Average Speed – Outward and return with different speeds: Rani travels from the origin to a point by car at 5 km/h and returns by scooter at 2 km/h. What is her average speed over the entire round trip?

Introduction / Context:
Average speed for a round trip with equal distances but different speeds is the harmonic mean of the two speeds. Direct arithmetic averaging is incorrect because travel times per leg differ.

Given Data / Assumptions:

• Outward speed v1 = 5 km/h.
• Return speed v2 = 2 km/h.
• Distances are equal for both legs.

Concept / Approach:
Average speed over equal distances is 2*v1*v2 / (v1 + v2). Derivation uses total distance divided by total time, with each leg's time equal to distance over speed.

Step-by-Step Solution:
Average speed = 2*5*2 / (5 + 2) = 20/7 km/h.

Verification / Alternative check:
Let distance each way be 1 km. Total distance = 2 km. Time = 1/5 + 1/2 = 0.2 + 0.5 = 0.7 h. 2 / 0.7 = 20/7 km/h.

Why Other Options Are Wrong:
Values like 26/7 and 36/7 arise from incorrect arithmetic means or misapplied formulas.

Common Pitfalls:
Averaging the speeds ( (5 + 2)/2 ) rather than using the harmonic mean for equal distances.

Final Answer:
20/7 km/h