Growth, decay and compound interest

Compound interest and exponential growth/decay are tested on OCR J560 Papers 2 and 3. The formula approach is the most efficient. Higher-tier questions may involve finding the interest rate or the time period.

Compound interest formula

A = P(1 + r/100)ⁿ

where:

• A = final amount
• P = principal (initial amount)
• r = annual interest rate (%)
• n = number of years

Example: £2,000 invested at 3.5% compound interest for 4 years. A = 2000 × (1.035)⁴ = 2000 × 1.1475… = £2295.05 (to nearest penny).

Depreciation (compound decay)

A = P(1 − r/100)ⁿ

Example: Car bought for £12,000 depreciates by 15% per year. Value after 3 years: A = 12000 × (0.85)³ = 12000 × 0.614125 = £7369.50.

General exponential growth and decay

Growth: y = a × (multiplier)^n where multiplier > 1. Decay: y = a × (multiplier)^n where 0 < multiplier < 1.

These are the same formula — just different multipliers.

Inverse problems: finding rate or time

Finding the rate: given A, P and n, solve for r. A/P = (1 + r/100)ⁿ → (A/P)^(1/n) = 1 + r/100 → r = [(A/P)^(1/n) − 1] × 100.

Finding time n: requires logarithms at A-level; at GCSE, trial and improvement or systematic iteration.

Example: How many years until £500 exceeds £600 at 4% compound interest? Year 1: 500 × 1.04 = 520. Year 2: 520 × 1.04 = 540.8. … Year 5: ≈ 608.33. Answer: 5 years.

Simple interest (for comparison)

Simple interest: I = PRT/100 (interest the same each year — not compounded).

Compound always grows faster than simple for positive rates.

Common OCR exam mistakes

1. Using simple interest instead of compound — they specifically ask for compound.
2. Not converting r to a decimal multiplier: using 3.5 instead of 1.035.
3. Depreciation: using (1 + 0.15) instead of (1 − 0.15) = 0.85.
4. Rounding intermediate steps — keep full calculator accuracy until the final answer.