A reverse percentage calculation runs the arithmetic in the direction most people find hardest: you know the figure after a percentage was applied, and you want the figure it started from. The jacket cost $120 in the sale and the sign said 20% off — what was the original price? The invoice total is $1,320 including 10% tax — what was the amount before tax? You scored 85% and got 34 marks — how many marks were available?
The instinct is to apply the same percentage in the opposite direction, and that instinct is wrong every time. You do not add 20% back to $120. You divide by the factor that was applied. $120 is 80% of the original, so the original is 120 ÷ 0.8 = $150.
When you need this
Finding a pre-discount price. To check whether a sale is as good as advertised, or to work out the real margin on a discounted sale.
Stripping tax out of a gross figure. This is the everyday accounting version. If a total includes 10% tax, divide by 1.1 — do not take 10% off, which gives a different and wrong answer. See how sales tax maths works for the full treatment.
Recovering a total from a part and a rate. 34 marks is 85% of the paper, so the paper was out of 34 ÷ 0.85 = 40 marks. Same operation, different dressing.
Worked examples
$120 after 20% off. The price paid represents 100% − 20% = 80% of the original, so the factor is 0.8. Divide: 120 ÷ 0.8 = $150. Check it forwards: 20% of 150 is 30, and 150 − 30 = 120. Correct.
$1,320 including 10% tax. The total represents 110% of the pre-tax figure, so the factor is 1.1. Divide: 1,320 ÷ 1.1 = $1,200. The tax itself is $120. Note that taking 10% off $1,320 would give $1,188 — a $12 error, and one that appears constantly in hand-prepared bookkeeping.
34 marks is 85%. 34 ÷ 0.85 = 40 marks available.
A figure after a 150% increase. If a value is now 500 after growing 150%, the factor was 2.5, so the original was 500 ÷ 2.5 = 200. Growth of 150% means ending at 250% of where you started, which is the step people most often get wrong.
Where people go wrong
Adding the percentage back instead of dividing. Adding 20% to $120 gives $144, not $150. The 20% discount was 20% of the original $150, which is $30 — but 20% of $120 is only $24. You cannot take a percentage of the wrong base and expect it to reverse.
Subtracting the tax rate from the gross. As above: dividing by 1.1 and subtracting 10% are different operations with different answers. The larger the rate, the larger the gap — at 20% tax the error is over 3% of the total.
Getting the factor wrong for increases versus decreases. After a decrease, divide by (1 − rate). After an increase, divide by (1 + rate). Mixing these up is easy when you are working quickly, and the result always looks plausible.
Reversing stacked percentages one at a time in the wrong order. If a price had 30% taken off and then a further 10%, the combined factor is 0.7 × 0.9 = 0.63, and you divide by 0.63. Reversing them separately in either order works too, as long as you divide each time — but dividing by 0.6 (from adding the rates) does not.
Assuming a round original. Reverse calculations frequently produce untidy numbers, and that is usually correct rather than a sign of error. A $99.99 price after 35% off started at $153.83, not $154.
The related calculators
If you want to go forwards instead — original price and rate, find the result — use decrease by a percentage or increase by a percentage. Our reverse percentages explained guide walks through the reasoning in more depth, with the tax and discount cases treated separately.