math · 2 min read · India
You cannot take 18% back off by subtracting 18%
To undo a percentage increase, divide rather than subtract. A total of ₹5,900 that already includes 18% GST came from a base of ₹5,000, because 5,900 ÷ 1.18 = 5,000. Subtracting 18% from 5,900 gives 4,838, which is ₹162 short. The same applies in reverse: an item selling at ₹800 after a 20% discount was ₹1,000, because 800 ÷ 0.80 = 1,000, not 800 + 20% = 960.
An invoice says ₹5,900 including 18% GST and you need the figure before tax. The obvious move is to take 18% off ₹5,900, which gives ₹4,838. The real answer is ₹5,000, and the difference is ₹162.
The obvious move fails because the 18% was never charged on ₹5,900. It was charged on ₹5,000, and 18% of ₹5,000 is a smaller amount than 18% of ₹5,900. Taking a percentage off the total removes too much, every time.
Divide by what the total represents
If a base grew by 18%, the total is 118% of the base, or 1.18 times it. Undoing that means dividing by 1.18 rather than subtracting anything: 5,900 ÷ 1.18 = 5,000. If a price fell by 20%, what remains is 80% of the original, so divide by 0.80. An item at ₹800 after a fifth off started at ₹1,000.
| What you know | Original | Subtracting instead | Off by |
|---|---|---|---|
| ₹5,900 including 18% GST | ₹5,000 | ₹4,838 | ₹162 |
| ₹1,120 after a 12% rise | ₹1,000 | ₹985.60 | ₹14.40 |
| ₹800 after 20% off | ₹1,000 | ₹960 | ₹40 |
| ₹1,000 after 50% off | ₹2,000 | ₹1,500 | ₹500 |
The error grows with the percentage
At 12% the subtraction method is off by about one and a half percent of the answer. At 50% it is off by a quarter: an item reduced to ₹1,000 in a half-price sale was ₹2,000, and adding 50% back gives ₹1,500. Push it to a 100% increase and the method collapses entirely, because subtracting 100% of the total leaves nothing at all.
This is the same mechanism as successive discounts, seen from the other end. Both go wrong when a percentage is read as a fixed quantity rather than a ratio applied to whatever number it was applied to.
The rule, in one line each
- After a rise of p%: divide the total by (1 + p/100).
- After a fall of p%: divide the total by (1 − p/100).
- Removing 18% GST: divide by 1.18, never subtract 18%.
- The tax or discount amount is then the difference between the two figures.
Worth checking the answer by going forwards again. Take ₹5,000, add 18%, and you should land on ₹5,900. If you do not, you divided by the wrong thing, and that check costs less than the mistake does.