Reverse Percentages: Original Price Before a Discount or Increase

To find an original price from a final price and a percentage change, divide by the change multiplier. After a 30% discount, 70% of the original price remains, so divide the sale price by 0.70. After a 12% increase, the final price is 112% of the original, so divide by 1.12.

This is called a reverse percentage calculation. You know the result and work backward to the starting amount. The examples below use illustrative prices; the same arithmetic works in any currency.

The reverse percentage formula

Let r be the percentage change written as a number, such as 30 for 30%. Choose the formula that matches the direction of the change:

Change	Final price represents	Original price formula
Discount or decrease of r%	(100 − r)% of the original	Final ÷ (1 − r / 100)
Increase of r%	(100 + r)% of the original	Final ÷ (1 + r / 100)

The original price represents 100%. For a discount, the retained percentage is the share of that original price still payable. The Open University's reverse percentages lesson explains the equivalent method: find the value of 1% from the known percentage, then multiply by 100.

Find the original price before a discount

A desk lamp costs $84 after a 30% discount. To recover the price before the sale:

Find the retained percentage: 100% − 30% = 70%.
Write it as a decimal: 70 ÷ 100 = 0.70.
Divide the sale price by that decimal: $84 ÷ 0.70 = $120.

Check forward from $120. The discount is 30% of $120, which is $36. Subtracting it gives $120 − $36 = $84, exactly the sale price. Alternatively, $120 × 0.70 = $84.

The 1% method reaches the same result: $84 ÷ 70 = $1.20 per percentage point of the original price; $1.20 × 100 = $120. Divide by 70 and multiply by 100, or simply divide by 0.70.

Find the original price before an increase

A service package now costs $179.20 after a 12% increase. Its new price represents 112% of the old price, not 100%.

Original price = $179.20 ÷ 1.12 = $160.

Check the result by applying the increase to $160: 12% of $160 is $19.20, and $160 + $19.20 = $179.20. The starting amount and the increase together reproduce the new price.

The same method works for increases above 100%. For example, a 150% increase makes the final amount 250% of the original, so the multiplier is 2.50. A final price of $300 would therefore come from $300 ÷ 2.50 = $120.

Why adding the same percentage back fails

Adding 30% to the discounted $84 does not recover the lamp's original price. It gives $84 × 1.30 = $109.20. That calculation takes 30% of the smaller sale price, whereas the original discount took 30% of $120.

The Open University's lesson on calculating original amounts identifies the key question: which amount is the percentage based on? Reversing the direction does not restore the original base.

Subtracting 12% from the increased $179.20 fails for the same reason. It gives $157.696 before rounding, rather than $160. To undo multiplication by 1.12, divide by 1.12. To undo multiplication by 0.70, divide by 0.70.

Check your answer with the percentage calculator

ToolNimbly's percentage calculator does not have a dedicated reverse-percentage mode that directly finds the original price from the final price and change rate. Do the division above first, then use its existing modes to check your answer:

X% of Y: enter 30 and 120. The discount amount is 36; subtract it from 120 to confirm 84.
X is what percent of Y: enter 84 and 120. The result is 70%, confirming the retained share.
Percentage change: use 120 as the starting value and 84 as the final value. The result is a 30% decrease.

Rounding and cases the formula cannot resolve

Keep the multiplier and division result unrounded until the final step. For example, $49.99 after a 15% discount gives $49.99 ÷ 0.85 = $58.811764… before rounding, or approximately $58.81. Checking $58.81 × 0.85 gives $49.9885, which rounds to $49.99. If the displayed sale price was rounded, the calculation recovers an estimate consistent with it, not necessarily a unique exact original amount.

0% change: the multiplier is 1, so the original and final prices are equal.
100% discount: the multiplier is zero. Every original price becomes zero, so the original cannot be recovered from the free price alone.
Discount above 100%: the formula produces a negative multiplier. That is outside an ordinary nonnegative-price discount example; check what the offer means.
Other adjustments: a fixed delivery charge, multiple discounts or a rounded percentage needs additional information. This single-change formula assumes the stated percentage is the only adjustment.