Successive percentages
SAT glossary · Math
Definition
Percentage changes multiply, they never add: +20% then +30% is 1.20 × 1.30 = 1.56, a 56% increase, not 50%. A rise of p% followed by a fall of p% gives 1-p^2 < 1 — equal-and-opposite changes always lose, because the second percentage applies to a bigger base. Order does not matter, since multiplication commutes; the loss comes from the asymmetry of the bases.
Successive percentages in plain English
When you apply two or more percentage changes back to back, you do NOT just add the percentages together — you MULTIPLY the growth factors instead. A 20% increase followed by a 30% increase is 1.20×1.30=1.56, meaning a total 56% increase, not the 50% you'd get by naively adding 20+30. And here's a sneaky related fact: if something rises by p% and then falls by that exact SAME p%, you end up with a NET LOSS overall — 1-p^2 is always less than 1 — because the second percentage change applies to a DIFFERENT (and different-sized) starting amount than the first one did. Interestingly, the ORDER doesn't matter (rising then falling gives the same final result as falling then rising), since multiplication can be done in any order — the loss comes purely from the fact that percentages of different-sized bases aren't symmetric.
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