Problem 16 · 1997 AJHSME
Hard
Fractions, Decimals & Percents
successive-percent
Penni buys $100 of stock in each of three companies: AA, BB, and CC. After one year AA is up 20%, BB is down 25%, and CC is unchanged. In the second year AA drops 20% from its new value, BB rises 25% from its new value, and CC is unchanged. If A, B, C are the final values, which ordering is correct?
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Answer: E — B < A < C.
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Hint 1 of 2
The big trap: a 20% gain then a 20% loss does NOT bring you back to $100 — the loss is taken on a bigger amount. Picture each percent change as MULTIPLYING by a factor, and notice the factors don't undo each other.
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Hint 2 of 2
Turn every percent change into a multiplier (up 20% → ×1.2, down 25% → ×0.75) and multiply them. ×1.2 then ×0.8 = ×0.96, a net loss.
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Approach: convert each change to a multiplier and multiply
- AA: ×1.2 then ×0.8 → overall ×0.96, so 100 → 96. The +20%/−20% pair lands BELOW the start because the 20% drop is off the higher $120.
- BB: ×0.75 then ×1.25 → overall ×0.9375, so 100 → 93.75. CC: unchanged at 100.
- Ordering the finals: 93.75 < 96 < 100, i.e. B < A < C.
- Why this transfers: percent changes never simply cancel — a +x% then −x% always leaves you at ×(1 − x²) of the start, a small loss. Multipliers, not addition, govern stacked percents.
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