Compound Interest
Compound interest generates earnings on both the original principal and previously earned interest, creating exponential growth over time. The formula A = P(1 + r)^n calculates the final amount where P is the principal, r the annual rate as a decimal, and n the number of years. This concept appears in Year 10 GCSE mathematics when students solve savings and investment problems.
Why it matters
Compound interest forms the foundation of personal finance and investment strategies throughout adult life. A £10,000 investment at 7% annual compound interest grows to approximately £38,697 after 20 years, demonstrating how small rate differences create substantial wealth variations. Pension funds, ISAs, and index funds all rely on compound growth to build retirement savings. Understanding this concept helps evaluate mortgage costs, where compound interest works against borrowers — a £200,000 mortgage at 5% over 25 years costs approximately £351,000 total. Banks use compound interest calculations for savings accounts, credit cards, and loans. The mathematical principle extends beyond finance into population growth models, radioactive decay, and scientific applications where exponential functions describe natural phenomena.
How to solve compound interest
Compound Interest
- Compound interest earns interest on both the original principal AND on previously earned interest — that's why the curve bends upward over time.
- Annual: A = P(1 + r)n, where P is the principal, r the rate as a decimal, n the number of years.
- Monthly compounding: A = P(1 + r/12)12n.
- With monthly contributions PMT: future value = PMT × [((1 + i)n − 1) / i], where i = r/12 and n is the number of months.
- Index funds and savings accounts both rely on this — small early differences in rate, time, or starting age compound to outsized differences at the end.
Example: £10,000 at 7% for 20 years: A = 10000 · 1.07²⁰ ≈ £38,697.
Worked examples
You deposit £4,000.00 in a savings account paying 4% interest per year. How much is in the account after one year?
Answer: 4160
- Calculate the interest → 4000 × 4/100 = 160 — Interest = principal × rate. £4,000.00 × 4% = £160.00.
- Add the interest to the principal → 4000 + 160 = 4160 — After one year: £4,000.00 + £160.00 = £4,160.00.
You invest £10,000.00 at 6% compound interest per year. What is the value after 2 years?
Answer: 11236
- Use the compound interest formula → A = P(1 + r)^n — P is the principal, r the rate as a decimal, and n the number of years.
- Plug in the values → A = 10000 × (1 + 0.06)^2 — P = £10,000.00, r = 0.06, n = 2.
- Compute the growth factor → (1 + 0.06)^2 = 1.1236 — Raise 1 + r to the power n.
- Multiply by the principal → A ≈ £11,236.00 — £10,000.00 × 1.1236 ≈ £11,236.00 after rounding.
You invest £10,000.00 in an index fund returning 8% per year. What is the value after 7 years, assuming returns are reinvested?
Answer: 17138
- Apply A = P(1 + r)^n → A = 10000(1 + 0.08)^7 — Reinvested returns compound — the formula treats each year's gain as next year's principal.
- Compute the result → A ≈ £17,138.00 — After 7 years, £10,000.00 grows to approximately £17,138.00 — a gain of £7,138.00.
Common mistakes
- Confusing simple and compound interest calculations, where £5,000 at 4% for 3 years yields £600 simple interest but £624.32 compound interest — the difference of £24.32 comes from interest earning interest.
- Applying the wrong compounding frequency, such as using annual compounding A = P(1 + r)^n when the problem specifies monthly compounding A = P(1 + r/12)^(12n), leading to incorrect results.
- Forgetting to convert percentage rates to decimals, calculating £1,000 × (1 + 5)^2 = £36,000 instead of £1,000 × (1 + 0.05)^2 = £1,102.50.