Two miles upstream from his starting point, a canoeist passed a log
floating in the river's current.  After paddling upstream for on more
hour, he paddled back and reached his starting point just as the log
arrived.  Find the speed of the current.

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Clarification of Question by sedaniellexy-ga on 14 Oct 2005 17:15 PDT

use distance=rate*time

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Clarification of Question by sedaniellexy-ga on 14 Oct 2005 19:10 PDT

sorry..spelling error...

"After paddling upstream for ONE more hour..."

Hello.

Let's say that:

 T = Total time it takes from when the canoeist first sees the log
until they both reach the canoeist's original starting point.

 C = Rate of the current in miles per hour
 R = Rate that the canoeist paddles in miles per hour
 X = Distance in miles that the canoeist travels upstream in the one
hour after he sees the log

Distance=Rate*Time, so: 
2 = C * T because the log (floating at the rate of the current) travels 2 miles. 

If 2 = C * T, rearranging  T = 2/C

Now, T also is equal to the time it takes for the canoeist to travel
upstream distance X and then back downstream distance X and then the
final 2 miles.

The canoeist's upstream rate is R-C (because the current is working
against him). The canoeist's downstream rate is R+C (because he's
going with the flow).
Distance=Rate*Time and Time=Distance/Rate.

So:  X/(R-C) is the time it takes for the canoeist to travel upstream distance X;
X/(R+C) is the time it takes for the canoeist to go back downstream
distance X;  2/(R+C) is the time it takes for the canoeist to go
downstream the final 2 miles.

Thus: 

T = X/(R-C) + X/(R+C) + 2/(R+C)

Now, we're told that the time it takes for the canoeist to travel
upstream distance X is one hour, so X/(R-C) = 1.

 X/(R-C) = 1
Or: X = R-C

Now, getting back to:
T = X/(R-C) + X/(R+C) + 2/(R+C)

Substituting:
T = 1   +  (R-C)/(R+C) + 2/(R+C)
Or:
T = 1   +  (R-C+2)/(R+C)

Since 1=(R+C)/(R+C)...

T = (R+C)/(R+C) +  (R-C+2)/(R+C)

Or:
T = (R+C+R-C+2)/(R+C)
Or:
T = (2R+2)/(R+C)

Now, remember from the beginning that T also equals 2/C because the
log travels the 2 miles in T time, so...

T =   (2R+2)/(R+C)
2/C = (2R+2)/(R+C); cross-multiplying, we see that:

C(2R+2) = 2(R+C); or
2CR+2C = 2R+2C; 

Subtracting 2C from each side, we see that:
2CR = 2R;  Dividing each side by 2R, we finally see that:
C=1

Thus, the speed of the current is 1 mile per hour.

A solution similar to this, as well as an alternate solution are on
this page Chaos Factors Discussion Forum page cached by Google:
http://66.102.7.104/search?q=cache:8QyetxiyqEIJ:130.212.34.151/forum/forum_posts.asp%3FTID%3D16977%26PN%3D19+&hl=en

I hope this helps. If anything is unclear, please let me know via the
"request clarification" feature.