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May 4, 2020

Ratio and Inequalities | AIME I, 1992 | Question 3

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Ratio and Inequalities.

Ratio and Inequalities - AIME I, 1992


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.

  • is 107
  • is 164
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Ratios

Inequalities

Check the Answer


Answer: is 164.

AIME I, 1992, Question 3

Elementary Algebra by Hall and Knight

Try with Hints


First hint

Let x be number of matches she has played and won then \(\frac{x}{2x}=\frac{1}{2}\)

Second Hint

and \(\frac{x+3}{2x+4}>\frac{503}{1000}\)

Final Step

\(\Rightarrow 1000x+3000 > 1006x+2012\)

\(\Rightarrow x<\frac{988}{6}\)

\(\Rightarrow\) x=164.

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