Compound Interest and Depreciation
Albert Einstein (allegedly) called compound interest the "eighth wonder of the world." Whether he said it or not, the maths is genuinely powerful. Your money earns interest, and then that interest earns interest, and it snowballs. Depreciation is the sad reverse — things lose value over time.
Simple vs Compound Interest
Simple interest: you earn interest only on the original amount. £1,000 at 5% simple interest earns £50 every year, forever. Boring but predictable.
Compound interest: you earn interest on the original amount AND on previously earned interest. After year 1 you have £1,050. In year 2 you earn 5% of £1,050 = £52.50. It keeps growing faster and faster. This is exponential growth.
Key Formulas
- Compound Interest: A = P(1 + r/100)^n
- Depreciation: A = P(1 - r/100)^n
- P = principal (starting amount), r = rate (%), n = number of years
- A = final amount
- Multiplier for growth: 1 + r/100 (e.g., 3% growth → multiply by 1.03)
- Multiplier for decay: 1 - r/100 (e.g., 15% depreciation → multiply by 0.85)
Worked Examples
Example 1 — £5,000 invested at 4% compound interest for 3 years
- Multiplier = 1 + 4/100 = 1.04
- A = 5000 x 1.04^3
- 1.04^3 = 1.124864
- A = 5000 x 1.124864 = £5,624.32
Answer: £5,624.32
Interest earned: £624.32. With simple interest it would only be £600.
Example 2 — A car costs £18,000 and depreciates by 15% per year. What is it worth after 4 years?
- Multiplier = 1 - 15/100 = 0.85
- A = 18000 x 0.85^4
- 0.85^4 = 0.52200625
- A = 18000 x 0.52200625 = £9,396.11
Answer: £9,396.11
The car has lost almost half its value in just 4 years!
Example 3 — How many years for £2,000 to exceed £3,000 at 6% compound interest?
- We need: 2000 x 1.06^n > 3000
- 1.06^n > 1.5
- Using trial and improvement or logarithms:
- n = 6: 1.06^6 = 1.4185... (not enough)
- n = 7: 1.06^7 = 1.5036... (yes!)
Answer: 7 years
Using logs: n = ln(1.5)/ln(1.06) = 6.96, so 7 complete years.
Example 4 — After 2 years at 5% compound interest, an account holds £2,205. What was the original investment?
- A = P x 1.05^2, so P = A / 1.05^2
- 1.05^2 = 1.1025
- P = 2205 / 1.1025 = £2,000
Answer: £2,000
Why This Matters in Real Life
Compound interest affects your savings account, your student loan, your mortgage, and inflation. Understanding the formula means you can compare bank accounts, predict how long it takes to double your money (roughly 72/rate years — the "Rule of 72"), and understand why starting to save early makes such a massive difference.
Practice Questions
- £3,000 at 5% compound interest for 2 years
- A laptop worth £800 depreciates by 20% per year. Value after 3 years?
- How many years for £1,000 to double at 8% compound interest?
- After 3 years at 4%, an account holds £5,624.32. Find the original amount.
- A painting appreciates by 12% per year. It is worth £4,000 now. What will it be worth in 5 years?
Answers: 1) £3,307.50 2) £409.60 3) 9 years 4) £5,000 5) £7,049.37