Problem:
Gary purchased a large beverage, but drank only of this beverage, where and are
relatively prime positive integers. If Gary had purchased only half as much and drunk twice as
much, he would have wasted only as much beverage. Find .
Solution:
Suppose Gary purchased amount and drank amount . Then the amount going to waste was . If Gary had purchased and drunk , he would have wasted . Thus , which simplifies to , implying that . The requested value is .
The problems on this page are the property of the MAA's American Mathematics Competitions