AMC 8 2022 Problem 22
AMC8 2022 年第 22 题
行程问题★★★☆☆
A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?
一辆公共汽车从一个站开到下一个站需要 2 分钟，在每个站乘客上下车需要等待 1 分钟。Zia 从一个公共汽车站走到下一个站需要 5 分钟。当 Zia 到达一个公共汽车站时，如果公共汽车是在上一站或已经离开上一站，那么她会等待公共汽车。否则，她将走向下一站。假设公共汽车和 Zia 同时出发前往图书馆，公共汽车落后 3 站。问多少分钟以后，Zia 将登上公共汽车？
- A.17
- B.19
- C.20
- D.21
- E.23
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Initially, suppose that the bus is at Stop 0 (starting point) and Zia is at Stop 3. We construct the following table of 5 -minute intervals:
$\begin{array}{c||c|c} \textbf{Time} & \textbf{Bus's Location} & \textbf{Zia's Location} \\ \hline \boldsymbol{5} \ \textbf{Minutes} & \text{Stop} 2 \ \text{(Waiting)} & \text{Stop} 4 \\ \boldsymbol{10} \ \textbf{Minutes} & \text{Midpoint of Stops} 3 \ \text{and} 4 & \text{Stop} 5 \\ \boldsymbol{15} \ \textbf{Minutes} & \text{Stop} 5 \ \text{(Leaving)} & \text{Stop} 6 \end{array}
$ Note that Zia will wait for the bus after 15 minutes, and the bus will arrive 2 minutes later. Therefore, the answer is $15+2=17.$Since Zia will wait for the bus if the bus is at the previous stop, we can create an equation to solve for when the bus is at the previous stop. The bus travels of a stop per minute, and Zia travels of a stop per minute. Now we create the equation, (the -1 accounts for us wanting to find when the bus reaches the stop before Zia's). Solving, we find that m=15. Now Zia has to wait 2 minutes for the bus to reach her, so our answer is 15+2=17.
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