AMC 8 2023 Problem 13

AMC8 2023 年第 13 题

比与比例★★★☆☆

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?

如下图所示，在自行车比赛路线的起点和终点之间均匀分布着7个饮用水补给站. 另外，在起点和终点之间还均匀分布着2个维修站.第三个饮用水补给站在第一个维修站后2英里处：问比赛的全程是多少英里？

A.8
B.16
C.24
D.48
E.96

正确答案 · Correct Answer

解析 · Solution

Solution 1

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

\displaystyle \frac{3d}{8} = \frac{d}{3} +2,

from which $$$$

\displaystyle \begin{aligned} \frac{9d}{24} &= \frac{8d}{24} +2 \\ \frac{d}{24} &=2 \\ d&=48. \end{aligned}

$$$$

, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM Solution😆 S___ 💧___ 💧__*_ 💧___ 💧___ 💧_*__ 💧___ 💧___E Step 1: 7 💧 —> 8 sections; 2 repair—> 3 sections; ( Devisable by 24) Step 2, by 1st repair is 2 miles away to 3rd 💧; 3rd 💧 Is

\displaystyle \frac{3}{8}

to starting point; 1st repair Is

\displaystyle \frac{1}{3}

to starting point; Step 3: calculate!

\displaystyle 2/(\frac{3}{8}-\frac{1}{3}) =48

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