Question

Gary 买了一大杯饮料，但只喝了其中的 $\frac{m}{n}$ ，其中 $m$ 和 $n$ 是互质的正整数。如果 Gary 只买一半那么多、并且喝掉两倍那么多，他浪费的饮料量就只有原来的 $\frac{2}{9}$ 。求 $m + n$ 。

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

答案

答案：37

知识点：一次方程比与比例

难度评级：1710

解答：

设 Gary 买了 $x$ 的量，喝了 $y$ 的量，于是浪费 $x - y$ 。在第二种情况中，他会买 $\frac{x}{2}$ 的量并喝掉 $2y$ ，于是浪费 $\frac{x}{2} - 2y$ 。条件为 $\frac{x}{2} - 2y = \frac{2}{9}(x - y).$

两边乘以 $18$ 得 $9x - 36y = 4x - 4y$ ，所以 $5x = 32y$ ，从而 $\frac{y}{x} = \frac{5}{32}$ 。因为 $\gcd(5, 32) = 1$ ，答案是 $5 + 32 = 37$ 。

Say Gary purchased an amount $x$ and drank an amount $y,$ wasting $x - y.$ In the second scenario he would have purchased $\frac{x}{2}$ and drunk $2y,$ wasting $\frac{x}{2} - 2y.$ The condition is $\frac{x}{2} - 2y = \frac{2}{9}(x - y).$

Multiplying by $18$ gives $9x - 36y = 4x - 4y,$ so $5x = 32y$ and $\frac{y}{x} = \frac{5}{32}.$ Since $\gcd(5, 32) = 1,$ the answer is $5 + 32 = 37.$

2011 AIME II 第 1 题

答案：37

难度评级：1710

解答：

其他年份的第 1 题