0580

Interest, Growth and Decay

Number

How it differs

  • 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 time, compound interest always exceeds simple interest

Calculating it

  • 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

Changing rates

  • 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
Exam tip

The compound interest formula is not in the List of Formulas on page 2 — that page carries mensuration results, the quadratic formula and the trigonometry rules — so it has to be memorised. Write the multiplier raised to the power before evaluating, and read whether the question wants the final balance or only the interest earned.