Compound Interest (Growth/Decay)

GCSE Percentages compound growth decay
Practice this formula All formulas
\( A=P\left(1+\tfrac{r}{100}\right)^{n} \)

Statement

The compound interest formula models repeated growth or decay where each step is based on the previous total. The general form is:

\[ A = P\left(1 + \frac{r}{100}\right)^n \]

Here, \(A\) is the final amount, \(P\) is the initial principal (starting value), \(r\) is the percentage rate per period, and \(n\) is the number of periods.

Why it’s true

  • In simple interest, the increase each time is fixed. In compound interest, each increase is calculated from the new total.
  • After 1 period, the amount is \(P\left(1+\tfrac{r}{100}\right)\).
  • After 2 periods, the amount is \(P\left(1+\tfrac{r}{100}\right)\left(1+\tfrac{r}{100}\right)=P\left(1+\tfrac{r}{100}\right)^2\).
  • Repeating for \(n\) periods gives the formula.

Recipe (how to use it)

  1. Identify the starting amount \(P\).
  2. Decide whether it is growth (\(r>0\)) or decay (\(r<0\)).
  3. Convert the percentage rate into a multiplier: \(1+\tfrac{r}{100}\).
  4. Raise this multiplier to the power of the number of periods \(n\).
  5. Multiply by the starting amount \(P\).

Spotting it

Look for words such as “compound interest”, “grows by x% per year”, “decays by x% each cycle”, or “repeated percentage change”. These signal the use of the formula.

Common pairings

  • Finance (bank accounts, loans, investments).
  • Population growth and decay models.
  • Depreciation of assets (cars, machinery).

Mini examples

  1. \(P=1000\), \(r=5\%\), \(n=3\): \(A=1000(1.05)^3≈1157.63\).
  2. \(P=500\), \(r=-10\%\), \(n=2\): \(A=500(0.9)^2=405\).

Pitfalls

  • Using simple interest instead: Don’t just do \(P+nr\); compound means repeated multiplication.
  • Forgetting decay: If it’s decay, use \(1-\tfrac{r}{100}\).
  • Wrong percentage format: Always convert the percentage to a decimal fraction (e.g. 5% = 0.05).

Exam strategy

  • Check carefully if the problem says simple or compound.
  • Underline the rate and number of periods in the question.
  • Write out the multiplier step (e.g. \(1.05\), \(0.92\)) to avoid mistakes.
  • Round at the end, unless instructed otherwise.

Summary

The compound interest formula models repeated percentage change. Growth uses multipliers greater than 1, while decay uses multipliers less than 1. The formula \(A=P\left(1+\tfrac{r}{100}\right)^n\) ensures that each step is applied to the current total, not just the starting amount. This makes it essential for real-world growth and decay contexts.