Percentage and GrowthEdexcel GCSE Maths: Revision notes

A percentage change describes a multiplicative relationship between two quantities — one quantity is a fraction or multiple of the other, rather than simply a difference. Expressing a relationship multiplicatively (as a ratio or fraction) is what allows percentage change, scaling and growth to be calculated using a single multiplier.

Example: if a price rises from £80 to £100, the multiplicative relationship is 100/80 = 1.25, meaning the new price is 1.25 times the old one (a 25% increase).

When one quantity grows in direct proportion to another, it follows the rule y = kx, where k is a constant multiplier. As x increases, y increases at the same constant rate — this is the simplest model of growth.

Example: if the cost of fuel is directly proportional to the number of litres bought, doubling the litres doubles the cost.

Direct proportion problems are solved by first finding k (using one known pair of values), then using y = kx to find any other value.

Growth relationships are not always simple linear proportion. Other common proportion equations include:

• y = kx² (y proportional to the square of x — growth speeds up rapidly)
• y = k√x (y proportional to the square root of x — growth slows down as x increases)
• y = k/x and y = k/x² (inverse proportion — y decreases as x increases)

To solve these, substitute a known pair of values to find k, then use the equation to find unknown values, just as with y = kx.

To express quantity A as a fraction of quantity B, calculate A ÷ B. This fraction can be:

• Less than 1 (A is smaller than B) — equivalent to a percentage below 100%
• Greater than 1 (A is bigger than B) — equivalent to a percentage above 100%

Example: if A = 120 and B = 80, then A as a fraction of B is 120/80 = 1.5, i.e. 150% — A is one and a half times the size of B, showing growth beyond the original amount.

Growth and percentage problems often require several steps, mixing fractions, decimals and ratios. A reliable method:

1. Identify what multiplier or fraction links the quantities
2. Convert all values to the same form (decimals are usually easiest for multi-step calculations)
3. Apply each step in order, keeping track of what each intermediate answer represents
4. Check the final answer makes sense in context (e.g. growth should increase the value)

Example: a quantity increases by 20% then decreases by 10%. Overall multiplier = 1.20 × 0.90 = 1.08, an overall 8% increase (not 10%, since the two percentage changes apply to different amounts).