Reversing a percentage means working backward from a final number to find the original value before a percentage was added or subtracted. The key principle: you divide by the percentage relationship, not subtract or add the percentage from the result. This one distinction trips up most people and leads to wrong answers.
Why You Can’t Just Subtract the Percentage
Say a store marks up a $200 item by 25%, making the price $250. If you try to reverse that by calculating 25% of $250 and subtracting it, you get $250 minus $62.50, which is $187.50. That’s wrong. The original price was $200.
The error happens because 25% of $250 is not the same as 25% of $200. The percentage was originally applied to the smaller number, but you’re now applying it to the larger one. This same logic applies in reverse for discounts: 30% of a reduced price is not the same as 30% of the original price. You need a different method entirely.
The Core Formula
Every reverse percentage problem follows one formula:
Original Value = Final Value ÷ (Percentage as a Decimal)
The “percentage as a decimal” depends on whether the change was an increase or a decrease:
- After a percentage increase: divide by (1 + the percentage as a decimal). A 25% increase means you divide by 1.25.
- After a percentage decrease: divide by (1 – the percentage as a decimal). A 20% discount means you divide by 0.80.
That’s the entire method. The rest is just applying it to different situations.
Reversing a Percentage Increase
Suppose a population grew by 15% and is now 46,000. To find the original population, recognize that 46,000 represents 115% of the starting value (100% plus the 15% growth). Divide 46,000 by 1.15 to get 40,000.
Another way to think about it: the final number equals 115% of the original number. So the original number equals the final number divided by 1.15. You can use this with any increase. A 6% raise brought your salary to $74,200? Divide $74,200 by 1.06 to find your previous salary of $70,000.
Reversing a Percentage Decrease
A coat is reduced by 20% and now costs £120. The sale price represents 80% of the original (100% minus the 20% discount). To find the original price, divide £120 by 0.80.
£120 ÷ 0.80 = £150. The coat originally cost £150.
You can verify this: 20% of £150 is £30, and £150 minus £30 is £120. It checks out. If you had mistakenly calculated 20% of £120 (which is £24) and added it back, you’d get £144, which is wrong.
The Step-by-Step Method
If dividing by decimals feels unintuitive, there’s an alternative approach that breaks the problem into smaller steps:
- Step 1: Figure out what percentage the final value represents. If something increased by 25%, the final value is 125%. If it decreased by 30%, the final value is 70%.
- Step 2: Find 1% by dividing the final value by that number. If the final value is $350 and it represents 70%, then 1% equals $350 ÷ 70 = $5.
- Step 3: Multiply by 100 to get the original value (which is 100%). So $5 × 100 = $500.
This method and the division method give identical results. Use whichever clicks for you.
Removing Sales Tax From a Total
One of the most common real-world uses for reverse percentages is stripping tax out of a total price. If you paid $53.50 for something and the sales tax rate was 7%, the total represents 107% of the pre-tax price. Divide $53.50 by 1.07 to get $50, the price before tax.
The same logic works for value-added tax (VAT) in other countries. A 20% VAT rate means you divide the total by 1.20. A reduced 5% rate means dividing by 1.05. If a receipt shows £240 including 20% VAT, the pre-tax price is £240 ÷ 1.20 = £200, and the tax portion is £40.
Finding the Original Price After a Markup
Retailers mark up products from their wholesale cost. If you know the retail price and the markup percentage, the reverse percentage formula tells you what the store paid. A product selling for $78 with a 30% markup means $78 is 130% of the wholesale cost. Divide $78 by 1.30 to get $60.
This also works for successive changes. If something was marked up 50% and then discounted 10%, you reverse each step separately, working backward. Start with the final price, divide by 0.90 to undo the 10% discount, then divide that result by 1.50 to undo the 50% markup.
Quick Reference for Common Percentages
For percentages you encounter regularly, it helps to know the divisor off the top of your head:
- 5% increase: divide by 1.05
- 10% increase: divide by 1.10
- 15% increase: divide by 1.15
- 20% increase: divide by 1.20
- 25% increase: divide by 1.25
- 50% increase: divide by 1.50
- 10% decrease: divide by 0.90
- 20% decrease: divide by 0.80
- 25% decrease: divide by 0.75
- 33% decrease: divide by 0.67
- 50% decrease: divide by 0.50
The pattern is simple: for increases, add the percentage (as a decimal) to 1. For decreases, subtract it from 1. Then divide your final number by that result. Every reverse percentage problem, no matter how complex it sounds, comes back to this single operation.