Curriculum fit: GCSE Maths Ratio, proportion and rates of change (Foundation and Higher). US: Grade 7 and Algebra I percent problems, exponential models.
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What Is It?
Sometimes you know the price after a percentage change and need to find the original. This is a reverse percentage problem. The key is to write the final amount as a percentage of the original, then divide.
Compound interest means each year's interest is added to the balance, so the next year's interest is calculated on a larger amount.
Key Formulas and Ideas
- Original = final amount ÷ multiplier.
- Multiplier for an increase of r%: 1 + r/100. For a decrease: 1 − r/100.
- Compound interest: A = P × (1 + r/100)ⁿ, where P is the principal, r the yearly rate and n the number of years.
Worked Examples
Reverse After a Decrease
After a 30% sale discount a jacket costs £42. What was the original price?
- The multiplier for a 30% decrease is 0.7.
- Original = 42 ÷ 0.7 = 60.
✓ Answer £60
Reverse After an Increase
After a 25% increase a price is £75. Find the original.
- Multiplier = 1.25.
- Original = 75 ÷ 1.25 = 60.
✓ Answer £60
Compound Interest
£2000 earns 3% compound interest a year. Find the amount after 4 years.
- Multiplier = 1.03.
- Amount = 2000 × 1.03⁴ = 2000 × 1.12550881.
- = 2251.02 (nearest penny).
✓ Answer £2251.02
Common Mistakes to Avoid
- Working out 30% of the sale price and adding it back, which gives the wrong original.
- Using the wrong multiplier, such as 0.3 instead of 0.7.
- Using simple interest by multiplying the interest by the number of years.
- Rounding the multiplier before finishing.
Practice Questions
Frequently Asked Questions
Why can't I add 30% to the sale price?
The 30% was taken from the original price, not from the sale price, so 30% of the sale price is a different amount.
How is compound interest different from simple interest?
Simple interest is always based on the original amount. Compound interest includes interest already earned.
Keep Going With These Guides
Percentages: How to Find, Increase and Decrease Them
Work out a percentage of an amount, increase or decrease by a percentage and calculate percentage change, with worked examples for KS3 and GCSE.
KS3GCSEMiddle SchoolExponential Growth and Decay: Formulas and Worked Examples
Model repeated percentage change with y = a × bˣ, including compound growth, depreciation and doubling, with step-by-step examples.
GCSEA-LevelHigh SchoolFractions, Decimals and Percentages: How to Convert Them
Switch between fractions, decimals and percentages, including recurring decimals, with worked examples and practice questions to check your answers.
KS3GCSEMiddle SchoolHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.