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?

$\dfrac{1}{16}$

$\dfrac{1}{8}$

$\dfrac{3}{16}$

$\dfrac{1}{4}$

$\dfrac{5}{16}$

Solution:

The picture covers the arc within $\dfrac{1}{8}$ lap of the start on each side. At $600$ s Rachel has run $6\tfrac{2}{3}$ laps, $30$ s short of the line; a quarter lap takes her $22.5$ s, so she is in view between $30 - 11.25 = 18.75$ s and $30 + 11.25 = 41.25$ s of the $10$ th minute.

At $600$ s Robert is $40$ s from the line; a quarter lap takes $20$ s, so he is in view between $30$ and $50$ s. Both appear between $30$ and $41.25$ s, a window of $11.25$ s out of $60,$ giving probability $\dfrac{11.25}{60} = \dfrac{3}{16}.$

Thus, the correct answer is C.

2009 AMC 12B Problem 18

Solution:

Problem 18 in Other Years