A boat covers 36 km downstream in 6 hours, which is 3 hours less than the time taken to cover the same distance upstream. What is the speed of the stream (in km/h)?

Introduction / Context:
This question compares downstream and upstream travel times for the same distance. The downstream time is given, and the upstream time is said to be 3 hours more. Using this, we can find the effective downstream and upstream speeds, then work back to the boat speed in still water and the stream speed. This is a typical relative speed problem in river boat scenarios.

Given Data / Assumptions:

• Downstream distance = 36 km.
• Time downstream = 6 hours.
• Upstream time is 3 hours more than downstream time.
• Thus upstream time = 6 + 3 = 9 hours.
• We assume uniform speeds and steady current.

Concept / Approach:
We first find the downstream and upstream speeds using distance divided by time. Then we use the standard relations:

• Downstream speed = b + c.
• Upstream speed = b - c.

Where b is boat speed in still water and c is stream speed. Solving these two equations will give c, the required stream speed.

Step-by-Step Solution:
Downstream speed = 36 / 6 = 6 km/h. Upstream time = 9 hours, so upstream speed = 36 / 9 = 4 km/h. Let b be boat speed and c be stream speed. Then b + c = 6 and b - c = 4. Add the equations: 2b = 10 so b = 5 km/h. Subtract the equations: 2c = 2 so c = 1 km/h. Therefore, the speed of the stream is 1 km/h.

Verification / Alternative check:
If the boat speed is 5 km/h and the stream speed is 1 km/h, downstream speed is 5 + 1 = 6 km/h. Time to cover 36 km is 36 / 6 = 6 hours, which matches the given data. Upstream speed is 5 - 1 = 4 km/h, and time to cover 36 km is 36 / 4 = 9 hours, which is indeed 3 hours more than 6 hours. This confirms consistency.

Why Other Options Are Wrong:
If the stream speed were 1.5 km/h, the speeds and times would no longer be 6 hours and 9 hours. For 0.75 km/h or 0.5 km/h, recalculated upstream times would not differ by exactly 3 hours from the downstream time. Only a stream speed of 1 km/h produces the required difference in travel times.

Common Pitfalls:
A frequent misunderstanding is to subtract times directly from speeds without using the distance relation. Another error is to misread the phrase "3 hours less" or "3 hours more" and reverse the relationship between upstream and downstream times. Carefully translating the verbal condition into an equation and then applying the boats and streams formulas avoids these issues.

Final Answer:
The speed of the stream is 1 km/h.