Problem 8 · 2009 AMC 8
Easy
Fractions, Decimals & Percents
percent-multiplier
The length of a rectangle is increased by 10% and the width is decreased by 10%. What percent of the old area is the new area?
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Answer: B — 99%.
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Hint 1 of 2
Tempting trap: "+10% then −10% cancels to 0." It doesn't — the 10% you take off is computed from the BIGGER length, so the drop slightly outweighs the gain.
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Hint 2 of 2
Turn each percent change into a multiplier (up 10% → ×1.1, down 10% → ×0.9) and just multiply them. Area = length × width, so the area's multiplier is the product.
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Approach: multiply the multipliers
- New area ÷ old area = 1.1 × 0.9 = 0.99 = 99%.
- Intuition for the 1% loss: 1.1 × 0.9 = (1 + 0.1)(1 − 0.1) = 1 − 0.1² = 1 − 0.01. The leftover is the square of the percent — tiny, and always a LOSS.
- You'll see it again: stacked percent changes always multiply, never add. "Raise then drop by the same %" always ends below where you started, by exactly (that %)².
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