Compound growth & decay

Description

Growth and decay problems involve quantities that increase or decrease by a fixed percentage repeatedly over time. The formula uses a multiplier raised to a power.

$$\text{Amount} = P \times r^n$$

where $P$ is the initial value, $r$ is the multiplier (e.g. $1.05$ for 5% growth, $0.92$ for 8% decay), and $n$ is the number of time periods.

Compound interest example: £2000 invested at 3% per year for 5 years. $$A = 2000 \times 1.03^5 \approx 2000 \times 1.1593 = £2318.55$$

Depreciation example: A car worth £12,000 depreciates at 15% per year. Value after 4 years: $$V = 12000 \times 0.85^4 \approx 12000 \times 0.5220 = £6264$$

This is not the same as simple interest/decrease (which adds the same amount each time) — compound growth is multiplicative.

Common error: adding the percentage change each year instead of multiplying by the power. Also: using the wrong multiplier (e.g. 0.15 instead of 0.85 for 15% decrease).

Links

Percentage change Basic indices

Use the $x^n$ key: enter $1.03$ then $x^5$ to compute $1.03^5$ in one step.