Interest - Simple and Compound Explained
Interest is the cost of borrowing money or the reward for lending it. When you borrow, you pay interest. When you save or invest, you earn it. Understanding the two types of interest – simple and compound – is one of the most practically useful skills in financial mathematics.
Simple Interest
Simple interest is calculated only on the original amount (the principal). It does not grow over time – the same amount of interest is added each period.
I = P × R × T
Where: I = interest, P = principal, R = annual interest rate (as a decimal), T = time in years.
Total amount = P + I = P(1 + RT)
Compound Interest
Compound interest is calculated on the principal plus the interest already earned. Each period, the interest is added to the balance, and the next period's interest is calculated on that larger amount – the snowball effect.
A = P(1 + r/n)nt
Where: A = final amount, P = principal, r = annual rate (decimal), n = number of compounding periods per year, t = years.
Compounding Frequency
| Frequency | n (per year) | Effect |
|---|---|---|
| Annual | 1 | Interest added once a year |
| Semi-annual | 2 | Every 6 months |
| Quarterly | 4 | Every 3 months |
| Monthly | 12 | Every month |
| Daily | 365 | Every day – highest return |
The more frequently interest compounds, the greater the final amount – though the difference between monthly and daily is usually small in practice.
Simple vs Compound: A Comparison
| Year | Simple interest (5%) | Compound interest (5%) |
|---|---|---|
| Start | £1 000 | £1 000 |
| 1 | £1 050 | £1 050.00 |
| 5 | £1 250 | £1 276.28 |
| 10 | £1 500 | £1 628.89 |
| 20 | £2 000 | £2 653.30 |
| 30 | £2 500 | £4 321.94 |
Annual Percentage Rate (APR)
APR is the standardised annual interest rate that includes fees and charges, used to compare the true cost of borrowing across different loans and credit products. A higher APR means a more expensive loan. By law, lenders must display the APR so consumers can make fair comparisons.
Worked Examples
I = 4 000 × 0.06 × 5 = £1 200.
Total = £4 000 + £1 200 = £5 200.
A = 4 000 × (1.06)5 = 4 000 × 1.3382 = £5 352.90.
Compound earns £152.90 more than simple interest over 5 years.
A = 2 500 × (1 + 0.04/12)12 × 3 = 2 500 × (1.003333)36 = 2 500 × 1.12749 = £2 818.73.
I = 1 040 − 800 = £240.
R = I / (P × T) = 240 / (800 × 3) = 240 / 2 400 = 0.10 = 10% per year.
Finding Principal, Rate, or Time
Rearranging I = PRT:
P = I / (RT) R = I / (PT) T = I / (PR)
Key Takeaways
- Simple interest: I = PRT. The same amount is added each period.
- Compound interest: A = P(1 + r/n)nt. Interest earns further interest.
- More frequent compounding = higher final amount.
- APR is the standardised rate for comparing borrowing costs.
Practice Questions
- Calculate the simple interest on £6 000 at 3.5% per year for 4 years.
- Find the compound amount on £1 500 at 7% per year for 10 years (annual compounding).
- How many years does it take for £2 000 to grow to £2 600 at 5% simple interest per year?
- Find the compound amount on £5 000 at 4.8% per year compounded quarterly for 2 years.
- Lena invests £3 000 for 8 years. Account A offers 5% simple interest. Account B offers 4.5% compound interest (annual). Which account gives a higher return, and by how much?