Question

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $90$ seconds, and Robert runs clockwise and completes a lap every $80$ seconds. Both start from the start line at the same time. At some random time between $10$ minutes and $11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?

$\dfrac{1}{16}$

$\dfrac18$

$\dfrac{3}{16}$

$\dfrac14$

$\dfrac{5}{16}$

Solution:

The picture spans $\dfrac18$ lap on each side of the start. After $600$ seconds Rachel is $30$ seconds short of the line; running $\dfrac14$ lap in $22.5$ seconds, she is in view between $30-11.25=18.75$ and $30+11.25=41.25$ seconds of the $10$ th minute.

After $600$ seconds Robert is $40$ seconds from the line; running $\dfrac14$ lap in $20$ seconds, he is in view between $30$ and $50$ seconds.

Both appear between $30$ and $41.25$ seconds, a window of length $11.25$ out of $60,$ so the probability is $\dfrac{11.25}{60}=\dfrac{3}{16}.$

Thus, the correct answer is C.

2009 AMC 10B Problem 23

Solution:

Problem 23 in Other Years