Something rises 25%, then falls 20%. Intuition says those should roughly cancel — a rise and a fall, opposite in direction, similar in size. They do not cancel. In fact they land you back exactly where you started, and the reason is one of the most common quiet errors in everyday percentage arithmetic.
Why 25% up and 20% down are actually the same move
Start with 100. A 25% rise takes it to 125. A 20% fall from 125 takes away 20% of 125 — which is 25 — landing back at exactly 100. The two percentages look different, 25 versus 20, but they describe the identical distance in opposite directions, because each percentage is measured against a different base. The rise is 25% of the original 100; the fall is 20% of the new, larger 125. Run this through the percentage calculator and both directions come out exact, not approximate — this is not a rounding coincidence, it is what has to happen whenever a percentage rise is exactly reversed by the correct percentage fall.
The rule hiding underneath
Whenever a value goes up by some percentage and you want to know what percentage fall gets you back to the start, the fall is always a smaller percentage than the rise was — because the fall is being measured against the new, larger number. A 100% rise (doubling) needs only a 50% fall to reverse it, not 100%: doubling 100 gives 200, and a 50% fall from 200 is exactly 100 again. A 10% rise needs only about a 9.1% fall. The bigger the rise, the bigger this gap grows between the rise percentage and the fall percentage that undoes it.
Where this actually matters
This is not a trivia point — it shows up anywhere something moves twice and gets summarized in percentages. A stock that falls 50% and then rises 50% is not back to even — it fell to half its value, and a 50% rise from half only gets you to three-quarters. A discount of 30% followed by a “restocking” markup of 30% does not return to the original price — the markup, applied to the already-discounted price, is a smaller absolute amount than the discount was. Anywhere you see two percentage changes applied in sequence, check whether they are being measured against the same base before assuming they offset.
The direction trap below zero
Percentage change keeps its ordinary meaning even when the starting number is negative, which is where a second common error creeps in. Going from −50 to −25 is a genuine rise — the number got bigger — even though both values are negative and the naive instinct is to think in terms of “smaller magnitude” rather than “larger value.” The calculator handles this by dividing against the absolute value of the starting number, which is what keeps the direction (rise or fall) correctly signed no matter which side of zero the numbers sit on.
The same trap in investment return claims
This exact error shows up constantly in how investment performance gets summarized. A portfolio that loses 50% in one year and gains 50% the next is not back to even — it fell to half its starting value, and a 50% gain from that halved value only recovers to three-quarters of the original. The return on investment calculator handles exactly this kind of multi-period comparison correctly, because averaging percentage returns arithmetically (adding them and dividing) systematically overstates real performance whenever returns vary from period to period — the mathematically correct method compounds each period’s actual return rather than averaging the percentages, for precisely the reason a rise and fall of matched percentage are not really a wash.
A simple test that catches the error before you make it
Before combining two percentage changes by adding or subtracting them, ask one question: were both percentages measured against the same starting number? If a rise is measured against the original value and a later fall is measured against the new, already-changed value, they are not directly comparable by simple addition or subtraction — the correct approach multiplies the growth factors together (1.25 × 0.80 for a 25% rise then a 20% fall) rather than adding the percentages, and 1.25 × 0.80 = 1.00 exactly, confirming the return-to-start result without needing to trace through the dollar amounts by hand each time.
This same multiply-the-factors approach extends cleanly to three or more sequential changes, which addition or subtraction cannot handle correctly at all: a 10% rise, then a 10% rise again, then a 10% fall is 1.10 × 1.10 × 0.90 = 1.089, an 8.9% net increase — not the 10% a naive running tally might suggest, and not zero, which a rise-rise-fall pattern might carelessly be assumed to net out to.
How to use this
- Never assume a rise and a later fall of the same percentage cancel out — check which number each percentage was actually measured against.
- To find the fall that exactly reverses a rise, remember it will always be a smaller percentage than the rise was, because it is measured from the larger, already-risen number.
- When comparing two states, describe both as percentages of the same reference point if you want the arithmetic to actually add up the way it looks like it should.
