2023 AMC 8 Problems/Problem 13

Problem

Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also 2 repair stations evenly spaced between the start and finish lines. The 3rd water station is located 2 miles after the 1st repair station. How long is the race in miles?

$\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96$

Solution

Knowing that there are $7$ equally spaced water stations they are each located $\frac{d}{8}$ , $\frac{2d}{8}$ ,… $\frac{7d}{8}$ of the way from the start. Using the same logic for the $3$ station we have $\frac{d}{3}$ and $\frac{2d}{3}$ for the repair stations. It is given that the 3rd water is $2$ miles ahead of the $1$ st repair station. So setting an equation we have $\frac{3d}{8} = \frac{d}{3} + 2$ getting common denominators $\frac{9d}{24} = \frac{8d}{24} + 2$ so then we have $d = \boxed{\text{(D)}48}$ from this.