What Is Compound Interest?
Compound interest is interest calculated on both the original amount (the principal) and any interest already earned. Unlike simple interest, which is only calculated on the principal, compound interest leads to exponential growth because you earn interest on your interest.
The formula for compound interest is:
A = P(1 + r/100)ⁿ
Where A is the final amount, P is the principal (initial amount), r is the annual interest rate as a percentage, and n is the number of years.
This formula appears on the IGCSE formula sheet, but understanding how to use it and when to apply it is what earns you the marks.
Simple Interest vs Compound Interest
With simple interest, the interest earned each year is the same because it is always calculated on the original principal. With compound interest, the interest grows each year because it is calculated on the accumulated total.
Consider $1000 invested at 5% for 3 years:
Simple interest: Interest = $1000 × 0.05 × 3 = $150. Total = $1150.
Compound interest: Year 1: $1000 × 1.05 = $1050. Year 2: $1050 × 1.05 = $1102.50. Year 3: $1102.50 × 1.05 = $1157.63.
Or using the formula: A = 1000 × 1.05³ = 1000 × 1.157625 = $1157.63.
The compound interest earns $7.63 more than simple interest over 3 years. This difference grows dramatically over longer periods.
Applying the Formula
Example 1: $5000 is invested at 4% compound interest for 6 years. Find the total amount.
A = 5000 × (1 + 4/100)⁶ = 5000 × 1.04⁶ = 5000 × 1.26532… = $6326.60 (to nearest cent).
Example 2: A painting is bought for $12,000 and increases in value by 3.5% per year. Find its value after 10 years.
A = 12,000 × 1.035¹⁰ = 12,000 × 1.41060… = $16,927.22.
Example 3: A car worth $30,000 depreciates by 12% per year. Find its value after 5 years.
For depreciation, the rate is negative, so the multiplier is less than 1:
A = 30,000 × (1 − 12/100)⁵ = 30,000 × 0.88⁵ = 30,000 × 0.52773… = $15,831.83.
Finding the Number of Years
Some questions give you the final amount and ask you to find how many years it takes to reach that amount. This requires either trial and improvement or logarithms.
Example: $2000 is invested at 6% compound interest. After how many complete years will the investment first exceed $3000?
We need 2000 × 1.06ⁿ > 3000, so 1.06ⁿ > 1.5.
Try values: 1.06⁶ = 1.4185…, 1.06⁷ = 1.5036…
After 7 complete years, the investment first exceeds $3000.
At IGCSE level, trial and improvement is the expected method unless you are familiar with logarithms. List the accumulated amounts year by year until the target is reached.
Finding the Interest Rate
If you know the principal, final amount, and time, you can find the interest rate.
Example: An investment of $4000 grows to $5500 over 8 years with compound interest. Find the annual interest rate.
5500 = 4000 × (1 + r/100)⁸
1.375 = (1 + r/100)⁸
Take the 8th root: 1 + r/100 = 1.375^(1/8) = 1.0409…
r = 4.09% (to 3 sf).
On a calculator, find the 8th root using the power 1/8: 1.375^(0.125).
Compound Interest with Different Compounding Periods
Some problems specify that interest is compounded monthly, quarterly, or daily instead of annually. The adjusted formula is:
A = P(1 + r/(100k))^(kn)
Where k is the number of compounding periods per year.
For monthly compounding with r = 6% and n = 2 years: A = P × (1 + 6/(100×12))^(12×2) = P × (1.005)^24.
More frequent compounding produces a slightly higher return because interest is earned on interest more often.
Connecting to Other Topics
Compound interest connects to several IGCSE topics:
- Geometric sequences: The yearly amounts form a geometric sequence with first term P and common ratio (1 + r/100)
- Exponential growth and decay: The compound interest formula is a special case of the exponential function
- Percentage changes: The multiplier method for percentages leads directly to the compound interest formula
- Indices: Evaluating the formula requires confident use of powers and roots
Real-World Applications
Understanding compound interest is genuinely useful beyond the exam:
- Savings accounts and investments: How your money grows over time
- Loans and mortgages: How debt accumulates if not repaid
- Inflation: How prices increase over time, eroding purchasing power
- Population growth: How populations grow when the growth rate is proportional to the current size
- Radioactive decay: How substances diminish over time
Questions in these contexts test not just your ability to substitute into the formula but also your understanding of what the formula represents.
Common Mistakes
- Confusing simple and compound interest. Read the question carefully to determine which type is required.
- Using the interest rate as a decimal incorrectly. In the formula A = P(1 + r/100)ⁿ, r is the percentage (e.g., 5 for 5%), not the decimal (0.05).
- Forgetting to subtract P when asked for the interest earned. The formula gives the total amount. Interest = A − P.
- Rounding the multiplier before raising to the power. Keep full precision in the multiplier.
- Incorrect number of years. “After how many complete years” means you need a whole number answer, often found by trial.
- Using the formula for depreciation without adjusting the sign. Depreciation uses (1 − r/100), not (1 + r/100).
Practice Approach
Start with straightforward substitution questions. Progress to finding the number of years (trial and improvement) and finding the interest rate (using roots). Then tackle word problems that require you to set up the formula from a contextual description. Past papers provide excellent practice with varying levels of difficulty.
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