Problem:
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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
After 10 min. sec., Rachel will have completed 6 laps and\
be 30 seconds from the finish line. Because Rachel runs one-fourth of a lap in 22.5 seconds, she will be in the picture taking region between
seconds of the 10th minute. After 10 minutes Robert will have completed 7 laps and will be 40 seconds from the starting line. Because Robert runs one-fourth of a lap in 20 seconds, he will be in the picture taking region between 30 and 50 seconds of the 10 th minute. Hence both Rachel and Robert will be in the picture if it is taken between 30 and 41.25 seconds of the 10 th minute. The probability that the picture is snapped during this time is
The problems on this page are the property of the MAA's American Mathematics Competitions