Reverse Percentages — Original Price from a Sale Price, Pre-Raise Pay from Post-Raise Pay
What was the original price of an item that costs $80,000 after 20% off? Why adding the percentage back to a changed value is wrong, and the correct division formula, with examples.
When you only know a value after a percentage has already been applied — a sale price, a raised salary, a tax-inclusive price — finding the value before it was applied is called a reverse percentage calculation. The most common mistake is applying the percentage once more in the opposite direction.
The wrong way and the right way
If the price after 20% off is $80,000, what was the original price?
- Wrong: 80,000 × 1.2 = $96,000 → 20% off $96,000 is $76,800, which does not match.
- Right: 80,000 ÷ 0.8 = $100,000
The sale price is original × (1 − discount rate), so to go back you must divide by what was multiplied.
Original value = changed value ÷ (1 ± rate)
Divide by (1 + rate) if it increased, or by (1 − rate) if it decreased.
Everyday examples
- Sale price → original price: $52,000 after 35% off → 52,000 ÷ 0.65 = $80,000
- Post-raise → pre-raise salary: $3,360,000 after a 5% raise → 3,360,000 ÷ 1.05 = $3,200,000
- Tax-inclusive → pre-tax: $22,000 including 10% VAT → 22,000 ÷ 1.1 = $20,000
- After fees → principal: $970,000 received after a 3% fee → 970,000 ÷ 0.97 = $1,000,000
- After growth → before growth: this year's sales of $15 million, up 25% from last year → 15 ÷ 1.25 = $12 million
How much to charge to receive a set amount
If you want $1,000,000 in hand after a 3.3% fee, you should charge not 1,000,000 × 1.033 but 1,000,000 ÷ 0.967 ≈ $1,034,126. Charging $1,033,000 means a fee of $34,089 is deducted and you receive only $998,911.
When several steps are stacked
If the price after 20% off and then an extra 10% off is $72,000, the original price is 72,000 ÷ (0.8 × 0.9) = 72,000 ÷ 0.72 = $100,000. Multiply the factors of every step and divide once.
A 100% decrease cannot be reversed
A price after 100% off is $0, so the original value cannot be known (division by zero). In that case this calculator shows 'Cannot be calculated' instead of a result.
Make checking a habit
Apply the original rate to your reverse result and check that you get the starting value back. 20% off $100,000 → $80,000. The fourth row of the Percent tab (increased/decreased by) does this check instantly.
Try it in the calculator
In the fifth row of the Percent tab, enter the rate, the direction (increase or decrease) and the changed value to see the original value and the formula.