jmc

Question

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 .500. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than .503. What's the largest number of matches she could've won before the weekend began?

Accepted Answer

Let n be the number of matches she won before the weekend began. Since her win ratio started at exactly .500 = 12, she must have played exactly 2n games total before the weekend began. After the weekend, she would have won n+3 games out of 2n+4 total. Therefore, her win ratio would be (n+3)/(2n+4). This means that n+32n+4 > .503 = 5031000.Cross-multiplying, we get 1000(n+3) > 503(2n+4), which is equivalent to n < 9886 = 164.6. Since n must be an integer, the largest possible value for n is 164.