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 $0.500.$ During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $0.503.$ What’s the largest number of matches she could’ve won before the weekend began?

Answer

Answer: 164

Concepts:ratio and proportion inequality percentage

Difficulty rating: 1640

Small Hint:

If she had $w$ wins initially, a $0.500$ ratio means she had played $2w$ matches

Big Hint:

Translate the final ratio into a strict inequality before taking the largest integer $w$

Solution:

If she initially had $w$ wins, then she had played $2w$ matches. The final condition is $\frac{w+3}{2w+4}\gt\frac{503}{1000}.$ Cross-multiplication gives $1000w+3000\gt1006w+2012,$ so $6w\lt988$ and $w\lt164\frac23.$ The largest possible integer is $164.$

1992 AIME Problem 3

Answer: 164

Difficulty rating: 1640

Small Hint:

Big Hint:

Solution:

Problem 3 in Other Years