A motorist travels from one place to another 150 km away at an average speed of 50 km/h and returns to the starting point at an average speed of 30 km/h. What is his average speed for the entire round trip in km/h?

Introduction / Context:
This question is about average speed for a round trip where the distance is the same in both directions but the speeds are different. Many students go wrong by taking the simple average of the two speeds. However, for equal distances, the correct average speed is the harmonic mean of the two speeds, which accounts for different times taken at each speed.

Given Data / Assumptions:

One way distance between the two places is 150 km.
Speed while going from start to destination is 50 km/h.
Speed while returning is 30 km/h.
The total journey is a round trip of 300 km.
We must find the overall average speed for the entire journey.

Concept / Approach:
Average speed is defined as total distance divided by total time. For a two leg journey with equal distances but different speeds, we can either compute the total time for each leg and divide total distance by total time, or use the harmonic mean formula: average speed = 2 * v1 * v2 / (v1 + v2) where v1 and v2 are the two speeds. Both approaches yield the same result, but the harmonic mean formula is faster.

Step-by-Step Solution:
Step 1: Compute time taken for each leg of the journey. Time to go = distance / speed = 150 / 50 = 3 hours. Time to return = 150 / 30 = 5 hours. Step 2: Compute total distance and total time. Total distance for round trip = 150 + 150 = 300 km. Total time = 3 + 5 = 8 hours. Step 3: Compute average speed using total distance and total time. Average speed = total distance / total time = 300 / 8 = 37.5 km/h.

Verification / Alternative Check:
Using the harmonic mean formula for equal distances, average speed = 2 * v1 * v2 / (v1 + v2). Here v1 = 50 and v2 = 30. So, average speed = 2 * 50 * 30 / (50 + 30) = 3000 / 80 = 37.5 km/h. This matches our previous calculation, confirming the correctness of the answer.

Why Other Options Are Wrong:
Option 35 km/h would result from incorrect handling of the time or an arithmetic slip in division.
Option 36 km/h may come from wrongly averaging times or using a rough estimate instead of exact calculation.
Option 38.2 km/h does not correspond to any standard formula for combining speeds over equal distances.

Common Pitfalls:
The most common mistake is taking the arithmetic mean of the two speeds, that is (50 + 30) / 2 = 40 km/h, which is wrong because it ignores the different times spent at each speed. Another common error is forgetting to double the distance for the round trip or miscalculating the time for one of the legs. Always remember that average speed must be based on total distance divided by total time.

Final Answer:
The average speed for the entire round trip is 37.5 km/h.