Simple and Compound Interest

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

1. Simple interest

Interest is money added to a sum at regular intervals, increasing savings or increasing a debt
Simple interest is worked out on the starting amount alone, so the same payment recurs every period
The balance grows by equal steps, which is what separates it from compound interest

Find the stated percentage of the starting amount to get one period's interest
Multiply by the number of periods for the total interest
Add that total to the starting amount only if the question asks for the final balance

Worked example

Simple interest over several years

£4500 is invested at 3% simple interest per year for 7 years. Find the total interest earned and the final balance.

Solution:

Simple interest is worked out on the starting amount every year, so each year earns exactly the same
One year's interest is 3% of £4500
3% as a decimal is 0.03, so the interest is 0.03 × 4500 = £135
That same £135 is added in each of the 7 years, so the interest never grows
Total interest = 135 × 7 = £945
The balance is the original sum plus the interest it has earned
Final balance = 4500 + 945 = £5445
Read the question carefully: the interest (£945) and the balance (£5445) are different answers, and each is asked for separately
Both answers are exact, so neither needs rounding

With compound interest each period's payment is worked out from the balance as it then stands, so earlier interest goes on to earn interest of its own
The balance therefore grows by increasing steps rather than equal ones
Over the same rate and more than one period, compound interest always exceeds simple interest

Applying a percentage increase repeatedly means multiplying by the same multiplier repeatedly
Raising the multiplier to the power of the number of periods does this in one step
- A balance P at r% for n periods becomes P × (1 + r/100)ⁿ
The result is the final balance, not the interest, so subtract the original amount when the interest is what was asked for

Where the rate changes partway through, treat each stretch as its own calculation
Carry the balance from the end of one stretch into the start of the next

Worked example

Compound interest over several years

£4500 is invested at 3% compound interest per year for 7 years. Find the final balance and the interest earned.

Solution:

Compound interest is worked out on the balance at the time, so each year's interest earns interest of its own afterwards
An increase of 3% leaves 103% of the amount, so the multiplier is 1.03
Applying that multiplier once per year for 7 years means raising it to the power 7
Balance = 4500 × 1.03⁷
1.03⁷ = 1.2298739…, and keeping the full figure in the calculator matters here
Balance = £5534.43 to the nearest penny
The interest earned is the growth above the original sum, not the balance itself
Interest = 5534.43 − 4500 = £1034.43
Compare that with the £945 simple interest at the same rate over the same 7 years: the extra £89.43 is what compounding adds
Rounding 1.03⁷ to 1.23 partway through would give £5535, which is out by more than half a pound