A man can row at a speed of 4.5 km/h in still water. He rows from a point to another point upstream and then rows back to the starting point downstream in a river that flows at 1.5 km/h. What is his average speed for the entire round trip?

Introduction / Context:
This question covers average speed for a round trip in a river where the effective speed differs upstream and downstream. The man rows against the current in one direction and with the current in the other, so we must compute the average speed based on total distance and total time, not simply by averaging the two speeds.

Given Data / Assumptions:

• Speed of man in still water = 4.5 km/h.
• Speed of river current = 1.5 km/h.
• Upstream speed = 4.5 - 1.5 = 3 km/h.
• Downstream speed = 4.5 + 1.5 = 6 km/h.
• Distance from starting point to turning point = d km (same both ways).
• We need the average speed for the entire upstream and downstream journey.

Concept / Approach:
Average speed over a journey is total distance divided by total time, not the simple arithmetic mean of speeds. For a round trip with equal distances, the effective average speed is the harmonic mean of the upstream and downstream speeds, which is 2uv / (u + v), where u and v are the two speeds. We apply this formula using the upstream and downstream speeds.

Step-by-Step Solution:
Let d be the one way distance between the two points on the river. Upstream speed u = 3 km/h, so upstream time t_up = d / 3 hours. Downstream speed v = 6 km/h, so downstream time t_down = d / 6 hours. Total distance for round trip = d (upstream) + d (downstream) = 2d km. Total time for round trip = t_up + t_down = d / 3 + d / 6. Compute total time: d / 3 + d / 6 = 2d / 6 + d / 6 = 3d / 6 = d / 2 hours. Average speed V_avg = total distance / total time = 2d / (d / 2) = 2d * 2 / d = 4 km/h. Therefore, the man's average speed for the whole journey is 4 km/h.

Verification / Alternative check:
We can also use the harmonic mean formula directly: V_avg = 2uv / (u + v) = 2 * 3 * 6 / (3 + 6) = 36 / 9 = 4 km/h. This matches the result obtained using total distance and total time, confirming the correctness of the calculation.

Why Other Options Are Wrong:
The simple arithmetic mean (3 + 6) / 2 = 4.5 km/h is not valid here because the times spent at each speed are different. Options such as 5, 7, 3 or 6 km/h do not match the ratio of total distance to total time for equal upstream and downstream distances. Only 4 km/h correctly represents the average speed for the entire trip.

Common Pitfalls:
A frequent mistake is to compute the average of the two speeds without considering that the distance is the same in both directions. Another error is to ignore the effect of different times spent at each speed. Always use total distance over total time or the harmonic mean formula when dealing with round trips with equal distances.

Final Answer:
The average speed for the entire round trip is 4 km/h.