A boat has a speed of 9 km/h in still water. It goes 12 km downstream and then returns 12 km upstream, taking a total of 3 hours for the round trip. What is the speed of the stream (in km/h)?

Introduction / Context:
This problem is about a boat that travels downstream and then upstream over the same distance. We know the speed of the boat in still water and the total time for the round trip. The objective is to find the speed of the river current. The concept relies on understanding that downstream and upstream speeds are different due to the current and that total time is the sum of times for each leg.

Given Data / Assumptions:

• Boat speed in still water = 9 km/h.
• Distance downstream = 12 km.
• Distance upstream = 12 km.
• Total time for the trip = 3 hours.
• Speed of stream = c km/h (unknown).
• Speeds are constant and the river current is uniform.

Concept / Approach:
If b is the speed of the boat in still water and c is the speed of the current, then:

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

Time taken downstream is distance divided by downstream speed, and time upstream is distance divided by upstream speed. Their sum is the given total time. With b known and total time known, we can set up an equation in c and solve it.

Step-by-Step Solution:
Let b = 9 km/h and c be the speed of the stream. Downstream speed = 9 + c km/h. Upstream speed = 9 - c km/h. Time downstream = 12 / (9 + c) hours. Time upstream = 12 / (9 - c) hours. Total time = 12 / (9 + c) + 12 / (9 - c) = 3 hours. Multiply both sides by (9 + c)(9 - c): 12(9 - c) + 12(9 + c) = 3(81 - c^2). Left side simplifies to 108 - 12c + 108 + 12c = 216. So 216 = 3(81 - c^2). Divide both sides by 3: 72 = 81 - c^2. So c^2 = 81 - 72 = 9 and hence c = 3 km/h.

Verification / Alternative check:
If c = 3 km/h, downstream speed is 9 + 3 = 12 km/h and upstream speed is 9 - 3 = 6 km/h. Downstream time is 12 / 12 = 1 hour and upstream time is 12 / 6 = 2 hours. The total is 1 + 2 = 3 hours, which matches the given total time. Therefore the calculated current speed 3 km/h is correct.

Why Other Options Are Wrong:
If the stream were only 1 km/h or 4 km/h, the total time would not match 3 hours when recalculated. A current speed of 5 km/h would produce an upstream speed of only 4 km/h, which would make the upstream leg take too long. Thus 3 km/h is the only value that produces a consistent round trip time.

Common Pitfalls:
One common error is to simply average the speeds or times without using the correct equation. Another mistake is to treat 3 hours as the time for one way instead of the round trip. Some students also forget that upstream speed is boat speed minus stream speed. Setting up the time equation correctly and solving the resulting quadratic in a careful way avoids these mistakes.

Final Answer:
The speed of the stream is 3 km/h.