GCSE Maths Practice: percentages

Question 5 of 10

Population growth at a fixed percentage is compound. Give your final answer to the nearest whole number.

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\( \begin{array}{l} \textbf{A population is } 5000.\\ \textbf{It grows by } 15\% \textbf{ per year for 3 years.}\\ \textbf{What is the population after 3 years?} \end{array} \)

Choose one option:

Use the compound multiplier (1 + 0.15)^3. Expect an answer a little above 7250.

Compound growth (Higher Tier)

When a quantity grows by the same percentage each year, each increase is applied to the latest value, not the original. This is compound growth. The multiplier for a growth rate of r% per year is \(1+\tfrac{r}{100}\), raised to the number of years.

Formula

\[ N=P(1+r)^t, \] where P is the initial amount, r is the decimal rate per period, and t is the number of periods.

Year-by-year check

Year 1: \(5000\times1.15=5750\).
Year 2: \(5750\times1.15=6612.5\).
Year 3: \(6612.5\times1.15=7604.375\Rightarrow 7604\) (nearest whole).

Estimation

Simple 15% three times would suggest roughly \(5000+0.45\times5000=7250\). Compound growth must be a bit higher than this; \(7604\) is reasonable.

Common mistakes

  • Adding the percentage three times to get \(7250\) (that’s simple, not compound).
  • Using the wrong earlier answer (e.g., \(6938\) comes from a different rate/time).
  • Rounding after each year; round only at the end.

Variants to practise

  • Decay: replace \((1+r)\) with \((1-r)\).
  • Mixed change: e.g., up 12% for 2 years then down 6% for 1 year: \(P\times1.12^2\times0.94\).
  • Reverse: given \(N\), find \(P=N/(1+r)^t\).