2011 AIME II Problems/Problem 1

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Problem 1

Gary purchased a large beverage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only 2/9 as much beverage. Find m+n.

Solution

Let $x$ be the fraction consumed, then $(1-x)$ is the fraction wasted. We have $1/2 - 2x = 2/9(1-x)$, or $9 - 36x = 4 - 4x$, or $32x = 5$ or $x = 5/32$. Therefore, $m + n = 5 + 32 = \fbox{37.}$

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