How do you reverse a percentage change?

Divide by the multiplier. After a 25% increase, divide by 1.25 to get the original.

How do areas and volumes scale in similar shapes?

If lengths scale by k, areas scale by k^2 and volumes by k^3.

What is the difference between direct and inverse proportion?

In direct proportion both quantities rise together at a fixed rate. In inverse proportion, one rises as the other falls and the product stays fixed.

TMUA Ratio, Proportion, Percentages and Units

Ratio and proportion questions use GCSE content in unfamiliar settings. Set up a relationship, solve with exact fractions, and check the units.

Paper 1 and Paper 2 topic. Specification codes: M1.1, M1.2, M3.3, M3.8, M3.9, M3.10. Last checked 12 November 2026.

Percentages

Use multipliers: an increase of 25% is multiplying by $1.25$ , and to reverse it you divide by $1.25$ , not subtract 25%.

Proportion

Direct proportion: $y=kx$ . Inverse proportion: $y=\dfrac kx$ . Find $k$ from the given values, then use it. Powers work the same way, as in $y=kx^2$ .

Similar shapes

If lengths are in ratio $1:k$ , then areas are in ratio $1:k^2$ and volumes in ratio $1:k^3$ .

Units and repeated change

Convert units before doing anything else. $72\text{ km/h}$ is $72\times\tfrac{1000}{3600}=20\text{ m/s}$ . Speed is distance over time, density is mass over volume and pressure is force over area.

Repeated percentage changes multiply: an increase of 10% followed by a decrease of 10% is $1.1\times0.9=0.99$ , a net fall of 1%.

For a ratio question, convert to a common unit and write each quantity as a share of the total number of parts. Check that your shares add back up to the total.

Worked examples

Example 1

After a 25% increase, a price is £60. What was the original price?

Show the worked solution

The original is multiplied by $1.25$ , so original $=\dfrac{60}{1.25}=48$ .

Answer: £48

Example 2

$y$ is inversely proportional to $x^2$ , and $y=8$ when $x=3$ . Find $y$ when $x=6$ .

Show the worked solution

$y=\dfrac k{x^2}$ with $k=8\cdot9=72$ . At $x=6$ , $y=\dfrac{72}{36}=2$ .

Answer: 2

Example 3

Two similar solids have volumes in the ratio $8:27$ . The smaller has surface area $20\text{ cm}^2$ . Find the larger surface area.

Show the worked solution

The length ratio is $2:3$ , so the area ratio is $4:9$ . Then $20\times\tfrac94=45\text{ cm}^2$ .

Answer: $45\text{ cm}^2$

Common mistakes

Reversing a percentage increase by subtracting the same percentage.
Using the length ratio for areas or volumes.
Mixing units, such as minutes with hours, in a rate.
Using direct proportion when the relationship is inverse.

Test yourself on this topic with our free practice questions.

Quick answers

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TMUA Ratio, Proportion, Percentages and Units

Percentages

Proportion

Similar shapes

Units and repeated change

Worked examples

Common mistakes

Quick answers

Practise it under timed conditions

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