Successive percentage changes never simply add, and this trap does more damage in real casing than any other. If a value rises by a percent and then by b percent, the naive guess is a total change of a + b percent. That guess is wrong. Multiply the two growth factors together and you get the exact relationship:
(1 + a)(1 + b) = 1 + a + b + ab
The naive guess drops ab, the product of the two rates. Written in percentage-point units, so you never have to convert to decimals halfway through, the exact rule is:
Combined % change = a + b + (a × b) ÷ 100
Worked example: two successive 10% rises
Try it on the cleanest case you can, two successive 10% rises. The naive guess is 10 + 10 = 20%. The correction is (10 × 10) ÷ 100 = 1. So the exact answer is 21%, and sure enough 1.10 × 1.10 = 1.21. That missing point is part of the exact answer and owes nothing to rounding. It's how "grew 10% two years running" compounds to 21% instead of 20%. The correction grows with the size of the two rates. When both are small single digits it's under a point, and often safe to ignore on a fast first pass. Once either rate clears roughly 10%, name the correction out loud, because by then it's more than noise.
The formula holds whatever the signs are, so it's worth memorizing instead of re-deriving live. Revenue up 30%, then down 10%: the naive guess is 30 − 10 = 20%, the correction is (30 × −10) ÷ 100 = −3, and the exact answer is 17% (check: 1.30 × 0.90 = 1.17). Now a price down 20%, then up 25%. The naive guess is −20 + 25 = 5% and the correction is (−20 × 25) ÷ 100 = −5, so the exact answer is 0%. The two moves cancel completely (0.80 × 1.25 = 1.00), and that's no accident. It leads straight into the next trap.
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What happens after an equal rise and fall?
The equal-and-opposite trap. Set b = −a in the identity above and something sharper falls out: (1 + a)(1 − a) = 1 − a². Go up a% and then down the same a%, in either order, and you end with a loss of a². You never land back on zero. A few anchors are worth holding in memory:
| Equal-and-opposite moves | Net change |
|---|---|
| ±10% | −1% |
| ±15% | −2.25% |
| ±20% | −4% |
| ±25% | −6.25% |
| ±50% | −25% |
Check the middle row as a story. A price rises 20% and later falls 20%. Intuition says you're back where you started. In fact 1.20 × 0.80 = 0.96, so the price ends four percent below the original. The loss grows with the square of the move. At ±5% it's a quarter of a point and feels negligible. At ±50% a full quarter of the value is gone, and nobody can ignore that. Keep this in mind the next time someone describes a volatile price as having "gone nowhere" after a sharp rise and an equally sharp fall.
How big a gain do you need to undo a fall?
The reciprocal recovery trap. This close cousin is arguably the most quoted "gotcha" in business writing. The percentage move needed to undo a decline is always larger than the decline, and the move needed to undo a gain is always smaller than the gain. To recover from a decline of d, you need a later gain of d ÷ (1 − d). To recover from a gain of g, you need a later decline of g ÷ (1 + g). Memorize the canonical case outright. A stock that falls 50% needs a 100% gain, not 50%, to get back to where it started (0.5 ÷ (1 − 0.5) = 1.00, and indeed 0.5 × 2.0 = 1.0). A milder version: a price rises 25% during a promotion, and the discount that fully unwinds it is 20%, not 25% (25 ÷ 125 = 0.20, matching the −20%/+25% cancellation you just checked). So "we lost 30% last year and gained 30% this year" isn't a wash, and this asymmetry is the arithmetic reason. Be ready to produce it on the spot whenever a client tells a decline-then-recovery story.