A T-shirt tag says "30% OFF — now ₹560." The question that should be easy but somehow isn't: what was the price before the discount?

Your instinct might be, "add 30\% back to ₹560," giving 560 \times 1.30 = 728. That is a popular answer. It is also wrong, and the reason it is wrong is one of the cleanest lessons in percentage grammar — when you undo a percentage change, the base changes, and undoing by adding the same percentage gets the base wrong.

The right way to reverse a discount

Let's call the original price P. A 30\% discount removes 30\% of P, leaving 70\% of P. So the discounted price is

\text{discounted price} = 0.70 \times P

You are told the discounted price is ₹560. So

560 = 0.70 \times P

Solving for P:

P = \frac{560}{0.70} = 800

The original price was ₹800. You can check: 30\% of 800 is 240, and 800 - 240 = 560. Perfect.

Why dividing by 0.70 and not multiplying by 1.30: the 30\% was taken off the original price P, not off the discounted ₹560. So ₹560 equals 70\% of the original, which means P is ₹560 divided by 0.70. Adding 30\% to 560 adds 30\% of the discounted price, which is a smaller base, and gives a different answer.

Why the naive "add 30% back" answer fails

Let me spell out why the intuitive reflex is wrong, because the logic of the mistake is instructive.

When you think "-30\% then +30\% should cancel," you are treating the two percentages as opposite additions. But they are not additions — they are multiplications, and the bases differ.

The two percentages act on different-sized bases. Taking 30\% off 800 is 240; adding 30\% to 560 is 168. Those are different amounts, so they do not cancel. This is the same phenomenon as "up 20% then down 20% doesn't return to start" — percentages compose by multiplication, not addition.

Reversing a thirty percent discount from five hundred sixty rupees to the original price of eight hundredThree horizontal bars stacked. The top bar is eight hundred rupees long and labelled original price P. The middle bar is five hundred sixty rupees long and shaded, labelled discounted price seventy percent of P. An arrow from the top to the middle is labelled multiply by zero point seven. A second arrow from the middle back to the top is labelled divide by zero point seven, showing how to recover the original. A third bar on the right of the middle shows the incorrect naive answer seven hundred twenty-eight, found by multiplying the discounted price by one point three. ₹800 (original P) × 0.70 ₹560 (70% of P) ÷ 0.70 correct: 560 ÷ 0.70 = 800 ✓ naive: 560 × 1.30 = 728 ✗ different bases — 30% of 800 is ₹240, 30% of 560 is only ₹168
Going forward from original to discounted price is multiplication by $0.70$. Going backward — recovering the original — is division by $0.70$, not multiplication by $1.30$. The naive reversal gets ₹$728$, which is less than the true ₹$800$ by exactly the "missing percentage" from operating on a smaller base.

The general rule

For any discount percentage d\%, the rule is the same.

For a 30\% discount, the factor is 0.70. For a 20\% discount, the factor is 0.80. For a 50\% discount, the factor is 0.50 — and dividing by 0.50 is the same as multiplying by 2, which matches the intuition that "the original was double the half-off price."

For a price increase (say a tax), the rule is dual: the factor is (1 + \tfrac{t}{100}). So to reverse an 18\% GST add-on, divide the after-tax price by 1.18.

A quick worked example

The sticker says "25\% OFF, final price ₹1800." What was the original price?

The remaining fraction after a 25\% discount is 0.75. So

P = \frac{1800}{0.75} = 2400

Check: 25\% of 2400 is 600, and 2400 - 600 = 1800. Correct.

Compare with the naive "add 25\% back": 1800 \times 1.25 = 2250, which is wrong. The gap between 2250 and 2400 — ₹150 — is exactly the accumulated asymmetry between the two bases.

A unitary-method view, for the same answer another way

You can also solve this with the unitary method, which may feel more natural than decimal multipliers.

The discounted price is 70\% of the original. So if 70\% is worth ₹560, then 1\% is worth \tfrac{560}{70} = ₹8. And 100\% — the original — is 100 \times 8 = ₹800. Same answer, same logic, just packaged as "find one percent, then scale up to one hundred."

Why this works: "70\% of P = 560" is the unitary setup — you know the value of some number of percent, and you want the value of one hundred percent. Dividing to find one percent, then multiplying to find a hundred, is exactly the same step as dividing by 0.70 once.

The key habit

Any time you see "reverse a percentage change," run this quick mental check: was the percentage taken off the unknown original? If yes, then the discounted price equals (\text{original}) \times (\text{remaining fraction}), and you recover the original by dividing the discounted price by the remaining fraction.

The dangerous shortcut — "add the same percentage back" — works only for very small percentages, and even then it is approximate. For any serious problem, use the division rule. Once it becomes reflex, reversing a discount is no harder than applying one.

Related: Percentages and Ratios · Price Up 20% Then Down 20% — Why You End Up Below Where You Started · Shop-Price Slider: Drag the Discount %, Watch the Price Bar Shrink · Why You Can't Just Add Percentages When Discounts Stack