Introduction
A reverse percentage helps you find a starting number when you only know the end number. This Reverse Percentage Calculator runs that math backward. Type in the percent and the number you already have, and it returns the original value.
Pick one of three cases:
- Decrease: a price after a discount was taken off. A coat costs $68 after 15% off, so it was $80 before.
- Increase: a total after tax or a markup was added. A bill of $115 with 15% tax was $100 before tax.
- Proportion: a percent of a number that you know. If 32% of a number is 160, the number is 500.
You get the answer, the formula, and each step of the working, so you can see how it fits together. You can choose a currency sign, set the decimal places, copy the answer, view a chart, or save the steps as an image. It is useful for shopping, sales tax, tips, pay raises, and math homework.
How to use our Reverse Percentage Calculator
Pick your scenario, type the percentage and the number you already know, and the calculator finds the original value with a formula, step-by-step working, and a chart.
Choose Your Scenario: Click Decrease if the number you know is what is left after a percent was taken off. Click Increase if the number already has a percent added on, like tax. Click Proportion if a known percent of the original equals your number.
Percentage Rate (%): Type the percent that was taken off, added on, or that your number stands for. Enter 15 for a 15% discount, not 85. For a decrease, the rate must be under 100.
Currency or unit symbol: Pick $, £, €, %, or None from the drop-down. This only labels your answer. It never changes the math.
Final Value: Type the number you already know, such as the sale price, the total bill, or the part. It cannot be zero.
Decimal Places in Result: Choose how many decimals you want, from 0 to 8. Money usually uses 2.
Calculate: Click this button to see the original value, the formula, each step, and the bar chart.
Random: Click this to fill the boxes with a practice problem.
Clear: Click this to empty all boxes and start over.
What Is a Reverse Percentage?
A reverse percentage works backward. You already know the final number after a percent was added or taken away. You want to find the original number you started with. This is also called finding the whole, the starting price, or the pre-discount amount.
Why You Can't Just Add the Percent Back
Adding the percent back does not undo it, and that is an easy mistake to make. If a coat costs $68 after a 15% discount, adding 15% to $68 gives $78.20, which is wrong. The answer is $80. Here's why: the 15% was taken off the original price, not off the sale price. Percents are always a share of the number they come from, so you must divide, not add.
The Three Main Types
- Decrease: A price dropped by a percent. A discount, a sale, or a loss in value. The final number is smaller than the original.
- Increase: A percent was added on. Tax, tips, markup, or a pay raise. The final number is bigger than the original.
- Proportion: You know that a certain percent of a number equals your value. Like "32% of a number is 160."
The Formulas
All three use the same idea: divide the final value by its multiplier.
- Decrease: Original = Final ÷ (1 − Rate ÷ 100)
- Increase: Original = Final ÷ (1 + Rate ÷ 100)
- Proportion: Original = Final ÷ (Rate ÷ 100)
A multiplier is just the percent turned into a decimal. A 15% discount means you paid 85%, so the multiplier is 0.85. A 15% tax means you paid 115%, so the multiplier is 1.15.
A Worked Example
A hotel bill is $115 and it includes 15% tax. What was the price before tax?
15 ÷ 100 = 0.15. Then 1 + 0.15 = 1.15. Now divide: 115 ÷ 1.15 = $100. Check it: 15% of $100 is $15, and $100 + $15 = $115. It works.
Where You Use This in Real Life
Reverse percentages show up more often than you think. Use them to find an item's list price before a sale, work out a bill before sales tax, find an old salary before a raise, figure out a car's value before it lost worth, or find a test's total marks when you only know your score and your percent.
Quick Tips
- Always enter the percent that changed, not the percent that's left. For a 15% discount, type 15, not 85.
- For a decrease, the rate must be under 100%. A 100% drop would leave you with zero.
- Check your answer by working forward. Apply the percent to your answer and see if you get back to the final value.
- For a decrease, the original is always bigger than the final. For an increase, it's always smaller.