A traveler covers half of the journey at 40 km/h, one-third at 60 km/h, and the remaining one-sixth at 30 km/h. What is the average speed for the entire journey?

Introduction / Context:
Average speed over non-uniform speeds must be computed as total distance divided by total time. When distances are fractional parts of the whole, computing the time for each segment and summing is the most reliable method.

Given Data / Assumptions:

• Fractions of total distance: 1/2, 1/3, 1/6 (which sum to 1).
• Speeds: 40 km/h, 60 km/h, 30 km/h respectively.
• Constant speeds on each leg; no stoppage time.

Concept / Approach:
Let total distance = D. Then times per leg are (D/2)/40, (D/3)/60, (D/6)/30. The average speed v_avg = D / (sum of times). D cancels, leaving a pure numeric result.

Step-by-Step Solution:
t1 = (D/2) / 40 = D/80.t2 = (D/3) / 60 = D/180.t3 = (D/6) / 30 = D/180.Total time T = D * (1/80 + 1/180 + 1/180) = D * (17/720).Average speed v_avg = D / T = 720/17 ≈ 42.35 km/h.

Verification / Alternative check:
Using any concrete D (e.g., 720 km) yields T = 17 h, so v_avg = 720/17 km/h as above.

Why Other Options Are Wrong:
45 or 50 km/h come from averaging speeds directly; 40 km/h ignores that the slowest segment consumes disproportionate time.

Common Pitfalls:
Taking a simple arithmetic mean of 40, 60, and 30 without distance weights, or incorrectly weighting by time rather than distance.

Final Answer:
42.35 km/h