AIME 2011 II · 第 1 题

AIME 2011 II — Problem 1

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

Gary purchased a large beverage, but only drank $\frac{m}n$ 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 $\frac{2}{9}$ as much beverage. Find $m+n$ .

解析

Solution

Let $x$ be the fraction consumed, then $(1-x)$ is the fraction wasted. We have $\frac{1}{2} - 2x =\frac{2}{9} (1-x)$ , or $9 - 36x = 4 - 4x$ , or $32x = 5$ or $x = \frac{5}{32}$ . Therefore, $m + n = 5 + 32 = \boxed{037}$ .

Solution 2

$AIME diagram$

Thus, the answer is $\boxed{37}$ ~idk123456