A boat sails 15 km upstream 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 to cover the same 15 km distance downstream?

Introduction / Context:
This boats and streams question examines how relative speeds change when moving upstream versus downstream. You are given the time taken to travel upstream and the relationship between the boat's speed in still water and the speed of the current. From this, you must find the time required to travel the same distance downstream. Such questions help strengthen understanding of relative motion in flowing water.

Given Data / Assumptions:

• Distance upstream = 15 km.
• Time taken upstream = 5 hours.
• Speed of 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.
• Let the speed of the current be c km/h = b/4.

Concept / Approach:
When moving in a river:

• Upstream speed = (b − c) km/h.
• Downstream speed = (b + c) km/h.

First we find b from the upstream journey, then compute the downstream speed and finally the time required downstream for the same distance. This uses the basic relation speed = distance / time.

Step-by-Step Solution:
Step 1: Upstream speed = distance / time = 15 / 5 = 3 km/h.Step 2: Using b as boat speed in still water and c = b/4 as current speed, upstream speed = b − c = b − b/4 = (3b/4).Step 3: We know upstream speed is 3, so 3b/4 = 3.Step 4: Solving for b: b = 3 * 4 / 3 = 4 km/h.Step 5: Then c = b/4 = 4 / 4 = 1 km/h.Step 6: Downstream speed = b + c = 4 + 1 = 5 km/h.Step 7: Time taken downstream for 15 km = distance / speed = 15 / 5 = 3 hours.

Verification / Alternative check:
We can check consistency by recomputing the upstream speed from the found values. Upstream speed with b = 4 and c = 1 is 4 − 1 = 3 km/h, which matches the speed calculated from the given upstream journey (15 km in 5 hours). This confirms that both the boat speed and current speed are correct, and therefore the 3-hour downstream time is reliable.

Why Other Options Are Wrong:
4 hours or 5 hours would correspond to downstream speeds smaller than or equal to the upstream speed, which is not possible because the current assists the boat downstream. 1.8 hours or 2.5 hours would require unrealistically high downstream speeds given that the boat itself moves at only 4 km/h in still water and the current is just 1 km/h.

Common Pitfalls:

• Mistaking upstream speed as (b + c) instead of (b − c).
• Incorrectly interpreting "one-fourth the speed of the boat" and taking current speed as 4b instead of b/4.
• Forgetting to use the same distance (15 km) for both directions.
• Trying to solve intuitively without forming and solving the equation for b.

Final Answer:
The boat will take 3 hours to cover the distance downstream.