A train covers k kilometres at 40 km/h and then another 2k kilometres at 20 km/h. What is the average speed of the train in km/h for the entire 3k kilometres of the journey?

Introduction / Context:
Average speed questions often appear in aptitude tests to check whether candidates understand that average speed is total distance divided by total time, not the simple average of two speeds. Here, a train travels different parts of its journey at different speeds. We must carefully calculate the total time taken and then use the average speed formula over the entire distance of 3k kilometres.

Given Data / Assumptions:

• First part of the journey: distance = k km, speed = 40 km/h.
• Second part of the journey: distance = 2k km, speed = 20 km/h.
• Total distance travelled = k + 2k = 3k km.
• Speeds are constant in each segment.

Concept / Approach:
Average speed is defined as total distance divided by total time. We first compute the time taken for each part of the trip using time = distance / speed. After adding these times, we divide the total distance, which is 3k km, by this total time. The variable k will cancel out, giving a numeric value for the average speed.

Step-by-Step Solution:
Step 1: Time for first part = distance / speed = k / 40 hours. Step 2: Time for second part = 2k / 20 hours. Step 3: Simplify second time: 2k / 20 = k / 10 hours. Step 4: Total time = k / 40 + k / 10. Step 5: Express with common denominator: k / 40 + 4k / 40 = 5k / 40 = k / 8 hours. Step 6: Total distance = 3k km. Step 7: Average speed = total distance / total time = 3k / (k / 8) = 3k * 8 / k = 24 km/h.

Verification / Alternative check:
We can choose a specific convenient value, for example k = 40 km. Then first leg: 40 km at 40 km/h takes 1 hour. Second leg: 80 km at 20 km/h takes 4 hours. Total distance = 120 km, total time = 5 hours. Average speed = 120 / 5 = 24 km/h, which matches the derived result and confirms that k cancels correctly.

Why Other Options Are Wrong:
A simple average of 40 and 20 is 30 km/h, which is incorrect because the train spends more time at the lower speed of 20 km/h. Values such as 26, 28, or 32 km/h do not satisfy the total distance and total time relationship when recalculated carefully, so they must be rejected.

Common Pitfalls:
The most common mistake is to take the arithmetic mean of 40 and 20 instead of using total distance divided by total time. Another error is mishandling algebra with k or failing to harmonise denominators when adding times. Always remember that average speed depends on how long the train travels at each speed, not just the numerical values of the speeds themselves.

Final Answer:
The average speed of the train for the entire journey is 24 kmph.