Growth and decay
The compound interest / growth formula
$$A = P imes left(1 + rac{r}{100} ight)^n$$
Where A = final amount, P = initial amount (principal), r = rate per period (%), n = number of periods.
Growth (r > 0): e.g. population, investment at interest. Decay (r < 0 or use 1 − r/100): e.g. depreciation, radioactive decay, drug elimination.
For decay: A = P × (1 − r/100)ⁿ.
Compound interest (calculator)
Example: £4,000 invested at 3.2% per annum for 7 years. A = 4000 × (1.032)⁷ = 4000 × 1.2457... ≈ £4,983.
On Edexcel Papers 2 and 3, give your answer to the nearest penny unless otherwise stated.
Depreciation
A car worth £18,000 depreciates at 14% per year. Find its value after 5 years. Value = 18,000 × (0.86)⁵ = 18,000 × 0.4704... ≈ £8,467.
Comparing simple and compound interest
Simple interest: interest = P × r × n / 100 (same interest each period). Compound interest grows faster — interest is earned on the growing balance.
Exponential growth and decay graphs
- Growth: y = Abˣ with b > 1. Curve increases, getting steeper.
- Decay: y = Abˣ with 0 < b < 1. Curve decreases, approaching zero.
- On a percentage-scale graph, compound growth appears as a straight line.
Edexcel Higher — solving for n (the time period)
"How long until the investment doubles?" requires solving: 2P = P × (1.05)ⁿ → 2 = (1.05)ⁿ → n = log(2)/log(1.05). Edexcel may ask you to use trial and improvement or recognise the form without requiring logarithms at GCSE.
⚠Common mistakes
- Using simple interest when compound is required (or vice versa).
- Wrong multiplier: for 12% depreciation, use 0.88, not 1.12 or 0.12.
- Forgetting to raise to the power n: the formula is (1 ± r/100)ⁿ, not × n.
- Rounding intermediate steps: keep full precision until the final answer.
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