Sheet № 232 · Higher only · AQA · Edexcel · OCR
Reverse Compound Interest –
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 for
§Key definitions
Question:
After 3 years of compound interest at 5% per year, an investment is worth £5,788.13. What was the original investment?
Answer:
The original investment was £5,000.
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?
§Formulas to memorise
A = P(1 + r/100)^n
P = A / (1 + r/100)^n
(1 + r/100) = (A/P)^(1/n)
n = log(A/P) / log(1 + r/100)
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.
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.
Worked example
After 3 years of compound interest at 5% per year, an investment is worth £5,788.13. What was the original investment?
Working:
⚠ 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.