Question 415415
Let the speed of the current X.
When the barge is traveling downriver, its net speed equals to the sum of the speed of the barge in still water and the speed of the current. That is, (X+6) km/h.
When the barge is traveling upriver, its net speed equals to the difference of the speed of the barge in still water and the speed of the current. That is (X-6)
km/h.
According to the time formula: time=distance/speed, you can set up an equation:
        51/(X+6)           +           51/(X-6)             =  114
time spent in downriver         time spent in upriver        total time
solve by multiplying (X-6)(X+6) to each side of the equation:
    (X-6)(X+6)*51/(X+6)    +  (X-6)(X+6)*51/(X-6)  = 114*(X-6)(X+6)
reduce and simplify:
         51*(X-6)           +       51*(X+6)        = 114(X^2-36)
         51*X-6*51          +       51*X +6*51      = 114*(X^2) - 114*36
Move every item to right side and combine like terms:
        114*(X^2) -102*X -114*36=0
Simplify by reducing a 6:
         19*(X^2) - 17X - 684 =0 
Use the formula 
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
you can get two answers, one is negative, not suitable in this case. The other is
       (17+sqrt(52273))/38=6.46 (rounded to the nearest hundredth)

So, the speed of the current is 6.46km/h.