A bus travels half of the total distance between two places at 40 km/h and the remaining half at 60 km/h. What is the average speed of the bus for the entire journey?

Introduction / Context:
This question is another example of an average speed problem where the distance is split into two equal halves with different speeds. It tests whether you know how to handle average speed correctly when distance segments, rather than time segments, are equal. The harmonic mean formula for equal distances is again the central idea.

Given Data / Assumptions:
The bus travels the first half of the distance at 40 km/h.
The bus travels the second half of the distance at 60 km/h.
Both halves represent equal distances, not equal times.
There are no intermediate stops or changes in speed apart from the two given speeds.
We are asked to find the average speed for the whole journey in km/h.

Concept / Approach:
When a vehicle covers equal distances at two different speeds v1 and v2, the average speed is not the arithmetic mean but the harmonic mean: average speed = 2 * v1 * v2 / (v1 + v2). This formula comes from adding the times for each half distance and dividing the total distance by the total time. Applying it correctly yields the average speed over the complete trip.

Step-by-Step Solution:
Step 1: Let the total distance be 2D km, so each half is D km.Step 2: Time for first half at 40 km/h = D / 40 hours.Step 3: Time for second half at 60 km/h = D / 60 hours.Step 4: Total distance = 2D km.Step 5: Total time = D / 40 + D / 60 hours.Step 6: Combine time: D * (1 / 40 + 1 / 60) = D * (3 / 120 + 2 / 120) = D * (5 / 120) = D * (1 / 24).Step 7: Average speed = total distance / total time = 2D / (D / 24).Step 8: Simplify: 2D * 24 / D = 48 km/h.

Verification / Alternative check:
Using the harmonic mean formula directly with v1 = 40 and v2 = 60 gives average speed = 2 * 40 * 60 / (40 + 60) = 4800 / 100 = 48 km/h. This matches the detailed derivation. Hence both methods confirm that the average speed is 48 km/h.

Why Other Options Are Wrong:
50 km/h is the arithmetic mean of 40 and 60, which is not applicable for equal distances.
46 km/h and 42 km/h are values that might arise from incorrect handling of the time formula or arithmetic mistakes.
52 km/h is too high and cannot be the overall average when the bus travels part of the journey at only 40 km/h.

Common Pitfalls:
The most common error is taking the simple average (40 + 60) / 2 = 50 km/h. Another frequent mistake is to assume equal times were spent at each speed, which is not given here. Always check whether distances or times are equal, and pick the correct method accordingly. For equal distances with two speeds, the harmonic mean approach is the safest and most accurate.

Final Answer:
The average speed of the bus for the entire journey is 48 km/h.