Question 123956
Let C be the speed of the current.  When going downstream, the speed of the current is added to the speed of the boat in still water.  So, when going downstream the speed over the ground is 13 + C.  When going upstream, the speed of the current is subtracted from the speed of the boat in still water, so the upstream speed over the ground is 13 - C.

Distance = Speed x Time, or Time = Distance/Speed.  The time it takes to go 14 miles downstream is therefore 14/(13+C).  The time it takes to go 8 miles upstream is 8/(13-C).  But we know the times are equal, so:

{{{8/(13-C) = 14/(13+C)}}}.

Multiply both sides by (13-C):

{{{(13-C)*8/(13-C)=(13-C)*14/(13+C)}}}

Replace the {{{(13-C)/(13-C)}}} on the left hand side by 1, and multipy both sides by (13+C):

{{{(13+C)*1*8=(13-C)*14*(13+C)/(13+C)}}}

Simplify by replacing the {{{(13+C)/(13+C)}}} by 1 to give:

{{{(13+C)*8=(13-C)*14}}}

Multiply this out to give:

{{{13*8+C*8=13*14-C*14}}} or 

{{{104+8C=182-14C}}}

Now, subtract 104 from both sides, and add 14C to both sides:

{{{104-104+8C+14C=182-104-14C+14C}}}

Again, simplify:

{{{0+22C=78+0}}}

Divide both sides by 22 to give the answer:

{{{C=3.545}}}

Now check the answer.  The speed downstream is 13+C, or 16.545.  To cover 14 miles downstream takes 14/16.545 hours, or .846 hours.  The speed upstream is 13-C, or 9.455.  To cover 8 miles upstream takes 8/9.455 hours, or .846 hours.  The times are the same, the solution works.