The exact batting average

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At some point during a baseball season, a player has a batting average of less than 80%.  Later during the season, his average exceeds 80%.  Prove that at some point, his batting average was exactly 80%.

Also, for which numbers other than 80% does this property hold?

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    Let us assume that the player’s batting average at some point during the season was less than 80%. Let x be the number of hits the player got in the first part of the season, and y be the number of at-bats he had. Then the player’s batting average in the first part of the season is x/y, which is less than 80%.

    Now, let z be the number of hits the player got in the later part of the season, and w be the number of at-bats he had in this period. Then the player’s batting average in the later part of the season is z/w, which is greater than 80%.
    Let us assume that the player’s overall batting average at some point in the season was never exactly 80%. Then, the player’s overall batting average at any point in the season is the total number of hits divided by the total number of at-bats. This can be expressed as:

    Total batting average = (x + z)/(y + w)

    Since we assumed that the player’s overall batting average was never exactly 80%, we have two cases:

    Case 1: The total batting average is always less than 80%

    In this case, we have: (x + z)/(y + w) < 0.8

    Multiplying both sides by (y + w), we get:  x + z < 0.8(y + w)
    But we know that: x/y < 0.8

    Multiplying both sides by y, we get: x < 0.8y
    Similarly, we know that: z/w > 0.8

    Multiplying both sides by w, we get:  z > 0.8w
    Adding these two inequalities, we get:  x + z < 0.8y + 0.8w

    But this contradicts the earlier inequality that we derived, which was: x + z < 0.8(y + w)
    Therefore, case 1 cannot hold, and the total batting average must have been equal to 80% at some point in the season.

    Case 2: The total batting average is always greater than 80%

    In this case, we have: (x + z)/(y + w) > 0.8

    Multiplying both sides by (y + w), we get: x + z > 0.8(y + w)
    But we know that: x/y < 0.8

    Multiplying both sides by y, we get: x < 0.8y
    Similarly, we know that:  z/w > 0.8

    Multiplying both sides by w, we get:  z > 0.8w
    Adding these two inequalities, we get:  x + z > 0.8y + 0.8w

    This does not contradict the earlier inequality that we derived, so case 2 can hold. In this case, there is no guarantee that the player’s batting average was ever exactly equal to any other number.

    Therefore, we can conclude that the property holds only for 80%.

    Moshe Expert Answered on 26th March 2023.
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