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Math Help - Race problem

  1. #1
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    Race problem

    A can beat B by 20 yards in a race of 200 yards. B can beat C by 10 yards in a race of 250 yards. By how many yards can A beat C in a race of 100 yards?
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  2. #2
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    Re: Race problem

    Hello, hisajesh!

    A can beat B by 20 yards in a 200-yard race.
    B can beat C by 10 yards in a 250-yard race.
    By how many yards can A beat C in a 100-yard race?

    Let \begin{Bmatrix}a &=& \text{A's speed} \\ b &=& \text{B's speed} \\ c &=& \text{C's speed} \end{Bmatrix}

    In the time A has run 200 yards, B has run only 180 yards.
    B's speed is \tfrac{180}{200} = \tfrac{9}{10} of A's speed: . b = \tfrac{9}{10}a .[1]

    In the time B gas run 250 yards, C has run only 240 yards.
    C's speed is \tfrac{240}{250} = \tfrac{24}{25} of B's speed: . c = \tfrac{24}{25}b .[2]

    Substitute [1] into [2]: . c \:=\:\tfrac{24}{25}(\tfrac{9}{10}a) \quad\Rightarrow\quad c \:=\:\tfrac{108}{125}a
    . . That is, C's speed is \tfrac{108} {125} of A's speed.

    When A has run 100 yards, C has run only \tfrac{108} {125}(100) \:=\:86.4 yards.

    Therefore, A wins by 13.6 yards.
    Thanks from MarkFL and hisajesh
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  3. #3
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    Re: Race problem

    @Soroban: Thanks for the detailed explanation!
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