let

c = the speed of the current, c>=0

v = the speed of the boat in still water, v>=0

d = 300 mi the distance downstream and back

d/2 = 300/2 = 150 mi the distance traveled in each direction

t1 = 6 hr the time of the downstream trip

t2 = 15 hr the time of the upstream trip

the speed of the boat when moving upstream is: v - c

the speed of the boat when moving downstream is: v + c

since speed = distance/time => distance = time*speed

t1(v + c) = d/2

6(v + c) = 150

t2(v - c) = d/2

15(v - c) = 150

by solving the system of equations:

6(v + c) = 150

15(v - c) = 150

we find:

v = 17.5 mph

c = 7.5 mph

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

the speed of the water current is 7.5 mph.