Scalars and vectors question

A man is rowing a boat at 2.5km/h. He sets off in a direction perpendicular to the bank. If the river is 200m wide and the current is 2.0km/h, with the help of a vector diagram, determine the resulting speed of the boat, and hence the distance he would have travelled down stream by the time he reaches the opposite bank.

Do I assume that the time taken for the resultant is equal to the time taken for the man to reach the opposite bank without the current?

Yes, his speed across the river is entirely determined by his rowing, and his speed down the river is entirely determined by the current. The time needed to reach the opposite parallel bank is the same with or without the current. So calculate that time, and then use it to figure out how far something would move at the river's velocity in that much time.

Vector Diagram : AB = w = width of the river = 200 m = 1/5 km ;
BC = d = distance travelled down stream ;
AC = l = resultant path travelled by the man with the boat
Vector AB + Vector BC = Vector AC
=> vector w + vector d = vector l
=> w² + d² = l²
A figure DEF similiar to ABC will represent the corresponding velocities
Speed of rowing the boat along AB = u = 2.5 km/h = (5/2) km/h ;
Distance travelled along AB = w = 1/5 km
Time taken = t = w / u = (1/5) / (5/2) = 2/25 hr = 288 s
Speed of the current parallel to BC = v = 2.0 km/h
Distance travelled along BC = d
Time taken to travel the distance (d) = t = 2/25 hr
BC = d = v*t = 2*(2/25) km = 4/25 km = 160 m
Resultant speed of the boat along AC = r = √(u² + v²) = √{(5/2)² + 2²} = √41 / 4 km/h

short answer = yes
The fact that the man in boat is moving with the river current doesn't change his time to row across the river.
(ie his time to cross = 0.2/2.5 = 0.08 h = 60(0.08) = 4.8 min = 4.8(60) = 288 s)
so
downstream distance = (2.0)(0.08) = 0.16 km = 160 m