2009 AMC 10A Problems/Problem 20

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Problem

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?

$\mathrm{(A)}\ 20\qquad\mathrm{(B)}\ 30\qquad\mathrm{(C)}\ 55\qquad\mathrm{(D)}\ 65\qquad\mathrm{(E)}\ 80$

Solution

Let their speeds in kilometers per hour be $v_A$ and $v_L$ . We know that $v_A=3v_L$ and that $v_A+v_L=60$ . (The second equation follows from the fact that $1\,\unit{\rm km/min} = 60\,\unit{\rm km/h}$ .) This solves to $v_A=45$ and $v_L=15$ .

As the distance decreases at a rate of $1$ kilometer per minute, after $5$ minutes the distance between them will be $20-5=15$ kilometers.

From this point on, only Lauren will be riding her bike. As there are $15$ kilometers remaining and $v_L=15$ , she will need exactly an hour to get to Andrea. Therefore the total time in minutes is $5+60 = \boxed{65}$ .

2009 AMC 10A (Problems • Resources)
Preceded by Problem 19	Followed by Problem 21
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2009 AMC 10A Problems/Problem 20

From AoPSWiki

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Solution

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