Divide, do not add back.
An item is 48 after 20% off. The original was 60 — because 48 is 80% of 60, so 48 ÷ 0.8 = 60.
Adding 20% back to 48 gives 57.60, which is wrong. It is always wrong, and always too low.
Why adding back fails
The 20% was taken from 60, so it was worth 12. When you add 20% to 48, you are calculating 20% of 48, which is only 9.60.
You are applying the same rate to a smaller base, so you get a smaller amount back than was removed. The gap never closes on its own.
The method
Work out what fraction of the original survived, then divide by it.
After a decrease of r%:
original = final ÷ (1 − r ÷ 100)48 after 20% off → 48 ÷ 0.8 → 60
After an increase of r%:
original = final ÷ (1 + r ÷ 100)130 after a 30% rise → 130 ÷ 1.3 → 100
One division either way. The only decision is whether to divide by something below 1 or above it.
Checking it
Run the change forwards on your answer and confirm you land back on the number you started with.
- Claim: 48 after 20% off means the original was 60
- Check: 20% of 60 is 12, and 60 − 12 = 48 ✓
Try the wrong answer: 20% of 57.60 is 11.52, giving 46.08 — not 48. The check fails instantly.
This takes seconds and catches every version of this error.
Where you actually need it
Removing tax from a total. A price includes 20% tax and you need the pre-tax figure. Divide by 1.2, not 0.8. On a 120 total that is 100, not 96.
Working out the real discount. Something is advertised at 45 “down from” a price you cannot see, and you know it was a third off. Divide 45 by ⅔ to get 67.50.
Recovering a base figure. Sales are 88 this quarter after a 12% fall. Last quarter was 88 ÷ 0.88 = 100.
Backing out commission. You received 940 after a 6% fee. The gross was 940 ÷ 0.94 = 1,000.
The one to memorise
For a rise of r%, dividing by 1 + r/100 is right and subtracting r% is wrong.
The temptation to subtract is strongest with tax, because “the tax is 20%, so take 20% off” feels self-evidently correct. It is the single most common version of this mistake, and it under-reports the original by a consistent margin every time.