SOLUTION: A man rowing upstream drops his hat at point M. Thirty minutes later at N, he notices its loss and rows back. He picks up his hat at P, 440 yards below M. What is the rate of the

First we think of all motion as being relative to the river, as if the river
were not flowing at all and as if the land were moving beside still water
instead. Then relative to the river, 

1.  the hat does not move at all. That is, relative to the river M and P are the
SAME point!! 

2.  The man goes away from his hat and back to his hat at the SAME speed!!

Now, when we think relative to the river being still, all that has happened is
that the man dropped his hat, went 30 minutes away from his hat, and then 
returned 30 minutes back to his hat, a total of 60 minutes. So he picked up his
hat 60 minutes or 1 hour after he dropped it.  So that tells us that he was gone
1 hour.

So the 1 hour he was away from his hat is the answer for the total elapsed time. 

[Notice that how fast he went away from his hat and returned to his hat does not
matter, other than the fact that his speed was constant going and coming,
relative to the river.  So there is no way to find N, for that would depend on
his speed, which doesn't matter.]

Now let's go back to the standard way of taking motion relative to the land.
During the 60 minutes he was away from his hat, his hat moved 440 yards. So the
rate of the stream is the speed at which his hat traveled from M to P, which is
440 yards/hour.

That's the answer.  The only thing left to do is change that to feet/minute and
miles/hour.

440 yd/hour = 440 (3 ft)/(60 minutes) = (440*3/60) ft/minute = 22 ft/minute

Since 88 ft/minute = 60 miles/hour, and 22 is 1/4 of 88, the speed of the
current is 1/4 of 60 or 15 miles/hour.

Edwin