Problem 10 · 2000 AMC 8
Medium
Fractions, Decimals & Percents
percent-multiplier
Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?
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Answer: E — 55 inches.
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Hint 1 of 2
The 60 inches is Shea's height AFTER a 20% growth — so 60 is 120% of the shared start, not the start itself. Undo the growth to find where they both began.
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Hint 2 of 2
Watch the units switch: Ara grew 'half as many *inches*,' not half the percent. Once you know Shea's inches gained, halve THAT number.
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Approach: undo the percent to find the start, then count inches
- 60 is 120% of the common starting height, so start = 60 ÷ 1.2 = 50 inches. That means Shea gained 60 − 50 = 10 inches.
- Ara grew half as many *inches*: 10 ÷ 2 = 5 inches. Ara is now 50 + 5 = 55 inches.
- The habit to build: the two people start equal but the problem mixes a percent (Shea) with raw inches (Ara). Always convert the percent into actual inches before comparing — reasoning in percents would have you compare 20% of one height with an inch count, which aren't the same kind of thing.
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