How to Use This Tool
A jacket costs £80 in the sale, marked “20% off”. What was the original price? The instinct is to add 20% back: 80 × 1.2 = 96. That is wrong, and it is wrong in a way that feels right, which is why it survives into spreadsheets, invoices and pricing decisions.
Why 96 is not the answer
The 20% was taken from the original price, not from the sale price. Twenty per cent of 100 is 20; twenty per cent of 80 is only 16. You removed 20 and added back 16, so you land four short.
The correct move is division. If 20% came off, what remains is 80% of the original, so:
original = 80 ÷ 0.8 = 100
Check it forwards: 100 − 20% = 80. Correct. This single reciprocal is the whole of reverse percentage — every situation below is the same operation with a different denominator.
The four situations
- After a discount. Divide by (1 − rate). £80 after 20% off → 80 ÷ 0.8 = £100.
- After an increase. Divide by (1 + rate). A salary of £42,000 after a 5% rise was 42000 ÷ 1.05 = £40,000.
- Removing tax. The same as an increase. A £120 total including 20% VAT is 120 ÷ 1.2 = £100 net, and the VAT is £20. Note that 20% of the gross is £24, which is the wrong answer and appears on a great many invoices.
- Finding the whole from a part. If 35 people are 14% of the group, the group is 35 ÷ 0.14 = 250.
The mistake in one sentence
Percentages are always taken of something, and after a change that something has changed. If you find yourself multiplying to undo a multiplication, stop and divide instead.
Where this costs real money
VAT is the big one. Charging 20% VAT on a net price and extracting 20% from a gross price are different sums, and getting the second one wrong understates your net revenue by 4% on every line. On a retail margin of 8%, that is half the margin. The panel above always shows both the correct figure and the tempting wrong one, precisely because the wrong one looks so plausible.
