Reverse percentages answer the question “What was the amount before the percentage change?” If you know the final price after a discount, or the total after tax, you can calculate the original amount by dividing by the right multiplier.
The basic idea is simple: after a decrease, use original = final / (1 - rate); after an increase, use original = final / (1 + rate). You can also use the same method for markups, tax, fees, and other percentage changes, and you can check the basics in this percentage reference.
What reverse percentages mean
A reverse percentage calculation works backward from the result you already know. Instead of asking “What is a percentage of this number?”, you ask “What number produced this final amount after a percentage change?”
This comes up in situations like:
- a sale price after a discount
- a bill total after tax
- a salary after a raise
- a product price after a markup
The key is to identify whether the percentage change made the amount smaller or larger. That tells you whether to subtract the rate from 1 or add it to 1.
The reverse percentage formula
Use one of these two forms:
- After a decrease: original = final / (1 - rate)
- After an increase: original = final / (1 + rate)
Here, the rate must be written as a decimal. So a 20% discount becomes 0.20, and an 8% tax becomes 0.08.
If you are unsure which form to use, ask this question: did the percentage change reduce the starting amount, or add to it? A discount lowers the price, while tax or a markup increases it.
If you want a broader refresher on percent calculations, the Percentage calculator guide can help you connect the reverse method to the ordinary forward method.
How to find the original price after a discount
A discount is a decrease, so you divide by 1 minus the discount rate.
Step-by-step method
- Write the sale price as the final amount.
- Convert the discount percent to a decimal.
- Subtract the decimal from 1.
- Divide the sale price by that result.
Example: sale price after a discount
Suppose a jacket costs $80 after a 20% discount. You want the original price.
Use the decrease formula:
original = final / (1 - rate)
original = 80 / (1 - 0.20)
original = 100
So the original price was 100.
You can check the result by reversing the operation: the discount amount is 20, and subtracting that from the original gives the sale price.
Why this works
A 20% discount means you kept 80% of the original price. In decimal form, 80% is the same as 80 percent, so the sale price equals original × 0.8. Reverse percentages simply undo that multiplication.
This same logic applies to any discount. If something is 30% off, you divide by 0.7 of the original price. If it is 15% off, you divide by 0.85 of the original price.
If you want a direct way to work with markdowns and sale prices, see how discounts are calculated.
How to find the pre-tax amount
If a price includes tax, the tax increased the amount, so you divide by 1 plus the tax rate.
Step-by-step method
- Write the total price as the final amount.
- Convert the tax rate to a decimal.
- Add the decimal to 1.
- Divide the total by that result.
Example: total with tax included
Suppose your receipt total is $54 and it includes 8% tax. You want the pre-tax amount.
Use the increase formula:
original = final / (1 + rate)
original = 54 / (1 + 0.08)
original = 50
So the pre-tax amount is 50.
The tax portion itself is the difference between the total and the pre-tax amount: 4.
Example: price before a markup
Reverse percentages are not just for taxes and discounts. Suppose an item sells for $120 after a 25% markup. To find the base price:
original = 120 / (1 + 0.25)
original = 96
So the original price was 96.
That means the markup added 24 to the original amount.
A fast way to think about the method
You can treat a percent change as a multiplier:
- discount means multiply by a number below 1
- tax means multiply by a number above 1
- markup means multiply by a number above 1
Reverse percentages just divide by that multiplier.
That gives you a quick mental check:
- if the amount was reduced, divide by something less than 1
- if the amount was increased, divide by something more than 1
This is why reverse percentage questions often feel like “undoing” the calculation. You are not finding the percentage itself first; you are reversing the effect of the percentage change.
Common mistake to avoid
A common mistake is dividing by the percentage rate itself instead of the multiplier.
For example, if a $80 sale price reflects a 20% discount, do not calculate 80 / 20%. That does not undo the price change. The correct setup is:
- discount: divide by 1 minus rate
- tax or markup: divide by 1 plus rate
Another frequent error is using the wrong direction. If the price went down, do not add the rate. If the price went up, do not subtract it. The sign depends on whether the percentage change was a decrease or an increase.
If you often mix up percent changes, the distinction explained in percentage vs. percentage points can help separate the idea of a percent change from a point change.
Practical checklist before you calculate
Before you divide, check these points:
- Is the final amount the price after a discount, tax, or markup?
- Did the percentage make the amount smaller or larger?
- Have you converted the percent to a decimal?
- Did you use 1 minus the rate for a decrease?
- Did you use 1 plus the rate for an increase?
If the problem gives a final amount and asks for the original, reverse percentages are usually the right tool. If it asks for the size of the discount or tax itself, you may need a separate percentage calculation after finding the original.
For recurring finance problems, a dedicated sales tax calculator can also save time once you know whether you need the pre-tax amount or the final total.
Frequently asked questions
- What is the formula for reverse percentages?
- Use original = final / (1 - rate) for a decrease, and original = final / (1 + rate) for an increase. The rate should be written as a decimal, not as a whole percent.
- How do I find the original price after a discount?
- Divide the sale price by 1 minus the discount rate. For example, if something is off by 20%, divide by 0.8. That gives the pre-discount price.
- How do I find the pre-tax amount from a total price?
- Divide the total by 1 plus the tax rate. If the tax rate is 8%, divide by 1.08. The result is the amount before tax.
- Why can’t I just subtract the percentage from the final amount?
- Because the percentage is based on the original amount, not the final one. After a discount or tax, the starting point has already changed, so subtraction alone does not undo it correctly.
- What is the most common mistake with reverse percentages?
- The most common mistake is dividing by the percentage itself instead of by the full multiplier. For a 20% discount, the correct divisor is the remaining share of the original price, not 20%.
- Can reverse percentages be used for salary raises or markups?
- Yes. If a salary or price increased by a percentage, divide the final amount by 1 plus the rate. That gives the original amount before the increase.
- How can I check if my reverse percentage answer is right?
- Multiply the original amount by the percent-change multiplier and see if you get the final amount back. If the problem was a discount, use the remaining share of the original price; if it was an increase, use the added share.
About this article
Written by Yahav Atia, WSO – the digital agency (wso.co.il) that owns and founded this site, with the help of AI, and passed automated quality checks before publication. Numbers in the worked examples were computed in code, not estimated.