let

v = the speed of the motorboat in still water

c = the speed of the current of the river

d = the distance traveled

the unpowered raft will only use the current of the river thus it will take d/c hours to pass the distance downriver

moving downriver the speed of the motorboat is: v + c

v + c = d/32 multiply both sides by 32/c

32(v/c) + 32 = d/c

(v/c) = (1/32)(d/c - 32)

mowing upstream the speed of the motorboat is: v - c

v - c = d/48 multiply both sides by 48/c

48(v/c) - 48 = d/c

plug (v/c) = (1/32)(d/c - 32) into the last equation:

48(1/32)(d/c - 32) - 48 = d/c

lets put x = d/c and solve the equation:

48(1/32)(x - 32) - 48 = x

by solving we find:

x = 192 hr

click here to see the step by step solution of the equation:

the unpowered raft needs 192 hours to cover the distance downstream.