let

b = the rate of the boat in still water

c = the rate of the current

t = 6 hr the time of the travel in one direction

d1 = 338 km the distance of the trip upstream

d2 = 384 km the distance of the trip downstream

the rate of the boar traveling upstream is: b - c

the rate of the boar traveling downstream is: b + c

since the rate = distance/time => distance = time*rate

d1 = t(b - c)

d2 = t(b + c)

or we have the following system

6(b - c) = 338

6(b + c) = 384

by solving the system of equations we find

b = 60.17 km/hr

c = 3.83 km/hr

click here to see the step by step solution of the system of equations

the rate of the boat in still water is 60.17 km/hr and the rate of the current is 3.83 km/hr.