A boat sails 15 km upstream on a river in 5 hours. If the speed of the current is one-fourth of the speed of the boat in still water, how long will it take the boat to cover the same distance downstream?

Introduction / Context:
Boat and stream problems are classic in quantitative aptitude, testing your understanding of relative speed in upstream and downstream motion. Here, the time taken upstream is given along with a relation between the speed of the current and the speed of the boat in still water. You need to determine the time taken to cover the same distance downstream.

Given Data / Assumptions:

Distance travelled upstream = 15 km.
Time taken upstream = 5 hours.
Speed of the current = one-fourth of the speed of the boat in still water.
Let the speed of the boat in still water be b km/h and the speed of the current be c km/h.
Upstream speed = (b - c), downstream speed = (b + c).

Concept / Approach:
First, we find the upstream speed from distance and time. Then we use the relation between b and c to express upstream speed in terms of b, solve for b, and then find downstream speed. Finally, time = distance / speed is used to compute the downstream time for the same 15 km. The key idea is that current reduces effective speed upstream and increases it downstream.

Step-by-Step Solution:
Step 1: Compute upstream speed. Upstream speed = distance / time = 15 / 5 = 3 km/h. Step 2: Use the relation between boat speed and current. Given c = (1/4)b, so upstream speed = b - c = b - b/4 = (3/4)b. But upstream speed is 3 km/h, so (3/4)b = 3. Step 3: Solve for b. b = 3 * (4/3) = 4 km/h. Then c = b/4 = 1 km/h. Step 4: Find downstream speed and time. Downstream speed = b + c = 4 + 1 = 5 km/h. Time downstream = distance / downstream speed = 15 / 5 = 3 hours.

Verification / Alternative check:
Check consistency: with b = 4 km/h and c = 1 km/h, upstream speed = 3 km/h and downstream speed = 5 km/h. Upstream time for 15 km: 15 / 3 = 5 hours (matches given). Downstream time: 15 / 5 = 3 hours, which agrees with our computed answer.

Why Other Options Are Wrong:
1.8 h would correspond to a downstream speed of 15 / 1.8 ≈ 8.33 km/h, which is impossible given the still water speed is only 4 km/h.
4 h and 5 h imply downstream speeds of 3.75 km/h and 3 km/h respectively, both less than or equal to the upstream speed, which cannot be true when the current aids the boat downstream.

Common Pitfalls:
One common mistake is to ignore the relation c = b/4 or to misinterpret it. Another is to assume that upstream and downstream times are symmetric or to average speeds incorrectly. Always derive actual speeds using distance/time and the given relationships, then apply time = distance / speed carefully.

Final Answer:
The boat will take 3 h to cover the same 15 km distance downstream.