Answer: 620

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 $$\frac{50}{r} + 245 = \frac{200}{3r} + 120,$$ so $$\begin{aligned} 125 &= \frac{200 - 150}{3r} \\ &= \frac{50}{3r}, \end{aligned}$$ and $r = \frac{2}{15}.$ The common time is $\frac{50}{2/15} + 245 = 375 + 245 = 620$ minutes.