A person travels from town K to town L at 50 km/h and returns from L to K by increasing his speed by 50 percent. What is his average speed (in km/h) for the entire journey?

Introduction / Context:
This is a standard round trip average speed problem where the speeds in the two directions are different. The forward speed is given, and the return speed is a percentage increase over the forward speed. The question tests whether you can compute the average speed over equal distances correctly using the harmonic mean idea or by explicitly using total distance and total time.

Given Data / Assumptions:

• Speed from K to L = 50 km/h.
• Return speed from L to K = 50 percent more than 50 km/h.
• Distances from K to L and L to K are equal.
• Speeds are constant on each leg of the journey.

Concept / Approach:
Find the return speed first. A 50 percent increase means multiplying 50 km/h by 1.5, giving 75 km/h. Let the one way distance be d km. Time going is d / 50 and time returning is d / 75. The total distance is 2d and total time is d / 50 + d / 75. Average speed is then (total distance) / (total time). For equal distances, this is equivalent to the harmonic mean: 2 * v1 * v2 / (v1 + v2).

Step-by-Step Solution:
Forward speed v1 = 50 km/h. Return speed v2 = 50 * 1.5 = 75 km/h. Using harmonic mean for equal distances: Average speed = 2 * v1 * v2 / (v1 + v2). Average speed = 2 * 50 * 75 / (50 + 75). Average speed = 7500 / 125 = 60 km/h.

Verification / Alternative check:
Let the one way distance be 150 km. Time going = 150 / 50 = 3 hours. Time returning = 150 / 75 = 2 hours. Total distance = 300 km, total time = 5 hours. Average speed = 300 / 5 = 60 km/h, which matches the harmonic mean result.

Why Other Options Are Wrong:

• 55 km/h, 58 km/h and 66 km/h do not equal 2 * 50 * 75 / 125 and do not match total distance divided by total time when any reasonable distance is assumed.

Common Pitfalls:
Some students mistakenly average the two speeds arithmetically, (50 + 75) / 2 = 62.5 km/h, which is incorrect for equal distances. Others compute the percentage increase incorrectly or confuse it with a percentage of the final speed. Always remember that average speed over equal distances must be calculated from the total distance and total time, not by simply averaging the two speeds.

Final Answer:
The average speed for the entire journey is 60 km/h.