Difference between revisions of "2011 AIME II Problems/Problem 1"
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== Problem == | == Problem == | ||
Gary purchased a large beverage, but only drank <math>m/n</math> of it, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only <math>2/9</math> as much beverage. Find <math>m+n</math>. | Gary purchased a large beverage, but only drank <math>m/n</math> of it, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only <math>2/9</math> as much beverage. Find <math>m+n</math>. |
Revision as of 14:44, 9 August 2018
Contents
[hide]Problem
Problem
Gary purchased a large beverage, but only drank of it, where and are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only as much beverage. Find .
Solution
Let be the fraction consumed, then is the fraction wasted. We have , or , or or . Therefore,
See also
2011 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.