Question 940709
First they went 32miles against the current given by:
{{{32miles/(12 mph-C)=Time upstream}}} where c is the speed of the current and 12mph is their normal speed
Then downstream given by:
{{{32miles/(12 mph+C)=Time downstream}}}
And for the total trip
{{{32miles/(12 mph-C)+ 32miles/(12 mph+C)=6 hours}}} Multiply both sides by 12mph-C
{{{12mph-C(32miles/(12 mph-C)+ 32miles/(12 mph+C))=(6 hours)12mph-C}}}
{{{(12mph-C)(32 miles)/(12mph-C)+(12mph-C)(32 miles)/(12 mph+C)=(6 hours)12mph-C}}} 
{{{32miles + (32 miles)(12mph - C) /(12 mph +C)=6 Hrs(12mph-C)}}}Multiply both sides by (12mph+C)
{{{(12mph+C)(32miles + (32 miles)(12mph - C) /(12 mph +C))=(12mph+C)6 Hrs(12mph-C)}}}
{{{32 miles(12 mph+C)+ (32 miles)(12 mph-C)=6 hrs(12mph+C)(12mph-C)}}}
{{{ (32 miles)((12mph+C)+(12mph-C))=6 HRS(12mph+C)(12mph-C)}}}
{{{ (32 miles)(24 mph)=6 Hrs(144-c^2)}}}Divide both sides by 6
{{{((32 miles)(24mph))/6=(6 hrs(144-c^2))/6}}} 
{{{ (32 miles)(4 mph)=144-c^2}}}
{{{128=144-c^2}}}  subtract 128 from both sides
{{{ 0=16-c^2}}}Add {{{c^2 }}} to both sides
{{{c^2=16}}} so c = plus/minus 4 So the current is 4 MPH

CHECK:
upstream + downstream= 6 hours
upstream={{{32 miles/(12mph-4mph)=32 miles/8 mph = 4 hrs}}}
downstream={{{(32 miles)/(12mph+4mph)=32 miles/16 mph= 2 hrs}}}
 
2Hrs + 4hrs= 6hrs