2023 AMC 8

Complete problem set with solutions and individual problem pages

Problem 13 Medium

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?

  • A.

    8

  • B.

    16

  • C.

    24

  • D.

    48

  • E.

    96

Answer:D

Suppose that the race is d miles long. The water stations are located at

\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}

miles from the start, and the repair stations are located at

\frac{d}{3}, \frac{2d}{3}

miles from the start.

We are given that \frac{3d}{8}=\frac{d}{3}+2, from which

\begin{align*} \frac{9d}{24}&=\frac{8d}{24}+2 \\ \frac{d}{24}&=2 \\ d&=\boxed{\textbf{(D)}\ 48}. \end{align*}

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