Reverse compound interest questions are Higher tier problems that require you to work backwards from a final amount. Instead of calculating what an investment grows to, you may need to find the original value before interest was applied, the interest rate, or the number of years. These questions build on the standard compound interest formula and test your ability to rearrange and reason with powers. They frequently appear on AQA, Edexcel and OCR Higher papers and can carry 3 to 5 marks. This guide covers all three reverse scenarios with clear methods.
What Is Reverse Compound Interest?
Standard compound interest uses the formula to find the final amount. Reverse compound interest starts with the final amount and works backwards to find a missing piece of information.
Key Formulas
The standard compound interest formula:
Where A = final amount, P = original (principal) amount, r = interest rate (%), n = number of years.
To find the original amount:
To find the rate, rearrange and take the nth root:
To find the number of years, use trial and improvement or logarithmic reasoning:
Step-by-Step Method
Finding the original amount
- Identify the final amount A, the rate r, and the number of years n.
- Calculate the multiplier: (1 + r/100)^n.
- Divide the final amount by the multiplier: P = A / multiplier.
Finding the interest rate
- Divide the final amount by the original: A/P.
- Take the nth root of this result.
- Subtract 1 and multiply by 100 to get the percentage rate.
Finding the number of years
- Calculate the multiplier per year: (1 + r/100).
- Use trial and improvement: keep multiplying by the annual multiplier until you reach or exceed the final amount. Count the steps.
Worked Example 1 — Foundation-style Higher
Question: After 3 years of compound interest at 5% per year, an investment is worth £5,788.13. What was the original investment?
Working:
Multiplier = (1.05)^3 = 1.157625
Original amount P = 5788.13 / 1.157625 = £5,000
Answer: The original investment was £5,000.
Worked Example 2 — Higher Level
Question: A painting was bought for £2,000. After 4 years it is worth £2,928.20. Find the annual percentage rate of increase. Give your answer to 1 decimal place.
Working:
Step 1: A/P = 2928.20 / 2000 = 1.4641
Step 2: Take the 4th root: (1.4641)^(1/4) = 1.4641^0.25 = 1.10
Step 3: Rate = (1.10 - 1) x 100 = 10.0%
Answer: The annual rate of increase is 10.0%.
Worked Example 3 — Exam Style
Question: £3,000 is invested at 4% compound interest per year. After how many complete years will the investment first exceed £4,000?
Working:
Year 1: 3000 x 1.04 = £3,120
Year 2: 3120 x 1.04 = £3,244.80
Year 3: 3244.80 x 1.04 = £3,374.59
Year 4: 3374.59 x 1.04 = £3,509.58
Year 5: 3509.58 x 1.04 = £3,649.96
Year 6: 3649.96 x 1.04 = £3,795.96
Year 7: 3795.96 x 1.04 = £3,947.80
Year 8: 3947.80 x 1.04 = £4,105.71
Answer: After 8 complete years the investment first exceeds £4,000.
Common Mistakes
- Dividing by the rate instead of the multiplier. To find the original, divide by (1 + r/100)^n, not by r/100 or by r alone.
- Forgetting to raise the multiplier to the power n. If interest compounds for 3 years at 5%, the multiplier is (1.05)^3 = 1.157625, not just 1.05.
- Confusing simple and compound interest methods. In reverse compound interest, you divide by a power of the multiplier, not subtract a simple percentage.
- Rounding intermediate values. Keep full calculator precision throughout and only round the final answer as instructed.
Exam Tips
- Write down the compound interest formula first and identify which variable you need to find. This earns a method mark.
- For "find the number of years" questions, trial and improvement is the expected method at GCSE. Set up a clear table showing each year.
- If you get a non-integer when finding a rate, check whether the question asks for a specific number of decimal places.
- The nth root can be calculated on a calculator using the power 1/n. For example, the 4th root of x is x^0.25.
Practice Questions
Q1 (Higher): After 2 years of compound interest at 6% per year, a savings account contains £2,247.20. Find the original amount.
Q2 (Higher): A house was bought for £150,000 and is now worth £181,500 after 3 years. Find the annual rate of increase to 1 decimal place.
Q3 (Higher): £5,000 is invested at 3% compound interest. After how many complete years will it first be worth more than £6,000?
Practise reverse compound interest with step-by-step solutions free on GCSEMathsAI.
Related Topics
Summary
- To find the original amount, divide the final amount by (1 + r/100)^n.
- To find the rate, divide A by P, take the nth root, subtract 1 and multiply by 100.
- To find the number of years, use trial and improvement by multiplying by the annual multiplier repeatedly.
- Always use the compound interest multiplier, not simple interest subtraction.
- Keep full calculator precision in intermediate steps.
- Write the compound interest formula at the start of your answer for method marks.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.