Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Ratio and Inequalities.
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. Find the largest number of matches she could have won before the weekend began.
Integers
Ratios
Inequalities
Answer: is 164.
AIME I, 1992, Question 3
Elementary Algebra by Hall and Knight
Let x be number of matches she has played and won then \(\frac{x}{2x}=\frac{1}{2}\)
and \(\frac{x+3}{2x+4}>\frac{503}{1000}\)
\(\Rightarrow 1000x+3000 > 1006x+2012\)
\(\Rightarrow x<\frac{988}{6}\)
\(\Rightarrow\) x=164.
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