Question

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 $50$ -mile mark at exactly the same time. How many minutes has it taken them?

Answer

Solution:

Let Rudolph bike at $r$ miles per minute. He rests after each of miles $1$ through $49,$ so his total time is $\frac{50}{r} + 49 \cdot 5 = \frac{50}{r} + 245$ minutes. Jennifer bikes at $\frac{3r}{4}$ miles per minute and rests after each of miles $2, 4, \ldots, 48,$ so her total time is $\frac{50}{3r/4} + 24 \cdot 5 = \frac{200}{3r} + 120$ minutes.

Setting the times equal gives $\frac{50}{r} + 245 = \frac{200}{3r} + 120, \qquad \text{so} \qquad 125 = \frac{200 - 150}{3r} = \frac{50}{3r},$ and $r = \frac{2}{15}.$ The common time is $\frac{50}{2/15} + 245 = 375 + 245 = 620$ minutes.

2008 AIME II Problem 2

Solution:

Problem 2 in Other Years