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 $3$ rd water station is located $2$ miles after the $1$ st 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

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 $\begin{align*} \frac{9d}{24}&=\frac{8d}{24}+2 \\ \frac{d}{24}&=2 \\ d&=\boxed{\textbf{(D)}\ 48}. \end{align*}$ ~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM

Video Solution (Animated)

https://youtu.be/NivfOThj1No

~Star League (https://starleague.us)

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=4439

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2023 AMC 8 Problems/Problem 13

Contents

Problem

Solution

Video Solution (Animated)

Video Solution by Magic Square

See Also