A steamer moves with a speed of 4.5 km/h in still water. It travels to a certain point upstream and then returns to the starting point downstream in a river whose current flows at 1.5 km/h. What is the average speed (in km/h) of the steamer for the entire journey?

Introduction / Context:
This question tests your understanding of average speed when travelling equal distances upstream and downstream in a river. The steamer has a known speed in still water, and the river has a known current speed. We must use relative speed to get effective upstream and downstream speeds, and then calculate the overall average speed for a round trip.

Given Data / Assumptions:

• Speed of steamer in still water = 4.5 km/h.
• Speed of river current = 1.5 km/h.
• Let the one-way distance to the upstream point be d km.
• Upstream speed = 4.5 - 1.5 = 3 km/h.
• Downstream speed = 4.5 + 1.5 = 6 km/h.
• The steamer covers distance d upstream and distance d downstream.

Concept / Approach:
For a round trip with equal distances travelled at two different speeds, average speed is total distance divided by total time. As in earlier questions, this leads to the harmonic mean of the two speeds. Here the two speeds are the effective upstream and downstream speeds. We can compute average speed using either the full distance-and-time method or the harmonic mean formula.

Step-by-Step Solution:
Step 1: Upstream speed = 4.5 - 1.5 = 3 km/h. Step 2: Downstream speed = 4.5 + 1.5 = 6 km/h. Step 3: Time taken upstream = d / 3 hours. Step 4: Time taken downstream = d / 6 hours. Step 5: Total distance for the complete journey = d + d = 2d km. Step 6: Total time = d / 3 + d / 6 = (2d / 6) + (d / 6) = 3d / 6 = d / 2 hours. Step 7: Average speed = total distance / total time = (2d) / (d / 2) = 2d * (2 / d) = 4 km/h.

Verification / Alternative check:
Using the harmonic mean formula directly: average speed = (2 * upstream speed * downstream speed) / (upstream speed + downstream speed) = (2 * 3 * 6) / (3 + 6) = 36 / 9 = 4 km/h. Both methods give the same value, so the result is confirmed.

Why Other Options Are Wrong:
Options such as 6, 10 or 12 km/h ignore the fact that the steamer spends more time moving slowly upstream than quickly downstream. The still water speed of 4.5 km/h is not itself the average because the current affects the effective speeds. Only 4 km/h matches the total distance divided by total time.

Common Pitfalls:
A common error is to take the arithmetic mean of 3 and 6 (which is 4.5) or mistakenly use the still water speed directly. Another mistake is to ignore that the upstream leg takes longer, which lowers the average. Always compute total distance and total time explicitly or apply the harmonic mean formula correctly.

Final Answer:
The average speed of the steamer for the entire journey is 4 km/h.