Aswimmer swims 3/5 the width of a river at one velocity, then swims the remainder of the river at half her initial velocity. what was the average speed across the river?

Let her initial velocity be U.

Let the width of the river be W.

She swims 3/5 the width of the river  at  U.

Remainder width =  (1 - 3/5) = 2/5.

She then swims 2/5 the width with velocity  U/2.

Average Speed  =  (Total Distance Traveled)  /  ( Total Time Taken).

Distance =  Speed * time

time  =  Distance /  Speed.

Time in first trip:    =    (3/5)W / U =   0.6W/U.

Time in second trip  =  (2/5)W / (U/2)  =  0.4W / 0.5 U =  0.8W/U

Total Distance Traveled =  W,  width of the river.

Average Speed  =      W  /  (0.6W/U  +  0.8W/U) =    W /  (1.4W/U)

                             =  W  *  U / 1.4W 
                              =    U/1.4
                              =  U * 10 / 14
                              =  (5/7) U.

Therefore Average speed is  (5/7) of the initial speed.