AMC 12 2009b Problem 18

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?

\textbf{(A)}\ \frac {1}{16} \textbf{(B)}\ \frac18 \textbf{(C)}\ \frac {3}{16} \textbf{(D)}\ \frac14 \textbf{(E)}\ \frac {5}{16}