A steamer moves with a speed of 4.5 km/h in still water, travelling to a point upstream and back to the starting point in a river flowing at 1.5 km/h. What is the average speed (in km/h) of the steamer for the entire journey?

Introduction / Context:
In this question, a steamer travels from a starting point to a point upstream and then back downstream to the starting point. We are given the speed of the steamer in still water and the speed of the river current. The task is to find the average speed for the entire round trip. Since the distance upstream and downstream is the same, average speed is determined by the harmonic mean of the upstream and downstream speeds.

Given Data / Assumptions:

• Steamer speed in still water b = 4.5 km/h.
• Speed of the current c = 1.5 km/h.
• Upstream speed = b - c = 4.5 - 1.5 = 3 km/h.
• Downstream speed = b + c = 4.5 + 1.5 = 6 km/h.
• Let the one way distance be d km.
• We must find the average speed for the round trip.

Concept / Approach:
For a journey where equal distances are covered at speeds u and v, the average speed is: Average speed = 2 * u * v / (u + v). Here u = 3 km/h and v = 6 km/h. This formula comes from total distance divided by total time, taking into account different speeds for equal distances. Alternatively, we can compute the time upstream and downstream separately and then find the total distance divided by total time.

Step-by-Step Solution:
Step 1: Let the one way distance be d km. Total distance = 2d km. Step 2: Compute the time upstream and downstream. Upstream time = d / 3 hours. Downstream time = d / 6 hours. Total time = d / 3 + d / 6 = (2d + d) / 6 = 3d / 6 = d / 2 hours. Step 3: Compute average speed. Average speed = total distance / total time = 2d / (d / 2) = 2d * 2 / d = 4 km/h.
Verification / Alternative check:
Using the harmonic mean formula: Average speed = 2 * 3 * 6 / (3 + 6) = 36 / 9 = 4 km/h. Both methods confirm the same result.
Why Other Options Are Wrong:
Speeds like 6 km/h, 10 km/h or 12 km/h ignore the fact that the steamer spends a longer time moving upstream at the slower speed of 3 km/h than downstream at 6 km/h. An average speed of 5 km/h would not match the total distance divided by the total time for the combination of the two legs.
Common Pitfalls:
A typical mistake is to average the speeds 3 km/h and 6 km/h directly to get 4.5 km/h, which is incorrect for equal distances. Another pitfall is to confuse the still water speed with the average speed, even though the current slows the boat upstream and speeds it up downstream.
Final Answer:
The average speed of the steamer for the total journey is 4 km/h.