1992 AIME Problems/Problem 3: Difference between revisions
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== Problem == | == Problem == | ||
A tennis player computes her win [[ratio]] by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math> | A tennis player computes her win [[ratio]] by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>.500</math>. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than <math>.503</math>. What's the largest number of matches she could've won before the weekend began? | ||
== Solution == | == Solution == | ||
Let <math> | Let <math>n</math> be the number of matches won, so that <math>\frac{n}{2n}=\frac{1}{2}</math>, and <math>\frac{n+3}{2n+4}>\frac{503}{1000}</math>. Cross [[multiply]]ing, <math>1000n+3000>1006n+2012</math>, and <math>n<\frac{988}{6}</math>. Thus, the answer is <math>164</math>. | ||
{{AIME box|year=1992|num-b=1|num-a=3}} | |||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
Revision as of 21:56, 11 November 2007
Problem
A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly 
. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than 
. What's the largest number of matches she could've won before the weekend began?
Solution
Let 
be the number of matches won, so that 
, and 
. Cross multiplying, 
, and 
. Thus, the answer is 
.
| 1992 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
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