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 .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 is the largest number of matches that she could have won before the weekend began?
Solution:
Let
W= the player’s number of wins at the start of the weekend
and
M=the number of matches played at the start of the weekend.
We are given MW​=.500 and (M+4)(W+3)​>.503. Thus M=2W and
W+3>.503(2W+4)=1.006W+2.012
It follows that W<(3−2.012)/.006=164.6. Finally, note that if W=164 and M=328, then W/M=.500 and (W+3)/(M+4)>.503. Hence, the largest number of matches that the player could have won before the start of the weekend is 164​.
The problems on this page are the property of the MAA's American Mathematics Competitions