Problem:
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every seconds, and Robert runs clockwise and completes a lap every seconds. Both start from the start line at the same time. At some random time between minutes and 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 ., Rachel will have completed laps and be seconds from the finish line. Because Rachel runs one-fourth of a lap in seconds, she will be in the picture taking region between
seconds of the th minute. After minutes Robert will have completed laps and will be seconds from the starting line. Because Robert runs one-fourth of a lap in seconds, he will be in the picture taking region between and seconds of the th minute. Hence both Rachel and Robert will be in the picture if it is taken between and seconds of the 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