AIME II #2 2008

[stupidityismygam]

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?

[Michael T]

<begin answer to really old post>

Rudolph runs at speed [unparseable or potentially dangerous latex formula] takes [unparseable or potentially dangerous latex formula] breaks (for a total of [unparseable or potentially dangerous latex formula] of breaktime), and the time he spends actually running the [unparseable or potentially dangerous latex formula] miles is represented by [unparseable or potentially dangerous latex formula] (I've forgotten latex, sorry).

Jennifer runs at speed [unparseable or potentially dangerous latex formula] takes [unparseable or potentially dangerous latex formula] breaks (for a total of [unparseable or potentially dangerous latex formula] of breaktime), and the time she spends actually running the [unparseable or potentially dangerous latex formula] miles is represented by [unparseable or potentially dangerous latex formula]

Since the total time running + breaktime must be equal for them to finish at the same time, we have:

[unparseable or potentially dangerous latex formula]
which gives us [unparseable or potentially dangerous latex formula]

Therefore Rudolph spends [unparseable or potentially dangerous latex formula] minutes running, plus [unparseable or potentially dangerous latex formula] minutes resting = [unparseable or potentially dangerous latex formula] minutes running the 50 miles.

[unparseable or potentially dangerous latex formula]

</begin answer to really old post>

[stupidityismygam]

I'll edit your post and put some fancy Latex in it