# Math Help - arithmetic (race problem)

1. ## 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 ?

2. First assign variables to the different speeds of each racer:

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

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

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

Sum of equation 1, 2, and 3 is

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

Sum of equation 4 and 5 is

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

Subtracting equation 7 from 6 gives

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

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:
$At_1 = d$ (From eq1)

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

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