AMC10 2009 A

AMC10 2009 A · Q20

AMC10 2009 A · Q20. It mainly tests Linear equations, Rates (speed).

Andrea and Lauren are 20 kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of 1 kilometer per minute. After 5 minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?

Andrea 和 Lauren 相距20千米。她们相向骑车，Andrea 的速度是 Lauren 的三倍，且它们之间的距离以每分钟1千米的速度减小。5分钟后，Andrea 因为爆胎停止骑车并等待 Lauren。从她们开始骑车起，经过多少分钟 Lauren 到达 Andrea 处？

(A) 20 20

(B) 30 30

(D) 65 65

(E) 80 80

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Let \(r\) be the rate that Lauren bikes, in kilometers per minute. Then \(r + 3r = 1\), so \(r = \frac{1}{4}\). In the first 5 minutes, the distance between Andrea and Lauren decreases by \(5 \cdot 1 = 5\) kilometers, leaving Lauren to travel the remaining 15 kilometers between them. This requires \[ \frac{15}{\frac{1}{4}} = 60 \] minutes, so the total time since they started biking is \(5 + 60 = 65\) minutes.

答案（D）：设 \(r\) 为 Lauren 骑行的速度（单位：千米/分钟）。则 \(r + 3r = 1\)，所以 \(r = \frac{1}{4}\)。在前 5 分钟内，Andrea 和 Lauren 之间的距离减少了 \(5 \cdot 1 = 5\) 千米，于是 Lauren 还需要骑完他们之间剩余的 15 千米。这需要 \[ \frac{15}{\frac{1}{4}} = 60 \] 分钟，因此从开始骑行到现在的总时间为 \(5 + 60 = 65\) 分钟。

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