Reverse percentages: find the original price before a discount

Start with the share that remains

A 20% discount leaves 80% of the original price. If the sale price is 80, the equation is original × 0.80 = 80. Dividing both sides by 0.80 gives an original price of 100. The numbers represent generic monetary units; no currency conversion or current retail offer is implied.

Adding 20% to the sale price would produce 96, because the 20% is then applied to 80 rather than 100. The mistake is a change of base, not an issue with rounding. Check a reverse calculation by applying the stated discount to your recovered original price.

Original before a discount = final / (1 − d/100)
80 / (1 − 20/100) = 100

Reverse an increase with a different denominator

If a bill becomes 120 after a 20% increase, divide 120 by 1.20 to recover 100. Subtracting 20% of 120 would produce 96 and would not undo the original increase. The multiplier is the cleanest way to keep the base explicit.

The same arithmetic can recover a pre-tax amount from a total only when a single known tax rate applies to the whole base. Real invoices may contain exempt items, multiple rates or separately rounded lines. The equation alone does not determine which tax treatment applies.

Original before an increase = final / (1 + r/100)

Undo successive discounts one multiplier at a time

Two discounts of 20% and 10% leave 0.80 × 0.90 = 0.72 of the original amount. A final price of 72 therefore came from 72 ÷ 0.72 = 100. Adding the discounts to get 30% off would be incorrect: the second discount acts on the already reduced price.

Write the full chain of multipliers before reversing it. If a fixed voucher is involved, its position matters. A 10-unit voucher before a percentage discount and the same voucher afterward need different equations. Keep shipping, tax and fixed fees separate unless the stated rules include them.

When the original cannot be recovered uniquely

A 100% discount leaves zero. Dividing by zero is undefined, and a final price of zero does not reveal a unique original price. Similarly, a rounded final price may correspond to a small interval of possible originals rather than a single exact cent amount.

The find-the-total mode calculates known part ÷ (known percentage / 100). After a 20% discount, the remaining share is 80%, so enter 80 as the known amount and 80 as the known percentage to recover 100. Entering the discount rate 20 instead would answer a different question. For a 20% increase, a final amount of 120 is 120% of its original, so enter 120 and 120. Two discounts of 20% and 10% leave 72%, not 70%; a final 72 then comes from an original 100. Verify the recovered amount by applying the changes forward.