SOLUTION: 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 fl

Algebra ->  Equations -> SOLUTION: 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 fl      Log On


   

Question 1181372: 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?
Found 3 solutions by mananth, josgarithmetic, ikleyn:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
Gabrielle rows a boat downstream for 192 miles.

The return trip upstream took 8 hours longer.

If the current flows at 2 mph,
Gabrielle speed in still water = x mph
downspeed speed =(x+2) mph
upstream speed = (x-2) mph
distance =192 miles
Time upstream - time downstream = 8
t=d/s
192/(x-2)- 192/(x+2) = 8
Simplify
Multiply the equation by (x+2)(x-2)
192(x+2) - 192(x-2) = 8(x+2)(x-2)
192x + 384 -192x +384 = 8x^2-32
8x^2=768+32
8x^2 = 800
x^2 = 100
x= 10 mph
Gabrielle speed in still water is 10 mph



Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
"The return trip upstream took 8 hours longer." 
                     SPEED             TIME                DISTANCE

DOWNSTREAM           r-2               192/(r+2)              192

UPSTREAM             r+2               192/(r-2)             192

192%2F%28r-2%29-192%2F%28r%2B2%29=8
.
.
.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
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?
~~~~~~~~~~~~~~~


            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. 


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%2F%28r%2B2%29 hours.

The effective rate upstream is (r-2) miles per hour and the time rowing downstream is  192%2F%28r-2%29 hours.


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

    192%2F%28r-2%29 - 192%2F%28r%2B2%29 = 8   hours.


or, dividing both sides by 8, for simplicity

    24%2F%28r-2%29 - 24%2F%28r%2B2%29 = 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%28100%29 = 10.


The problem is just solved.


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


Let's CHECK.  (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%2F%2810%2B2%29 = 192%2F12 = 16  hours.

                   The time rowing upstream   is  192%2F%2810-2%29 = 192%2F8 = 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 (!)

Solved and carefully explained.


///////////////


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