Excluding stoppages, the speed of a train is 120 km/h, but including stoppages its effective speed becomes 50 km/h. For how many minutes in each hour does the train stop?

Introduction / Context:
This problem explores the concept of effective speed when a train makes stoppages. The train moves at a higher speed while in motion, but due to stops its overall average speed reduces. By comparing the distance it could have travelled without stoppages to the actual distance covered including stops, we can determine the amount of time it spends stopped in each hour.

Given Data / Assumptions:

• Speed when the train is moving (excluding stops) = 120 km/h.
• Effective average speed including stops = 50 km/h.
• We seek the total stopping time per hour of clock time.
• Speed while moving is constant.

Concept / Approach:
Assume a reference period of 1 hour of clock time. During this hour, the train moves for some time and stands still for the rest. When moving, it covers distance at 120 km/h, but because of stoppages the overall distance in one hour is only 50 km. Let the running time be t hours and the stopping time be 1 - t hours. We use the relationship distance = speed * time with the effective speed condition to solve for t and then compute the stopping time in minutes.

Step-by-Step Solution:
Step 1: Let total clock period considered = 1 hour. Step 2: Let the train run for t hours and stop for (1 - t) hours in that hour. Step 3: Distance covered while running = 120 * t km. Step 4: Effective distance in 1 hour = 50 km (since effective speed is 50 km/h). Step 5: Therefore, 120 * t = 50. Step 6: Solve for t: t = 50 / 120 = 5/12 hours. Step 7: Stopping time = 1 - 5/12 = 7/12 hours. Step 8: Convert stopping time to minutes: (7/12) * 60 = 7 * 5 = 35 minutes.

Verification / Alternative check:
If the train runs for 5/12 hours (25 minutes) at 120 km/h, the distance covered is 120 * 5/12 = 50 km. In the remaining 7/12 hours (35 minutes) it is stationary and covers no distance, so the total distance in 1 hour is still 50 km, giving an effective speed of 50 km/h, which matches the problem statement.

Why Other Options Are Wrong:
Stopping times of 20, 25, 30, or 40 minutes would all give different effective speeds when recalculated. For example, stopping 20 minutes implies running for 40 minutes at 120 km/h, which would yield 80 km in an hour, not 50 km. Only a 35 minute stoppage per hour fits all the given data.

Common Pitfalls:
Some learners misinterpret the speeds and attempt to average them directly, while others forget that effective speed is based on total distance over total clock time. Confusing the fraction of time stopped with the fraction of distance lost also leads to errors. Treating 1 hour as a reference and carefully applying the distance formula avoids these problems.

Final Answer:
The train stops for 35 minutes in each hour.