Repeated percentage change

Objective. Apply a percentage multiplier repeatedly and interpret the result.

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Learn: the key idea

Repeated percentage change applies a percentage multiplier more than once, such as for several years of growth or decay, and is calculated by raising the multiplier to the power of the number of times it applies. For a value growing by 10% each year for 3 years, the overall multiplier is 1.1³, not 1.3, because each year's growth is calculated on the new, already-increased amount. For example, £200 growing by 10% for 3 years becomes £200 × 1.1³ = £266.20. This compounding effect means repeated percentage increases grow faster than simply adding the percentage each time.

Worked example

A population of 4000 grows by 5% each year. Find the population after 2 years of repeated percentage change.

1 $Write the annual multiplier: 100% + 5% = 105%, so 1.05$
2 $Apply it twice for two years: 4000 \times 1.05²$
3 $Calculate 1.05² = 1.1025$
4 $Multiply: 4000 \times 1.1025 = 4410$

Answer: 4410

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Common mistake. Pupils often multiply the original amount by 3 × 10% = 30% for three years of 10% growth, treating it like simple interest, instead of correctly using the multiplier 1.1³ which accounts for compounding.

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