Your shop friend says: "Relax, I'll mark the kurta up by 20\% for the festival, and after Diwali I'll mark it down by the same 20\%. The price comes right back to where it started." It sounds perfectly fair. It is not. The kurta that started at ₹1000 finishes at ₹960 — a quiet 4\% below where it began. The two 20s do not cancel, and the tower below shows you exactly why.

The tower

Three stacked bars showing a price of one thousand, then one thousand two hundred after a twenty percent increase, and finally nine hundred sixty after a twenty percent decreaseThree horizontal bars stacked vertically. The top bar is labelled starting price one thousand rupees and has a neutral fill. The middle bar is longer, labelled after plus twenty percent equals twelve hundred rupees, and is filled in accent colour. The bottom bar is shorter than the top, labelled after minus twenty percent equals nine hundred sixty rupees, and is filled in a contrasting colour. A dashed vertical guide line extends downward from the right edge of the top bar so the reader can see that the bottom bar falls short of the starting length by forty rupees. ₹1000 (starting price) ₹1200 (after +20%) +₹200 ₹960 (after −20%) −₹240 gap = ₹40 below the start (a 4% shortfall)
Starting at ₹$1000$, a $20\%$ increase adds ₹$200$ to reach ₹$1200$. A $20\%$ decrease then removes ₹$240$ — not ₹$200$ — because that second $20\%$ is taken off the *larger* ₹$1200$, not the original ₹$1000$. The bottom bar falls short of the top one by exactly ₹$40$.

Why the two twenties are not equal

The trap hides in one small word: "20\%." A percentage is never a standalone rupee amount. It is a rupee amount times a base. And the base silently changes between the two steps.

The first 20\% was computed from ₹1000. The second 20\% was computed from ₹1200. They were never the same rupee amount — one was ₹200, the other ₹240. Subtracting a bigger number than you added is exactly why you finish below the start.

Why: every percentage operation is a multiplication, and "of what" matters. "20\% of 1000" and "20\% of 1200" are two different numbers, even though both are "20\%". The percentage sign does not carry the base around with it — the base has to be supplied freshly each time.

The multiplier form

There is a cleaner way to see this whole dance. Every percentage change can be written as a single multiplier.

Chaining the two steps means multiplying by both:

1000 \times 1.20 \times 0.80 = 1000 \times 0.96 = 960

So the composite multiplier is 0.96, not 1. The "4\% loss" you see on the tower is the gap between 0.96 and 1. And this is not a rounding error — it is exact algebra:

(1 + p)(1 - p) = 1 - p^2

Plug in p = 0.20 and you get 1 - 0.04 = 0.96. The tail-end of the formula, -p^2, is the shortfall. For any percentage p (as a decimal), marking up by p and then marking down by p leaves you at 1 - p^2 of the original. You always end below.

The symmetry direction doesn't help

What if the shop friend had done it in the opposite order — 20\% off first, then 20\% up?

1000 \times 0.80 \times 1.20 = 1000 \times 0.96 = 960

Same answer. Multiplication is commutative, so swapping the order of a markdown and a markup gives the same final price. The two 20s cannot cancel no matter which one you fire first.

Example: +50% then −50% on a ₹$1000$ shirt

Start at ₹1000. Apply +50\% first: the shirt jumps to ₹1500 (you added ₹500). Now apply -50\%: you take half off ₹1500, which is ₹750 — and you finish at \textsf{₹}750, a full \textsf{₹}250 below where you started.

Check with the formula: 1000 \times 1.5 \times 0.5 = 1000 \times 0.75 = 750. Or directly 1 - 0.5^2 = 0.75. The bigger the percentage, the bigger the shortfall — because p^2 grows fast. A 50\% swing leaves a 25\% gap. A 10\% swing leaves only a 1\% gap.

The common confusion, named

"I added 20 and subtracted 20, so the two cancel." This reasoning is correct for rupee amounts but wrong for percentages. If the shop friend had written "I'll add ₹200 and then take off ₹200" the kurta really would return to ₹1000. The word "20\%" is what breaks the symmetry, because the rupee amount it stands for depends on what it is a percentage of.

This is also why a shopkeeper's "buy two, get 50\% off" often beats "50\% off everything, always" — the percentage alone does not tell you how much money changes hands. The base does.

Summary

Related: Percentages and Ratios · 50% Off Then 30% Off Is Not 80% Off · Why You Can't Add Successive Discount Percentages · Does a 200% Increase Mean Doubling?