With compound interest, each year's interest is added to the balance, so the next year earns interest on a bigger amount. Multiply by the same multiplier once for each year: £500 at 4% for 2 years is 500 × 1.04² = £540.80. Depreciation works the same way with a multiplier below 1. Simple interest, by contrast, pays the same amount every year.
Compound interest and depreciation
- Value after n years = original × multiplierⁿ
- Growth of r%: multiplier = 1 + r/100
- Depreciation of r%: multiplier = 1 − r/100
- Simple interest per year = r% of the original amount
- Interest earned = final value − original amount
How compound interest works
With compound interest, the interest is added to the account at the end of each year, so the next year's interest is calculated on the larger amount. That is why it grows faster than simple interest over time. Instead of working year by year, multiply by the multiplier once for every year: 5% for 3 years is × 1.05³.
Depreciation
Depreciation is compound decay: an item loses the same percentage of its value each year. A car losing 15% a year keeps 85% of its value, so the multiplier is 0.85. After n years its value is the original × 0.85ⁿ. Notice that it loses less money each year, because 15% of a smaller value is smaller.
Simple interest versus compound interest
Simple interest pays the same amount every year, a fixed percentage of the original amount. £2500 at 3% simple interest earns £75 a year, so £225 over 3 years. With compound interest the total after 3 years is 2500 × 1.03³ = £2731.82, so the interest is £231.82. Compound interest always earns more over two or more years.
Interest earned and final value
Read carefully whether a question asks for the final value or the interest earned. The interest is the final value minus the original amount. Give money answers to the nearest penny unless told otherwise.
Finding the number of years
To find how many years it takes to pass a target, multiply by the multiplier repeatedly, year by year, until the value crosses the target. A calculator's answer key makes this quick: type the starting value, then keep multiplying by the multiplier.
Worked examples
Example 1. £500 is invested at 4% compound interest per year. How much is it worth after 2 years?
- The multiplier is 1.04.
- 500 × 1.04² = 500 × 1.0816.
Answer: £540.80.
Example 2. A machine worth £8000 depreciates by 10% per year. What is it worth after 2 years?
- The multiplier is 0.9.
- 8000 × 0.9² = 8000 × 0.81.
Answer: £6480.
Example 3. £2500 is invested for 3 years at 3% per year. How much more interest does compound interest earn than simple interest?
- Simple: 3% of 2500 = 75 per year, so 225 over 3 years.
- Compound: 2500 × 1.03³ = 2731.8175, so interest = 231.82.
- Difference: 231.82 − 225.
Answer: £6.82 more.
Example 4. £1200 is invested at 6% compound interest. After how many whole years will it first be worth more than £1500?
- After 3 years: 1200 × 1.06³ = 1429.22.
- After 4 years: 1200 × 1.06⁴ = 1514.97.
Answer: 4 years.
Example 5. A phone bought for £640 loses 25% of its value each year. Work out its value after 3 years.
- The multiplier is 0.75.
- 640 × 0.75³ = 640 × 0.421875.
Answer: £270.
Common mistakes
- Using simple interest for a compound question. Compound interest multiplies by the multiplier each year.
- Multiplying by 4 instead of 1.04. A 4% increase uses the multiplier 1.04.
- Using 0.15 for a 15% depreciation. The multiplier is 0.85.
- Giving the final value when the interest is asked for. Interest = final value − original amount.
- Rounding each year. Keep full accuracy and round only the final answer.
How compound interest is examined
Compound interest and depreciation are tested at both tiers of Edexcel GCSE (1MA1) and in the International GCSE (4MA1), almost always on calculator papers. Questions ask you to:
- work out the value of an investment after a number of years
- work out the value of an item after depreciation
- compare simple and compound interest
- find how many years it takes to reach a target
Money-based growth and decay questions, with answers, are on the growth and decay worksheet; the number revision engine adds fresh ones each time.
This builds on percentages. For populations and science contexts, see compound growth and decay.
Frequently asked questions
Is there a formula for compound interest?
Final value = original × (1 + r/100)ⁿ, where r is the interest rate and n is the number of years.
What is the difference between simple and compound interest?
Simple interest pays the same amount every year; compound interest pays interest on the interest already earned.
How do you calculate depreciation?
Multiply by (1 − r/100) once for each year. A 20% loss per year uses 0.8ⁿ.