Why You Can't Just Add Percentages When Discounts Stack

A shop in Dilli Haat displays: "30\% off the marked price. Bank-card holders, extra 30\% off." The marked price of the jacket is ₹2000. You do the lazy mental arithmetic: "30 + 30 = 60, so I pay 40\% of ₹2000, which is ₹800." You walk to the counter confidently.

The counter person rings up ₹980. You blink. Where did the extra ₹180 come from?

The answer is the quiet reason students get stuck on percentage problems: percentages don't add when they sit on top of each other, because each "30\%" is silently a percentage of a different number. Once you see which number each percentage is measuring against, the whole paradox dissolves.

Tracing the rupees

Let's follow the money, step by step, exactly as the shopkeeper would.

Step 1: apply the first 30\% discount.

Subtract: ₹2000 - 600 = 1400.

Step 2: apply the second 30\% discount to the new price.

Here is the key step. The bank-card discount is not 30\% of the original ₹2000. It is 30\% of the already-discounted ₹1400. Why? Because the shop has already marked the jacket down; the card discount is computed on what you currently owe, not what you owed before.

Subtract: ₹1400 - 420 = 980.

That is the ₹980 on the bill. The ₹180 "missing" from your lazy estimate is exactly the difference between "30\% of 2000" and "30\% of 1400" on the second step:

The first discount was ₹600 because its base was ₹2000. The second was only ₹420 because its base had shrunk to ₹1400. Adding the two percentages would have correctly predicted the bill only if both percentages had the same base — which they don't.

Why the bases differ: a percentage is always attached to some reference quantity. The first discount's reference is the original price (2000). The second discount's reference is the price after the first discount (1400). Since the references differ, "30\%" in the two steps stands for different rupee amounts. The percentages are not wrong — they are just measuring different quantities.

What addition would require

Percentages are addable only when they are percentages of the same base. For example:

• "Sales tax is 18\% and a service charge is 5\%" — both applied to the same sub-total, so the total extra is (18 + 5)\% = 23\% of the sub-total. Adding works.
• "Discount is 30\% plus an extra 30\% off (both on the marked price)" — this phrasing would also permit adding, giving a 60\% discount on the marked price. But this is almost never what "extra 30\%" means in a real shop.

The language tells you. "30\% off, plus an additional 30\%" almost always means "the second 30\% applies after the first," i.e. to the reduced price. If a shop ever genuinely meant "two 30\%s on the marked price, so pay 40\%," they would say "60\% off the marked price" directly — because that is simpler.

The multiplier trick — adding turned into multiplying

There is a much cleaner way to handle stacked discounts: use multipliers instead of percentages.

• A 30\% discount corresponds to the multiplier 1 - 0.30 = 0.70.
• Two discounts of 30\% each correspond to the multiplier 0.70 \times 0.70 = 0.49.
• So you pay 49\% of the marked price, and the real discount is 100\% - 49\% = 51\%, not 60\%.

On ₹2000 the jacket costs 2000 \times 0.49 = 980. Matches the bill.

The multiplier form converts the hard rule ("percentages don't add") into a simple one ("multipliers multiply"). For any chain of percentage discounts p_1, p_2, \ldots, p_n, the final price is

and the overall discount is

Plug in p_1 = p_2 = 0.30 and you get 1 - 0.70 \times 0.70 = 1 - 0.49 = 0.51, which is the 51\% discount.

The gap is always a shortfall

Compare the two expressions for two stacked discounts:

The difference is the extra term -p_1 p_2. For p_1 = p_2 = 0.30, that extra term is -0.09 = -9\% — so the lazy sum overestimates the discount by 9 percentage points. And this is always a minus sign, because both p_i are positive. You will never guess less discount than there actually is by adding naively; you will always guess more. This is why shops love stacked percentages: they sound better than they are.

When you can add

There is one common situation where percentages really do add, and it is worth keeping in your pocket for contrast.

• Tax on the same sub-total. If GST is 18\% and a service charge is 5\% on a restaurant bill — both applied to the same sub-total — then together they add to 23\%. You can sanity-check by computing 1.18 \times 1.05 = 1.239, i.e. 23.9\% — slightly more than 23\% because of compounding, but close. In practice, Indian restaurants compute the service charge first, then GST on the service-charged subtotal, which gives exactly 1.05 \times 1.18 = 1.239. If both were genuinely applied to the same sub-total, the addition 18 + 5 = 23\% would be exact.

• Independent taxes on the same base. State tax and federal tax, each levied on the same sale price — these add exactly.

The rule is: percentages add when and only when they share a base. As soon as one percentage sits on top of another's outcome, you need multiplication, not addition.

The easy check — multiplier form always works

When in doubt, translate every percentage into a multiplier and multiply them together. This works for:

• Successive discounts. Multiply (1 - p_i) for each discount.
• Successive increases. Multiply (1 + p_i) for each increase.
• Mixed chains. Multiply each factor with its own sign. A 20\% increase followed by a 15\% decrease is 1.20 \times 0.85 = 1.02, a 2\% net rise — not a 5\% rise.

The multiplier form never needs you to remember whether percentages are adding or not. It handles every case uniformly. Most exam errors in percentage problems come from students trying to add the wrong percentages; switching to multipliers eliminates the whole class of mistakes.

Summary

• Percentages don't add when each percentage refers to a different base.
• Two stacked 30\% discounts give a 51\% total discount, not 60\%, because the second 30\% is computed from the already-reduced price.
• The multiplier form converts the puzzle into ordinary arithmetic: multiply (1 - p_i) for each discount.
• The correction for two stacked discounts is -p_1 p_2: the lazy sum always overestimates the real discount.
• Percentages do add when they share a base (e.g. two taxes computed on the same sub-total), but that is the exception, not the rule.

Related: Percentages and Ratios · 50% Off Then 30% Off Is Not the Same as 80% Off · Plus 20% Then Minus 20%: Why You End Up Below Where You Started · Does a 200% Increase Mean Doubling?