Compound Interest 2

Harder compound interest questions may ask for a missing multiplier or a missing number of years. Use \(\text{final amount} = \text{original amount} \times \text{multiplier}^{\text{time}}\), then rearrange or test powers carefully. A missing multiplier usually involves a root, while a missing number of years can be found by comparing powers.

Common mistake: Do not divide the final amount by the original amount and treat that as the yearly multiplier straight away. If the change happened for more than one year, you must also deal with the power.

Worked example:
£500 becomes £605 after 2 years of compound interest. Find the annual percentage increase.
  1. Let the annual multiplier be m.
    \[500m^2 = 605\]
    \[m^2 = 605 \div 500 = 1.21\]
    \[m = \sqrt{1.21} = 1.1\]
Answer:
The annual percentage increase is 10%.
Worked example:
£400 is invested at 6% compound interest per year. Find the smallest whole number of years until the value is more than £500.
  1. Step 1
    The multiplier for a 6% increase is 1.06.
    \[400 \times 1.06^n > 500\]
  2. Step 2
    Test powers of 1.06.
    \[400 \times 1.06^3 \approx 476.41\]
    \[400 \times 1.06^4 \approx 504.99\]
Answer:
The value is more than £500 after 4 years.