Reverse percentages: how to find the original number

To find the number before a percentage was added, divide by one plus the rate, not subtract the rate from the total. A total of 123 after a 23% increase began as 100, because 100 × 1.23 = 123, so working backwards means 123 ÷ 1.23. Subtract 23% from 123 instead and you get 94.71 — a wrong answer that feels so right it has its own name in maths classrooms: the reverse percentage error.

Why subtracting fails

The 23% was charged on the original, smaller number, not on the total you’re holding now. When you subtract 23% from 123, you’re taking 23% of the larger, already-inflated figure — a bigger slice than the one that was actually added. That’s why 123 minus 23% overshoots the correction and lands on 94.71 instead of 100. The increase and the decrease aren’t mirror images of each other, because the base changes between them. Going up, the rate applies to 100. Coming back down by subtraction, you’d be applying it to 123 — a different, larger number, so the arithmetic doesn’t undo itself.

The method

For an increase, divide the total by 1 plus the rate, expressed as a decimal. A 23% increase means dividing by 1.23. For a decrease, divide by 1 minus the rate. A price of 60 after a 20% discount started at 75, because 60 ÷ 0.8 = 75 — dividing by 1 minus 0.20, not adding 20% back onto 60.

The same logic scales to any rate. A total of 345 after a 15% increase started at 300, since 345 ÷ 1.15 = 300. Notice the pattern holds regardless of the size of the numbers or the size of the rate: increase, divide by 1 + rate; decrease, divide by 1 − rate. Once that rule is fixed, reverse percentages stop being a special case and become one division.

Where this shows up

Any price quoted after tax has been added hides an original, pre-tax figure that only division recovers correctly. Any figure reported after a period of growth — a population, a salary, a measurement — has an earlier value that subtraction will misstate. Any discounted price shown as the final, reduced amount conceals an original price that division alone gets back to exactly.

The sanity check

Once you’ve calculated an original figure, apply the percentage forward again and confirm you land back on the quoted total. Take the 100 recovered from 123: apply a 23% increase and you should get 123 again. Take the 75 recovered from 60: apply a 20% discount and you should get 60 again. If the forward calculation doesn’t return the number you started with, the reverse calculation was done wrong somewhere — usually because a subtraction crept in where a division belonged. This check costs one extra step and catches the exact mistake this page exists to prevent.

Reverse percentages sit next to percentage change and stacked discounts as the three places this arithmetic most often goes wrong, and it’s exactly what the percentage calculator’s reverse mode is built to handle directly.