1992 AIME Problem 3

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.