AMC8 2023

AMC8 2023 · Q13

AMC8 2023 · Q13. It mainly tests Linear equations, Ratios & proportions.

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?

在自行车赛的路线上，起点和终点之间有$7$个均匀分布的水站，如图所示。起点和终点之间还有$2$个均匀分布的维修站。第$3$个水站位于第$1$个维修站后$2$英里处。比赛总长多少英里？

(A) \ 8 \ 8

(B) \ 16 \ 16

(D) \ 48 \ 48

(E) \ 96 \ 96

Answer

Correct choice: (D)

正确答案：（D）

Solution

Suppose that it is $d$ miles long. The bathroom is located at \[\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}\] miles from the start, and the water fountains 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*}

假设总长为$d$英里。水站位于$\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}$英里处，维修站位于$\frac{d}{9}, \frac{2d}{9}$英里处。已知$\frac{3d}{8}=\frac{d}{9}+2$，由此 \begin{align*} \frac{27d}{72}&=\frac{8d}{72}+2 \\ \frac{19d}{72}&=2 \\ d&=\boxed{\textbf{(D)}\ 72}. \end{align*}

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