1992 AIME Problems/Problem 3
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 1
Let be the number of matches won, so that
, and
.
Cross multiplying, , so
. Thus, the answer is
.
Solution 2
Let be the number of matches she won before the weekend began. Since her win ratio started at exactly .
she must have played exactly
games total before the weekend began. After the weekend, she would have won
games out of
total. Therefore, her win ratio would be
This means that
Cross-multiplying, we get
which is equivalent to
Since
must be an integer, the largest possible value for
is
1992 AIME (Problems • Answer Key • Resources) | ||
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Followed by Problem 4 | |
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