2009 AMC 10A Problems/Problem 20
Andrea and Lauren are 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 kilometer per minute. After 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?
Let their speeds in kilometers per hour be and . We know that and that . (The second equation follows from the fact that .) This solves to and .
As the distance decreases at a rate of kilometer per minute, after minutes the distance between them will be kilometers.
From this point on, only Lauren will be riding her bike. As there are kilometers remaining and , she will need exactly an hour to get to Andrea. Therefore the total time in minutes is .
Because the speed of Andrea is 3 times as fast as Lauren and the distance between them is decreasing at a rate of 1 kilometer per minute, Andrea's speed is , and Lauren's . Therefore, after 5 minutes, Andrea will have biked .
In all, Lauren will have to bike . Because her speed is , the time elapsed will be
Since the distance between them decreases at a rate of kilometer per minute when they are both biking, their combined speed is kilometer per minute. Andrea travels three times as fast as Lauren, so they travel at speeds of kilometers per minute and kilometers per minute, respectively.
After minutes, the distance between them will have decreased by kilometers, so they will be kilometers apart when Andrea stops. Then, Lauren will take more minutes to reach Andrea.
They started to bike minutes before Andrea stopped, so the total time Lauren passed from the time they started biking to the time Lauren reached Andrea is minutes. Hence, the answer is . ~azc1027
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