Bob is having a nice camping/fishing trip along a river. He leaves his campsite early in the morning, and gets on his boat, heading full throttle upstream.

After going for exactly one mile, his hat flips off of his head, and starts floating downstream. Bob doesn't realize that his hat has fallen off for five minutes, but then he notices that it's missing, and turns full throttle downstream.

He finally catches the hat at exactly the same spot as he camped that morning. The question is, how fast was the water traveling?

(Assume that he travels the same speed the entire time and that there is no turn around time.)

First define some variables:

b=speed of Bob's boat in still water (miles/minute)

r=speed of river (miles/minute)

t=time Bob spent traveling downstream to get his hat back (minutes)

The distance Bob traveled upstream equals 5*(b-r)

The distance Bob traveled downstream equals t*(b+r)

The distance the hat traveled equals 1 or (t+5)*r

Two equations can be made: 5*(b-r) + 1 = t*(b+r) and (t+5)*r=1

Solving for these equals t = 5 minutes and river speed r = 0.1 miles/minute. The speed of Bob's boat can be any speed.