Find the original amount before a percentage increase
Find the original amount before a percentage decrease
Apply to real-life: pre-sale prices, pre-VAT prices, original salaries
Real-World Applications
Shopping: When an item is in a sale, the ticket shows the reduced price — reverse percentages let you work backwards to find the original price before the discount was applied.
Finance: VAT-inclusive prices are common on receipts — dividing by the correct multiplier gives the pre-tax price, which is important for businesses reclaiming VAT on purchases.
Business: If a product has been marked up before sale, working backwards from the selling price using reverse percentages reveals the original cost price paid by the retailer.
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A reverse percentage problem works backwards from a normal one: instead of knowing the original amount and finding the new amount after a percentage change, you know the new amount and have to find what the original was before the change happened.
You cannot simply subtract 20% from the new amount — that 20% is of the original, not the new price.
Example 1: After a 20% increase, a price is £60. What was the original?
Step 1
A 20% increase means the multiplier was 1.20.
Step 2
£60 = original × 1.20
Step 3
Original = £60 ÷ 1.20 = £50
Example 2: After a 30% increase, a salary is £39,000. What was the original salary?
Step 1
Multiplier = 1.30
Step 2
Original = £39,000 ÷ 1.30 = £30,000
⚠ Common mistake: Subtracting 20% from £60 gives £48, which is WRONG. Always divide by the multiplier.
✓ Quick Check 1 — What Reverse Percentages Mean
Reversing Decreases
For percentage decreases, the multiplier is less than 1. To reverse a 15% decrease, divide by 0.85.
Original = New amount ÷ multiplier | Increase: multiplier > 1 | Decrease: multiplier < 1
Example 3: A coat is in a 15% sale. Sale price is £68. What was the original price?
Step 1
15% decrease → multiplier = 0.85
Step 2
Original = £68 ÷ 0.85 = £80
Example 4: After a 20% reduction, a TV costs £320. What was the original price?
Step 1
Multiplier = 0.80
Step 2
Original = £320 ÷ 0.80 = £400
Example 5: A car has depreciated by 30%. It is now worth £14,000. What was the original price?
Step 1
Multiplier = 0.70
Step 2
Original = £14,000 ÷ 0.70 = £20,000
Quick check: After finding the original, multiply it by the multiplier — you should get back to the new amount.
Example 5b: Same problem, alternative method — the coat from Example 3 (sale price £68 after 15% off), using the unitary method
Step 1
A 15% reduction means the sale price is 85% of the original: 85% = £68.
Step 2
Find 1%: \(£68 \div 85 = £0.80\)
Step 3
Find 100%: \(£0.80 \times 100 = \mathbf{£80}\) — same answer as Example 3's divide-by-0.85 method.
💡 Two valid methods: Divide-by-multiplier (Examples 3-5) and the unitary method (find 1%, then ×100) always give the same answer — use whichever you find more natural. The unitary method can feel more intuitive when the percentage doesn't map to an obvious decimal, e.g. 85% → 1% → 100%.
✓ Quick Check 2 — Reversing Decreases
Real-Life Reverse Percentages
Three classic exam contexts: sale prices, VAT, and salary changes.
Example 6: A price tag says "30% off, now £56." What was the original price?
Step 1
After 30% decrease → multiplier = 0.70
Step 2
Original = £56 ÷ 0.70 = £80
Example 7: A bill is £120 including 20% VAT. What is the price before VAT?
Step 1
Price with VAT = original × 1.20
Step 2
Pre-VAT price = £120 ÷ 1.20 = £100
Example 8: A worker's salary after a 4% pay rise is £31,200. What was the original salary?
Step 1
After 4% rise → multiplier = 1.04
Step 2
Original = £31,200 ÷ 1.04 = £30,000
✓ Quick Check 3 — Real-Life Reverse Percentages
Practice: Reverse Percentages
#
Question
Answer
Result
1
After 20% increase: £60. Original?
2
After 15% sale: £68. Original?
3
Inc VAT 20%: £120. Pre-VAT?
4
After 10% rise: £55. Original?
5
After 25% off: £60. Original?
6
After 5% rise: £210. Original?
7
After 30% off: £56. Original?
8
After 4% rise: £31,200. Original?
9
After 8% rise: £270. Original?
10
After 40% off: £42. Original?
⚠ Things to Watch Out For
Finding a percentage of the sale price: If an item costs £85 AFTER a 15% reduction, do NOT find 15% of £85. £85 is NOT 100%.
Identifying what represents 100%: The ORIGINAL price is 100%. After a 15% reduction, the sale price is 85%. Set up: 85% = £85, so 1% = £1, so 100% = £100.
Multiplier method — using wrong multiplier: A 15% reduction means the price is 85% of the original, so divide by 0.85. Not by 0.15 (that gives 15% of the sale price).
Applying a percentage instead of reversing: Don't multiply the sale price by the percentage. You need to DIVIDE by the multiplier to reverse the change.
Checking the answer: Apply the original percentage change to your answer and check you get back to the given amount. If not, something went wrong.