A car travels from one place to another at 24 km/h in one direction and returns at 36 km/h. What is the average speed of the car for the entire journey?

Introduction / Context:
This question deals with average speed over a round trip where the distances in both directions are the same but the speeds are different. It tests your understanding that the average speed is not the simple arithmetic mean of the two speeds. Instead, a special harmonic mean formula must be used when the distance covered in each direction is equal.

Given Data / Assumptions:
Speed while going from the starting point to the destination = 24 km/h.
Speed while returning from the destination back to the starting point = 36 km/h.
The distance between the two places in each direction is the same.
The car moves without any stoppages or delays between the two speeds.
We are required to find the overall average speed for the complete journey, that is, going and coming back.

Concept / Approach:
When a vehicle travels the same distance at two different speeds, the average speed is computed using the harmonic mean: average speed = 2 * v1 * v2 / (v1 + v2). This arises because total time is distance over speed for each leg, and we must divide total distance by total time. Directly averaging the speeds gives the wrong answer and is a common trap in exams.

Step-by-Step Solution:
Step 1: Let the one way distance be D km.Step 2: Time taken to go from place 1 to place 2 = D / 24 hours.Step 3: Time taken to return = D / 36 hours.Step 4: Total distance for the round trip = D + D = 2D km.Step 5: Total time for the round trip = D / 24 + D / 36 hours.Step 6: Combine time: D * (1 / 24 + 1 / 36) = D * (3 / 72 + 2 / 72) = D * (5 / 72) hours.Step 7: Average speed = total distance / total time = 2D / (D * 5 / 72).Step 8: Simplify: 2D * 72 / (5D) = 144 / 5 = 28.8 km/h.

Verification / Alternative check:
The direct harmonic mean formula for equal distances is average speed = 2 * v1 * v2 / (v1 + v2). Substituting v1 = 24 and v2 = 36 gives 2 * 24 * 36 / (24 + 36) = 1728 / 60 = 28.8 km/h, which matches our detailed computation. This confirms that the answer is correct and the reasoning is consistent.

Why Other Options Are Wrong:
30 km/h is close to the arithmetic mean of the two speeds but not the correct harmonic mean, so it overestimates the true average speed.
27.6 km/h and 26.4 km/h do not match the result of the correct formula and arise from incorrect mixing of distance and time values.
31.2 km/h is higher than both the harmonic mean and the lower individual speed, which is impossible for a round trip where some part of the journey is done at only 24 km/h.

Common Pitfalls:
Many students mistakenly take the average of 24 and 36 as (24 + 36) / 2 = 30 km/h without considering that time spent at each speed is different. Remember that average speed is always total distance divided by total time, and for equal distances with two speeds, the harmonic mean formula should be applied. This understanding is critical for mastering time and distance problems.

Final Answer:
The average speed of the car for the entire journey is 28.8 km/h.