Problem:
Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the -mile mark at exactly the same time. How many minutes has it taken them?
Solution:
Suppose Rudolph bikes at miles per minute. He takes five-minute breaks in reaching the -mile mark, so his total time in minutes is . Jennifer bikes at miles per minute and takes five-minute breaks in reaching the -mile mark, so her total time in minutes is . Setting these two times equal gives , and hence . This yields a total time of minutes.
The problems on this page are the property of the MAA's American Mathematics Competitions