Question 1181372
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Could you please show me the steps to the solution of this problem:
Gabrielle rows a boat downstream for 192 miles. The return trip upstream took 8 hours longer. 
If the current flows at 2 mph, how fast does Gabrielle row in still water?
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            The setup equation in the post by @josgarithmetic is incorrect.


            The best what you can do with his post is to  IGNORE  it.


            I came to bring the correct solution and describe the setup and steps clearly and in detailed way.



<pre>
Ler  "r"  be the rate Gabrielle rows in still water (in miles per hour).

Then the effective rate downstream is (r+2) miles per hour and the time rowing downstream is  {{{192/(r+2)}}} hours.

The effective rate upstream is (r-2) miles per hour and the time rowing downstream is  {{{192/(r-2)}}} hours.


The difference of times is  8 hours, according to the condition.  It gives you this time equation

    {{{192/(r-2)}}} - {{{192/(r+2)}}} = 8   hours.


or, dividing both sides by 8, for simplicity

    {{{24/(r-2)}}} - {{{24/(r+2)}}} = 1.


To solve this equation, multiply both sides by (r-2)*(r+2) = r^2 - 4.  You will get

    24*(r+2) - 24*(r-2) = r^2 - 4.


Simplify step by step

    24r + 48 - 24r + 48 = r^2 - 4

    r^2 = 48 + 48 + 4 = 100

    r = {{{sqrt(100)}}} = 10.


The problem is just solved.


The <U>ANSWER</U> is: the speed Gabrielle rows in still water is 10 miles per hour.


Let's <U>CHECK</U>.  (in such problems, checking is very educative part of the solution.
               When you check, your understanding of the problems becomes MUCH BETTER (!) )

                   The time rowing downstream is  {{{192/(10+2)}}} = {{{192/12}}} = 16  hours.

                   The time rowing upstream   is  {{{192/(10-2)}}} = {{{192/8}}} = 24 hours.

                   The difference of times is  24 hours - 16 hours = 8 hours, exactly as stated in the problem.


Thus we checked and found out that the answer is CORRECT (!)
</pre>

Solved and carefully explained.



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@josgaritmetic made changes in his post after seeing my solution.