The sale board at the mall reads "Flat 50\% off, plus an additional 30\% off at the counter." You do the lazy arithmetic in your head — "50 + 30 = 80, so this is 80\% off" — and mentally plan the celebration. Then the bill comes and the kurta that was supposed to cost ₹200 costs ₹350. What went wrong? The two discounts compounded; they did not stack. The real discount was 65\%, not 80\%. The simulator below lines the two against each other so the gap is impossible to miss.

Two paths, one starting price

Comparing a fifty percent discount then thirty percent discount against a single eighty percent discount on a one thousand rupee itemA vertical stack of four bars, each labelled with a rupee value. The top bar is the marked price of one thousand rupees. Below it, a pair of progressively shorter bars shows the effect of applying a fifty percent discount (five hundred) and then a thirty percent discount on the reduced price, ending at three hundred fifty rupees. Beneath those, a single bar shows the result of an eighty percent flat discount, landing at two hundred rupees. Dashed vertical guides make the gap between the two endpoints explicit. ₹1000 (marked price) ₹500 (after −50%) ₹350 (after extra −30%) actual = 65% off ₹200 (flat −80%) lazy = 80% off gap = ₹150 — the stacked offer is worse than it sounded
The stacked "$50\%$ then $30\%$" offer lands at ₹$350$, while a flat $80\%$ discount would land at ₹$200$. The gap of ₹$150$ is the difference between adding percentages (wrong) and multiplying their multipliers (right). On a ₹$1000$ item that is a full $15\%$ of the marked price — real money.

Why stacking isn't adding

The mistake is to treat percentages like rupee amounts. Rupee amounts you can add: ₹500 off plus ₹300 off really is ₹800 off. Percentages, you cannot — because each percentage quietly refers to a different base.

The second discount shrank because the base it was computed from had already shrunk. "30\%" in the second step is not the same rupee amount as "30\%" in the first step would have been. That is the whole trick.

Why: a percentage sign does not carry its base around. "30\%" is just a multiplier — 0.30 — that has to be applied to some quantity to produce a rupee amount. When two percentage cuts happen in succession, the second multiplier sees a smaller quantity, so it produces a smaller rupee cut.

The multiplier form — adding made into multiplying

The cleanest way to handle any chain of percentage discounts is to convert each into a multiplier and multiply them together.

Stacking the first two:

1000 \times 0.50 \times 0.70 = 1000 \times 0.35 = 350

The composite multiplier is 0.35, meaning the customer pays 35\% of the marked price — so the total discount is 65\%. Not 80\%, not even close.

\text{actual discount} = 1 - (0.50 \times 0.70) = 1 - 0.35 = 0.65

The formula gives a tidy general rule: for two successive discounts of a\% and b\%, the combined discount is

1 - \left(1 - \frac{a}{100}\right)\left(1 - \frac{b}{100}\right) \;=\; \frac{a + b}{100} - \frac{ab}{10\,000}

The naïve answer (a + b)/100 — adding the percentages — is always an overestimate of the real discount. The correction is the ab/10{,}000 term, which is exactly what the compounded multiplication produces. For a = 50, b = 30, the correction is \tfrac{50 \times 30}{10{,}000} = 0.15 = 15\%. So the advertised "80\%" shrinks down to 65\%.

Spotting when the gap is big

The ab/10{,}000 correction is small when both percentages are small, and grows quickly as they climb.

The last row is instructive: two 50\% cuts do not make the product free. After half off, you are at ₹500; after half off again, you are at ₹250. Stacking 50s can never reach zero, because each step removes half of what remains — and there is always something left.

Example: will three 20% discounts beat a flat 50%?

Naïvely, three 20\%s add to 60\%, which beats 50\%. Is that real? The multiplier for the stack is

0.80 \times 0.80 \times 0.80 = 0.512

which is a 48.8\% discount. So the lazy arithmetic said "three 20s = 60\%" but the actual discount is 48.8\% — less than the flat 50\%. The correction term, grown over three steps, ate up more than a tenth of the headline discount.

Why shops love the stacked offer

A store can advertise "50\% off, plus an additional 30\% off" and a shopper's brain shortcuts to "80\% off." But the store knows that the math delivers only 65\% — so the stacked offer feels more generous than it actually is. This is the same pricing psychology behind "buy 3 get 1 free" (really a 25\% discount, not a 33\% discount), and behind stacking sale coupons on top of a markdown. Once you have the multiplier rule, the fog lifts instantly.

Summary

Related: Percentages and Ratios · Plus 20% Then Minus 20%: Why You End Up Below Where You Started · Why You Can't Add Successive Discount Percentages · Does a 200% Increase Mean Doubling?