Here is a statement that sounds perfectly reasonable: "if a price goes up 20\% and then comes down 20\%, it returns to where it started." It sounds like adding a positive and a negative of the same size, which should cancel. But it doesn't, and the reason it doesn't is one of the sharpest lessons in the grammar of percentages.

Take a ₹100 item. Up 20\%: ₹100 + 20 = 120. Down 20\% of 120: 120 - 24 = 96. You are left with ₹96, not ₹100. The "missing" ₹4 is the answer's fingerprint, and once you see where it comes from, the entire structure of compound percentages falls into place.

Where the missing money goes

The two percentages are not the same number. They look identical on paper — "20\% up, 20\% down" — but they are measured against different things.

The 20\% up is measured against the original ₹100. So it adds ₹20, taking the price to ₹120.

The 20\% down is measured against the new ₹120. So it subtracts 20\% of 120 = 24, not 20\% of 100 = 20. The rise added ₹20, the fall removed ₹24, and the gap is the missing ₹4.

Three bars showing a price of one hundred rising by twenty percent to one hundred twenty, then falling by twenty percent to ninety-sixThree horizontal bars stacked from top to bottom. The top bar represents one hundred rupees as the original price. The middle bar is twenty percent longer than the top, representing the price after a twenty percent rise to one hundred twenty rupees. The bottom bar is shorter than both, representing the price after a twenty percent fall from the new base, ending at ninety-six rupees. Labels beside each bar show the rupee amounts, and arrows between them show the absolute changes of plus twenty and minus twenty-four. ₹100 (start) original ₹120 (up 20% of 100) +₹20 ₹96 (down 20% of 120) −₹24 final price = 100 × 1.20 × 0.80 = ₹96
A price of ₹$100$ rises by twenty percent of its starting value (adding ₹$20$ to give ₹$120$), then falls by twenty percent of the new value (subtracting ₹$24$ to give ₹$96$). The two percentages look symmetric on paper but act on different bases, and the fall removes more than the rise added.

Why the bases differ: "up 20\%" means "add 20\% of the current price," and in step one the current price is ₹100. "Down 20\%" means "subtract 20\% of the current price," and by step two the current price has already changed to ₹120. The percentage is relative to whatever price it is applied to, not to some fixed historical value. So the two arrows move different absolute amounts, and their cancellation is only approximate.

The multiplier view

The cleanest way to see this is through multipliers. A 20\% increase multiplies the price by 1.20. A 20\% decrease multiplies by 0.80. The two operations together are a single multiplication:

100 \times 1.20 \times 0.80 = 100 \times 0.96 = 96.

The combined factor is 0.96, a 4\% net decrease. No matter what starting price you use, "up 20\% then down 20\%" always leaves you at 96\% of the original — exactly 4\% short.

Why 0.96 and not 1.00? Because 1.20 \times 0.80 = 0.96 and not 1.00. The arithmetic of multipliers is what it is; you cannot argue with it. The geometric identity behind this is

(1 + x)(1 - x) = 1 - x^2.

With x = 0.20, this gives 1 - 0.04 = 0.96. The "missing 4\%" is exactly x^2, the square of the percentage, expressed as a fraction. This is a pattern — not a coincidence — and it generalises.

The same trick with any pair

Let's test the pattern on other matched pairs.

Up 10\% then down 10\%. Multiplier: 1.10 \times 0.90 = 0.99. You end at 99\% of the original — a net 1\% loss. Indeed, x^2 = 0.10^2 = 0.01.

Up 50\% then down 50\%. Multiplier: 1.50 \times 0.50 = 0.75. You end at 75\% of the original — a 25\% loss. Indeed, x^2 = 0.50^2 = 0.25.

Up 100\% then down 100\%. Multiplier: 2.00 \times 0.00 = 0. You end with nothing. Indeed, x^2 = 1.00^2 = 1.00. A 100\% drop wipes the price out entirely, and "up 100\% first" doesn't save it.

The larger the percentage, the larger the gap — and the gap grows as the square of the percentage, not linearly. A 10\% wobble costs 1\%; a 20\% wobble costs 4\%; a 50\% wobble costs 25\%. Volatility is quadratically expensive.

Does the order matter?

Interesting question. Try it the other way around: down 20\% then up 20\%.

100 \times 0.80 \times 1.20 = 100 \times 0.96 = 96.

Same answer. Multiplication is commutative — 0.80 \times 1.20 = 1.20 \times 0.80 — so swapping the order of the two percentage operations leaves the net multiplier unchanged. The rupee amount changed during the process is different (you lose ₹20 first, then gain ₹19.20, ending at the same ₹96), but the final price is the same. The "missing 4\%" does not depend on which percentage you do first.

What students usually think, and why it's wrong

The intuition behind the mistake is "+20\% and -20\% are opposite operations, so they cancel." This is true when the operations are additions of fixed amounts: "add ₹20 then subtract ₹20" does return you to where you started. It fails for percentages because a percentage is not a fixed amount — it is a fraction of whatever you apply it to. The base changes between the two operations, so the two absolute amounts are different.

The same mistake shows up in more subtle places. Stock-market returns: "up 10\% one month, down 10\% the next" is a net 1\% loss, not a wash. Exchange rates: the rupee "falling 5\%" against the dollar and then "rising 5\%" does not restore parity. Productivity: "15\% faster then 15\% slower" ends up slower than starting. The pattern is everywhere, and spotting it becomes a reflex.

The deeper lesson

The real lesson is that percentage changes do not add; they multiply. You cannot sum "+20\%" and "-20\%" to get zero. You have to compose them as multipliers, 1.20 \times 0.80, and the answer is 0.96, not 1.00.

This is the single most important habit for anyone dealing with compound returns, inflation, discounts, interest rates, or exam percentages. A 50\% discount followed by a 30\% further discount is 1 \times 0.50 \times 0.70 = 0.35 — a 65\% discount, not 80\%. Two 10\% annual raises compound to 1.10^2 = 1.21, a 21\% raise, not 20\%. The missing piece — the cross-term, the square of the percentage — is the arithmetic of compound growth and the reason finance and inflation are taught as multiplicative, not additive.

Keep one rule: whenever two or more percentages are stacked on the same quantity, convert each to its multiplier and multiply them together. The rest is just arithmetic.

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