The rule
original = final ÷ (1 ± percentage ÷ 100)
Why it works
The percentage was applied to the original, so undoing it means dividing by the same factor, not adding the percentage to the final figure. Adding it back applies the percentage to the wrong base, which is smaller.
How to do it by hand
- Turn the percentage into a factor: 20 % off gives 0.8, 20 % on gives 1.2
- Divide the final value by that factor
- The result is the original value
- Check by applying the percentage forwards again
What is worth knowing
This is the single most common percentage error in commerce, and it always goes the same way. A price of 80 after 20 % off was originally 100, not 96 — adding 20 % to 80 gives 96, because that 20 % is taken from the smaller number. The same applies to tax: to strip 21 % VAT from a gross price you divide by 1.21, not multiply by 0.79. Those two give 82.6 % and 79 % of the gross, a gap of nearly four points, which on an invoice is money.