Time taken to cover (A+4) km in upstream is 3 times the time taken to cover…

Concept

Boats and streams: if upstream speed is u and downstream speed is d, then time = distance / speed, and the speed of the boat in still water is the average of the two, (u + d) / 2. Set up one equation per stated condition and solve.

Application

1. Let upstream speed = u and downstream speed = d. Given u : d = 1 : 2, so d = 2u.

2. Condition 1: time for (A+4) km upstream = 3 x time for (A-2) km downstream, i.e. (A+4)/u = 3(A-2)/d. Substitute d = 2u and cancel u: (A+4) = 3(A-2)/2.

3. Solve: 2(A+4) = 3(A-2) -> 2A + 8 = 3A - 6 -> A = 14.

4. Condition 2: time for (A+6) km downstream = 2 h, so (A+6)/d = 2 -> (14+6)/d = 2 -> d = 10 km/hr. Then u = d/2 = 5 km/hr.

5. Still-water speed = (u + d)/2 = (5 + 10)/2 = 7.5 km/hr.

Cross-check

Times: upstream (A+4)=18 km at u=5 gives 3.6 h; downstream (A-2)=12 km at d=10 gives 1.2 h; 3 x 1.2 = 3.6 h, matching condition 1. Downstream (A+6)=20 km at d=10 gives 2 h, matching condition 2.

Result

The still-water speed works out to 7.5 km/hr, which is not listed among 4.5, 5.5 or 4 km/hr, so the correct choice is "None of these".