Reverse percentages: how to find the original number
A price after a 20% rise is 96, so what was it before? How to work backwards from a percentage increase, discount or tax, and the common mistake that gives 76.8.
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Tools for this job: Percentage Calculator · Discount Calculator · Tax Calculator
A jacket costs 60 after a 25% discount. What was the full price? A bill is 150 including 20% VAT. How much is the VAT? These are reverse percentage questions: you know the number after the change and want the number before it. They trip people up because the obvious method gives the wrong answer.
The common mistake
Take the 150 bill with 20% VAT. The tempting move is to take 20% off: 150 − 30 = 120. But 20% of 120 is 24, and 120 + 24 = 144, not 150. The problem is that the 20% was calculated on the original price, not on the total. Taking 20% of the total uses the wrong base.
The method that works
Turn the percentage change into a multiplier, then divide by it.
- After an increase of 20%: the new value is 120% of the original, a multiplier of 1.20. Original = new ÷ 1.20.
- After a decrease of 25%: the new value is 75% of the original, a multiplier of 0.75. Original = new ÷ 0.75.
Worked examples
| Situation | Sum | Original |
|---|---|---|
| 96 after a 20% rise | 96 ÷ 1.20 | 80 |
| 60 after 25% off | 60 ÷ 0.75 | 80 |
| 150 including 20% VAT | 150 ÷ 1.20 | 125 (VAT 25) |
| 1,180 including 18% GST | 1,180 ÷ 1.18 | 1,000 (GST 180) |
| Salary 46,200 after a 10% raise | 46,200 ÷ 1.10 | 42,000 |
Check any answer by applying the percentage forwards: 80 plus 20% is 96. If the check does not land exactly on the number you started with, the method was wrong.
Let a calculator do it
- Discounts: the Discount Calculator has a Find original price mode: enter the price you paid and the discount that was applied.
- Tax: the Tax Calculator has a Remove tax from a total mode, with VAT and GST rates for several countries built in.
- Anything else: the Percentage Calculator handles percentage change in both directions, including the reverse change needed to get back to where you started.
Getting back to the start
A related trap: if a share price falls 20%, it needs to rise 25% to recover, not 20%. From 100, a 20% fall gives 80, and 80 needs another 20 to get back, which is 25% of 80. The bigger the fall, the bigger the gap: a 50% fall needs a 100% rise. The same applies to a 10% pay cut followed by a 10% pay rise, which leaves you 1% worse off.
Where reverse percentages come up
- Checking a sale really was the discount advertised.
- Splitting a receipt total into net price and tax for expenses.
- Working out a pre-raise salary or a pre-increase rent.
- Finding what a figure was last year from this year's figure and the growth rate.
For the forward direction, see how to calculate percentage change. For tax in particular, see how to add or remove VAT.