similarity (including trigonometric ratios). If two shapes are similar with a linear scale factor $k$, corresponding lengths are in ratio $1:k$.

To divide a quantity into parts using a ratio:

1. Add the parts of the ratio to find the total number of shares
2. Divide the total quantity by the number of shares to find the value of one share
3. Multiply each ratio part by the value of one share

A quantity can also be split and the split expressed as a ratio, and ratios can describe a part:part or part:whole relationship — be careful to identify which is being used.

Proportion means an equality of ratios — if two ratios are equal, the quantities are in proportion. Ratio and proportion are applied to a wide range of real contexts:

• Conversion (e.g. between currencies or units)
• Comparison and scaling
• Mixing and concentrations (e.g. paint or drink recipes)
• Best-buy problems, where you compare value for money between different pack sizes
• Scale factors, scale diagrams and maps, including geometrical problems involving scaled figures