Problem 9 · 2008 AMC 8
Easy
Fractions, Decimals & Percents
successive-percentages
In 2005 Tycoon Tammy invested 100 dollars for two years. During the first year her investment suffered a 15% loss, but during the second year the remaining investment showed a 20% gain. Over the two-year period, what was the change in Tammy's investment?
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Answer: D — 2% gain.
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Hint 1 of 2
The 20% gain is on the shrunken $85, not the original $100 — so −15% then +20% does NOT cancel to +5%.
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Hint 2 of 2
Turn each change into a multiplier (lose 15% = ×0.85, gain 20% = ×1.20) and multiply them in a row.
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Approach: chain the percentage multipliers
- Each year scales the money: a 15% loss is ×0.85, a 20% gain is ×1.20. Doing them in sequence means multiplying: 0.85 × 1.20 = 1.02.
- 1.02 means the money ended at 102% of the start — a 2% gain. The starting $100 never even matters.
- Why this transfers: percent changes compound (multiply), they never add — that's why the +20% can't undo the −15% to give +5%.
Another way — track the actual dollars:
- After year 1: 100 − 15% = $85. After year 2: 85 + 20% of 85 = 85 + 17 = $102.
- From $100 to $102 is a 2% gain.
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