Rajesh rows a boat in still water at 4.5 km/h to go to a place and then return. If the river flows at 1.5 km/h, what is his average speed (in km/h) for the entire round trip?

Introduction / Context:
This problem asks for the average speed of a round trip in a river, where Rajesh rows a boat to a certain point and returns. His speed in still water and the speed of the stream are given. Because downstream and upstream speeds differ, the effective average speed is not a simple average of the two speeds but rather total distance divided by total time. This illustrates the classic concept of harmonic type averaging in speed questions.

Given Data / Assumptions:

• Boat speed in still water = 4.5 km/h.
• Stream speed = 1.5 km/h.
• Distance from start to destination is d km (same on return).
• He rows the same route downstream and then upstream.
• We assume steady speeds throughout.

Concept / Approach:
We first compute downstream and upstream speeds. Then we find the time taken for each leg in terms of d. Total distance is 2d and total time is the sum of downstream and upstream times. Average speed for the whole journey is total distance divided by total time. Since d cancels, we get a numeric answer independent of the actual one way distance.

Step-by-Step Solution:
Downstream speed = 4.5 + 1.5 = 6 km/h. Upstream speed = 4.5 - 1.5 = 3 km/h. Let d be the one way distance in km. Time downstream = d / 6 hours. Time upstream = d / 3 hours. Total distance = 2d km. Total time = d / 6 + d / 3 = d / 6 + 2d / 6 = 3d / 6 = d / 2 hours. Average speed = total distance / total time = 2d / (d / 2) = 2d * 2 / d = 4 km/h.

Verification / Alternative check:
We can test with a specific value, for example d = 6 km. Downstream, Rajesh would take 6 / 6 = 1 hour. Upstream, he would take 6 / 3 = 2 hours. Total distance is 12 km and total time is 3 hours, giving an average speed of 12 / 3 = 4 km/h. This matches the value obtained algebraically and confirms the correctness of the solution.

Why Other Options Are Wrong:
An average speed of 6 km/h would suggest that the boat travelled at downstream speed the whole way, ignoring the slower upstream leg. An average of 2 km/h or 8 km/h has no basis in the given speeds and results in inconsistent time calculations. Only 4 km/h balances the downstream and upstream segments correctly over the entire journey.

Common Pitfalls:
Students often incorrectly take the simple arithmetic average of 6 km/h and 3 km/h as (6 + 3) / 2 = 4.5 km/h, which is not the correct average speed for a round trip over equal distances. The correct method always uses total distance divided by total time. Remember that when distances are equal, the average speed is closer to the smaller speed because more time is spent at that speed.

Final Answer:
Rajesh’s average speed for the entire trip is 4 km/h.