Question

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 $m + n.$

Answer

Answer: 37

Concepts:linear equation ratio and proportion

Difficulty rating: 1710

Small Hint:

Let $x$ be the amount purchased and $y$ the amount drunk; the wasted amount is $x - y$

Big Hint:

The second scenario wastes $\frac{x}{2} - 2y,$ so set $\frac{x}{2} - 2y = \frac{2}{9}(x - y)$ and solve for $\frac{y}{x}$

Solution:

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 Problem 1

Answer: 37

Difficulty rating: 1710

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years