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Thread: arithmetic (race problem)

  1. #1
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    arithmetic (race problem)

    In a race over d yards, A would beat B by 20 yards; B would beat C by 10 yards; and A would beat C by 28 yards. What is the value of d ?
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  2. #2
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    First assign variables to the different speeds of each racer:

    Let:
    Speed of A = $\displaystyle A $
    Speed of B = $\displaystyle B $
    Speed of C = $\displaystyle C $

    Let the time it takes A to run a distance of d be $\displaystyle t_1 $. Using information from your question:
    1. $\displaystyle At_1 = d$
    2. $\displaystyle Bt_1 = d-20 $
    3. $\displaystyle Ct_1 = d-28 $

    There is also a race between B and C. Let the time it takes B to run the distance d be $\displaystyle t_2$. Using information from your question:
    4. $\displaystyle Bt_2 = d$
    5. $\displaystyle Ct_2 = d-10$

    Sum of equation 1, 2, and 3 is

    6. $\displaystyle t_1(A+B+C) = 3d-48 $

    Sum of equation 4 and 5 is

    7. $\displaystyle t_2(B+C) = 2d-10 $

    Subtracting equation 7 from 6 gives

    8. $\displaystyle At_1 + (t_1-t_2)(B+C) = d-38 $

    You can solve for d from this equation. Aim to express every unknown variable (A, B, C, t1, and t2) in terms of d. Below are some hints to help you along if you get stuck:

    Spoiler:
    $\displaystyle At_1 = d$ (From eq1)

    Spoiler:
    $\displaystyle t_1 - t_2 = \frac{-20}{B} $ (From eq2 - eq4)

    Spoiler:
    $\displaystyle C = \frac{B(d-10)}{d}$ (From eq5/eq4)
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