Let u be the rate of the boat at still water and v be the current rate (in miles-per-hour).

Then the speed of the boat downstream is (u+v) mph, 
while the speed of the boat  upstream is (u-v) mph.


Hence, you have these two equations for two unknowns

 =        (1)   ("A man takes 5 hours and 20 minutes to row 6 miles up a river and back")

 =            (2)   ("he can row 3 miles downstream in the same time that he can row 1 mile upstream")


Simplify equation (1) in this way:

 =       (1') 

and replace    in  (1')  by , based on (2). You will get

 = ,    or

 = ,   or   (u+v) =  = . 


Then from (2)  u-v =  =  = .


You have now this system for u and v:

u + v = ,    (3)
u - v = .    (4)

Add the two equations. You will get

2u =  =  = 6.   Hence,   u =  = 3 miles per hour.

Now from (3),  v =  =  = 1.5 miles per hour.


Answer.  The boat speed in still water is 3 miles per hour.

         The current speed is 1.5 miles per hour.