Reverse percentages — finding the original amount
How do you do reverse percentages?
A reverse percentage question gives you the amount AFTER a percentage change and asks for the amount BEFORE it — the price before the sale, the bill before VAT, the salary before the rise. The method is to divide by the multiplier that produced the change. The trap is to take the percentage off the new amount, which gives the wrong answer every time.
Worked examples, step by step
A coat costs £60 in a 25% off sale. What was the original price?
- After 25% off, 75% of the original price remains, so the multiplier is 0.75
- Original price × 0.75 = 60
- Original price = 60 ÷ 0.75 = 80
- Check: 25% of 80 is 20, and 80 − 20 = 60. Answer: £80
A phone costs £300 including VAT at 20%. What was the price before VAT?
- Adding 20% VAT is a multiplier of 1.2
- Price before VAT × 1.2 = 300
- Price before VAT = 300 ÷ 1.2 = 250
- Check: 20% of 250 is 50, and 250 + 50 = 300. Answer: £250
Why is it wrong to find the price before VAT by taking 20% off £300?
- 20% of 300 is 60, and 300 − 60 = 240 — but that is not the answer
- Check it: 240 plus 20% VAT is 240 × 1.2 = 288, not 300
- The 20% was added to the ORIGINAL price (250), not to the final price (300), so it must be undone by dividing by 1.2, not by subtracting 20% of 300
- Answer: the correct original price is £250
After a 4% pay rise, Priya earns £31 200. What did she earn before the rise?
- A 4% rise is a multiplier of 1.04
- Old salary × 1.04 = 31 200
- Old salary = 31 200 ÷ 1.04 = 30 000
- Check: 4% of 30 000 is 1200, and 30 000 + 1200 = 31 200. Answer: £30 000
Now try it yourself
Frequently asked questions
How do you recognise a reverse percentage question?
The question gives you the amount after the change and asks for the amount before it — "the price in the sale is…, what was the original price?", "including VAT the bill is…", "after the rise she earns…". Whenever the unknown is the starting amount, it is a reverse percentage.
Why can't I just subtract the percentage?
Because the percentage was worked out on the original amount, which is the number you do not know yet. Subtracting 20% of the final amount takes off too much. The only way to undo a multiplication by 1.2 is to divide by 1.2.
What multiplier do I use?
For an increase of p%, the multiplier is 1 + p ÷ 100 (a 20% increase is 1.2). For a decrease of p%, it is 1 − p ÷ 100 (a 25% decrease is 0.75). Divide the final amount by that multiplier.
How do I check my answer?
Apply the percentage change to your answer and see whether you get back the amount in the question. If £80 with 25% off gives £60, then £80 is right. This check takes ten seconds and catches the subtract-instead-of-divide mistake every time.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specificationsLast updated: