Upstream Downstream Problem

I encountered this problem on the internet.

A raft and a motorboat left town A simultaneously and traveled downstream to town B. (The raft always moves at the same speed as the current, which is constant.) The motorboat arrived at town B, immediately turned back, and encountered the raft two hours after they had set out from A. How much time did it take the motorboat to go from A to B? (Assume that it travels at a constant rate of speed.)

 

 

Solution :

 Prima Facie the question seems tough as the number of unknowns seems apparently large.

    So, let us first reduce the number of unknowns. Let us take the time for  the motorboat to go from A to B be t1 and then from B to meeting point be t2. Let v be the velocity of the motorboat w.r.t. still water.

Clearly v is the relative velocity of motorboat w.r.t to raft from A to B and -v is relative velocity of motorboat w.r.t raft from B to meeting point. 

Now, displacement after 2 hours is 0 

      Therefore                  v/t1 + (-v)/t2=0 

                          =>             t1=t2

       Also                             t1+t2= 2 

                           =>          t1= 1 hour