Question 769071
<pre>
For a portion of the Green River in Utah, the rate of the river's current is 4 mph. A tour guide can row 6 mi down this river and back in 2 h. Find the rowing rate of the guide in calm water. 
I'm having a hard time recalling how o set up a problem like this. I have tried:
with current distance: 6mi
with current rate: x+4
with current time: 6/x+4 
Against current distance: 6mi
against current rate: x-4
against current time: 6/x-4 
Ans:
You have started on the right track. As you have calculated till now, if we
assume that x is the speed of the boat in still water.
Time for downstream = 6/(x+4)
Time for upstream = 6/(x-4)
It is given that the total time (up and down) is 2 hours. So we get the
equation
{{{6/(x+4) + 6/(x-4) = 2}}} Cross multiplying
{{{6*(x-4)+6*(x+4) = 2*(x+4)*(x-4)}}} Expanding the terms,
{{{6*x - 24 + 6*x + 24 = 2*(x^2 - 16)}}} 
{{{12*x = 2*x^2 - 32}}} Simplifying
{{{x^2 - 6*x - 16 = 0}}}
This is a standard quadratic equation that you can solve by factorizing.
You get
{{{x^2 - 8*x + 2*x - 16 = 0}}}
{{{x*(x - 8) + 2*(x - 8) = 0}}}
{{{(x - 8)*(x + 2) = 0}}}
x = 8 or x = -2
Since x cannot be negative, the speed of the boat in calm water is 8 mph.

Check for correctness:
If speed in still water is 8 mph
Time for downstream = 6/(8+4) = 0.5 hours
Time for upstream = 6/(8-4) = 1.5 hours
Total time = 0.5 + 1.5 = 2 hours.
Correct!

Hope you got it :)