Question 115098
HEY! Ease up on the ALL CAPS.


Remember the formula for distance, rate, and time:  {{{d=rt}}}.


We want to find the boat's speed through the water, so lets call that r.


The boat goes up river against the current, so the speed relative to the land is 5 miles per hour less than the speed through the water, or r - 5.


The boat goes down river with the current, so the speed is 5 mph more, or r + 5.


First let's rearrange the formula so that it is in terms of time:  {{{t=d/r}}}


The amount of time the boat traveled up river is then given by {{{52/(r-5)}}} and the time down river is given by {{{92/(r+5)}}}.  And we know from the problem statement that these two times are equal, so:


{{{92/(r+5)=52/(r-5)}}}


{{{92(r-5)=52(r+5)}}}, Cross-multiply the proportion


{{{92r-460=52r+260}}}, Distribute


{{{92r-52r=260+460}}}, Collect like terms
{{{40r=720}}}


{{{r=18}}}, Divide by 40


And the speed of the boat through the water is 18 miles per hour.


Last step, VERY important:  Check your answer.

{{{52/(18-5)=52/13=4}}}
{{{92/(18+5)=92/23=4}}}  Since both times (4 hours) are equal, the answer checks.


Hope that helps,
John