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GCSE Maths · Ratio & proportion
Reverse percentages GCSE Questions and Worked Answers
For a reverse percentage, the amount after a change is known and the original is missing. Identify what percentage of the original remains, then divide the final amount by that percentage as a decimal. Check by applying the change forwards.
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Start with the meaning
What you need to know about reverse percentages
A shop reduces a coat's price by 15%, and its new price is £68. We want the old price, not another discount. Think of the old price as the whole amount: 100%. After 15% is removed, 85% remains. So £68 represents 85% of the unknown original, not 100%.
See the idea first
Recover the whole from a known share
If 85 equal percentage parts cost £68, one part costs £68 ÷ 85 = £0.80. All 100 parts cost £80. The shorter calculator route is £68 ÷ 0.85 = £80, because 85% = 0.85 and original × 0.85 = final. Division undoes that multiplication.
| Share of the original | Value | Reason |
|---|---|---|
| 85% | £68 | 100% − 15% remains |
| 1% | £0.80 | £68 ÷ 85 |
| 100% | £80 | £0.80 × 100 |
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Recover an amount before a decrease
What the problem asks: A price is £72 after a 20% discount. Find the original.
How to solve it: 80% remains, so original = 72 ÷ 0.80 = £90. Check: 20% of £90 is £18, leaving £72.
Recover an amount before an increase
What the problem asks: A value becomes 132 after a 10% increase. Find the original.
How to solve it: The final amount is 110% of the original. Divide by 1.10 to get 120.
Find the whole from a share
What the problem asks: 35% of a number is 56. Find the number.
How to solve it: 56 represents 35%, so 1% is 56 ÷ 35 = 1.6 and 100% is 160. Equivalently 56 ÷ 0.35 = 160.
A reliable routine
Find the unknown original amount
Use this when you know a percentage share or changed value and need the original 100%. The equation is original × multiplier = known amount, so divide by the multiplier to reverse it.
- Label the original as 100%.
- Work out the percentage represented by the known amount.
- Divide by its decimal multiplier, or find 1% then 100%.
- Apply the stated percentage forwards to check your answer.
Check: Adding 20% to a discounted price does not undo a 20% discount: those percentages are taken from different starting amounts.
Fully worked
Reverse percentages GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
A discount
Question
A bag costs £54 after a 10% discount. Find its original price.
The remaining share is 90%, multiplier 0.9.
The original price was £60. Check: £60 − £6 = £54.
Example 2
A price increase
Question
A ticket costs £46 after a 15% increase. Find its original price.
The final price is 115% of the original.
The ticket originally cost £40. Check: 15% of £40 is £6.
Example 3
A given share
Question
24% of a number is 42. Find the number.
Find one percentage part:
Then the whole:
Check: 0.24 × 175 = 42.
Example 4
A decimal percentage
Question
A fee is £84 after a 12.5% discount. Find the original fee.
100% − 12.5% = 87.5%, so the multiplier is 0.875.
Original fee £96. Check: 12.5% of £96 is £12, leaving £84.
Example 5
Two successive changes
Question
A price increases by 20%, then falls by 10%, ending at £162. Find the starting price.
Both changes act in sequence:
Check: £150 → £180 → £162.
Example 6
Explain a wrong method
Question
A sale price is £80 after a 20% discount. A student adds 20% to £80 and says the original was £96. Explain the error and find the correct original price.
Adding 20% of the sale price uses the wrong base: the discount was a percentage of the original price. £80 represents 80% of that original, so
The correct original is £100. Check: 20% of £100 is £20, and £100 − £20 = £80.
10 original questions · total 25 marks
Reverse percentages GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 30 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Ten per cent off
A price is £63 after a 10% discount. Find the original.
Show worked answer
The original was £70.
Quarter off
A jacket costs £90 after 25% off. Find its original price.
Show worked answer
75% remains:
Original price £120.
Twenty per cent rise
A bill becomes £156 after a 20% increase. Find the original.
Show worked answer
The original was £130.
Five per cent rise
A population is 840 after a 5% increase. Find the previous population.
Show worked answer
The previous population was 800.
A share
40% of a number is 72. Find the number.
Show worked answer
The known 72 is 40%, not the whole.
A small share
8% of a number is 26. Find the number.
Show worked answer
The number is 325.
A fractional rate
A value becomes 258 after a 7.5% increase. Find the original.
Show worked answer
Check: 7.5% of 240 is 18.
Loss
A machine loses 30% of its value and is then worth £3500. Find its original value.
Show worked answer
70% remains:
Original value £5000.
Repeated discount
A price is reduced by 10% twice, ending at £64.80. Find its original price.
Show worked answer
Original £80; check £80 → £72 → £64.80. Two 10% reductions are not one 20% reduction.
Choose the operation
After a 40% increase an amount is 210. Explain whether to divide by 0.4, 1.4 or 0.6.
Show worked answer
Divide by 1.4, since 210 is 140% of the original.
0.4 represents only the increase, not the final amount. 0.6 would be the multiplier for a 40% decrease, which did not happen.
Examiner-style feedback
Common reverse percentages mistakes
The original is the reference 100%; identify which share the known value represents.
Reverse the multiplier by division instead.
Multiply the change factors in sequence.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Original = 100%.
- Find the known percentage.
- Divide by its multiplier.
- Check forwards.
Quick answers
Reverse percentages FAQ
Can I always use the 1% method?
Yes, when a known amount represents a known non-zero percentage of the original. It is equivalent to dividing by the decimal multiplier.
How do I recognise a reverse question?
Look for an amount after a change and a request for the original or previous value.
Content standards
Curriculum and rights review
R9 original-value percentage problems across tiers, including multistep problems marked harder rather than Higher-only. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references