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

Question

06-12-2022
Mathematics

Answered

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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

Sagot :

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Qwsun Qwsun · Answer 1 · 2022-12-11T04:52:42-05:00

The number of tennis matches that she could have won before the weekend began is 0.

Before the weekend, the player had a win ratio of 0.500, which indicates that she had won exactly as many games as she had lost. Call this number x for now. She has so lost x matches in addition to the x matches she won before to the weekend.

She competes in three matches during the course of the weekend and loses one, bringing her overall win total (after the weekend) to x + 3 and her overall loss total (after the weekend) to x + 1. Consequently, her weekend victory ratio is

[tex]\frac{(x + 3)}{(x + 1)}[/tex]

= [tex]\frac{(3x + 3)}{(x + 1) }[/tex]

= [tex]\frac{3x }{x + 1 }[/tex]

= [tex]\frac{3}{\frac{1}{x+1} }[/tex]

given that this value is greater than 0.503,

Thus,

[tex]\frac{3}{\frac{1}{x+1} }[/tex] > 0.503

Simplifying,

[tex]\frac{3}{\frac{1}{x+1} }[/tex]> 0.503 - 1

= -0.497

Further,

[tex]\frac{3}{\frac{1}{x+1} }[/tex] > -0.497

[tex]\frac{3}{\frac{1}{x+1} }[/tex] is equivalent to 3x

This inequality can be written as

3x > -0.497

3x is positive, dividing on both sides by 3 we get

x > -0.497 / 3

Simplifying,

x > -0.166

The least value of x that meets this inequality is 0, as x, the number of matches won, must be a positive integer. Thus, the maximum number of games the player might have won prior to the start of the weekend is x = 0, and the response is 0.

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