2008 AIME II Problems/Problem 2
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?
Let be the time Rudolph takes disregarding breaks and be the time Jennifer takes disregarding breaks. We have the equation Thus, the total time they take is minutes.
Let Rudolf bike at a rate , so Jennifer bikes at the rate . Let the time both take be .
Then Rudolf stops times (because the rest after he reaches the finish does not count), losing a total of minutes, while Jennifer stops times, losing a total of minutes. The time Rudolf and Jennifer actually take biking is then respectively.
Using the formula , since both Jennifer and Rudolf bike miles,
Substituting equation into equation and simplifying, we find
Let the total time that Jennifer and Rudolph bike, including rests, to be minutes. Furthermore, let Rudolph's biking rate be so Jennifer's biking rate is . Note that Rudolf takes 49 breaks, taking minutes, and Jennifer takes 24 breaks, taking minutes. Since they both reach the 50 mile mark, then by , the rate times time taken for Rudolph and Jennifer must equal. Hence, we disregard the breaks from the total time taken and get the equation yielding .
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