Question 173454
Still water doesn't have any speed.  That's the point.  Saying the speed of the boat in still water is the same as saying the speed of the boat if there were no current.  Since the current in a river flows in the downstream direction (by definition of downstream), when the boat is going downstream its speed relative to the bank of the river is the speed in still water plus the speed of the current.  Conversely, going upstream, the speed of the current is subtracted from the speed of the boat in still water to get the speed relative to an observer on the bank of the river.


So:


Start with the basic formula that relates distance, rate (or speed), and time:


{{{d=rt}}} which can also be expressed as {{{r=d/t}}}


Let's say that the relative speed for the upstream trip is {{{r[u]}}}, since we know that an upstream trip of 325 km takes 5 hours, we can say that {{{r[u]=325/5=65}}} km/hr.


Likewise, the downstream trip is given by {{{r[d]=534/6=89}}} km/hr.


Now, let's say the speed of the boat in still water is {{{r}}} and the speed of the current is {{{r[c]}}}.  Since the current subtracts from the boat speed going upstream and adds going downstream, we can write the following two equations:


{{{r-r[c]=65}}}


{{{r+r[c]=89}}}


Here we have a system of two linear equations in two variables.  Since the coefficients on the {{{r[c]}}} terms are additive inverses, just add the two equations, term by term:


{{{r+r-r[c]+r[c]=65+89}}}


{{{2r=154}}}


{{{r=77}}} km/hr.


So the speed of the boat in still water is 77 km/hr.


Since we know that {{{r+r[c]=89}}}, we can substitute:


{{{77+r[c]=89}}}


{{{r[c]=89-77=12}}} km/hr


Meaning the speed of the current is 12 km/hr.