A fast moving superfast express train crosses another passenger train in 20 seconds. The speed of the faster train is 72 km per hour and the speed of the slower train is 27 km per hour. The faster train is 100 metres long. If both trains are moving in the same direction on parallel tracks, what is the length of the slower train in metres?

Introduction / Context:
This is a standard relative speed problem involving two trains moving in the same direction on parallel tracks. The faster train overtakes or crosses the slower train in a given time. We know the speeds of both trains and the length of the faster train. When one train completely crosses the other, the relative speed and the sum of their lengths determine the time taken. From this, we can find the unknown length of the slower train.

Given Data / Assumptions:

• Speed of the faster train = 72 km per hour.
• Speed of the slower train = 27 km per hour.
• Length of the faster train = 100 metres.
• Time taken to cross the slower train completely = 20 seconds.
• Both trains run in the same direction on parallel tracks.

Concept / Approach:
When two objects move in the same direction, their relative speed is the difference of their speeds. Here, the faster train effectively moves past the slower one at this relative speed. The distance covered in the crossing process is equal to the sum of the lengths of the two trains, because the faster train has to pass the entire length of the slower train plus its own length. Using the basic relation distance = speed × time, we can solve for the unknown length of the slower train in metres.

Step-by-Step Solution:
Step 1: Calculate the relative speed when both trains move in the same direction. Step 2: Relative speed = 72 km/h − 27 km/h = 45 km/h. Step 3: Convert this relative speed into metres per second. Since 1 km/h = (5/18) m/s, we have 45 km/h = 45 × 5 / 18 = 225 / 18 = 12.5 m/s. Step 4: Let L be the length of the slower train in metres. The total distance covered during crossing is L + 100 metres. Step 5: Time taken to cross is 20 seconds, so distance = speed × time gives L + 100 = 12.5 × 20. Step 6: Compute the right side: 12.5 × 20 = 250. Step 7: Therefore L + 100 = 250, so L = 250 − 100 = 150 metres.

Verification / Alternative check:
We can verify by checking if the calculated length is consistent. The total length to be crossed is 100 + 150 = 250 metres. At 12.5 m/s, the time required is 250 / 12.5 = 20 seconds, which matches the given time. This confirms that the length of the slower train has been computed correctly.

Why Other Options Are Wrong:
If the slower train were 100 metres, the total length would be 200 metres, and time would be 200 / 12.5 = 16 seconds, not 20. For 125 metres, the total length is 225 metres, giving 18 seconds. For 175 metres, the total length is 275 metres, giving 22 seconds. A length of 200 metres would give 300 metres total and 24 seconds. None of these match the required 20 seconds; only 150 metres does.

Common Pitfalls:
Common mistakes include adding the speeds instead of subtracting them when trains move in the same direction, forgetting to convert speeds from km per hour to metres per second, or using only one train length instead of the sum of both lengths. Carefully distinguishing between opposite and same direction cases and being systematic with units avoids these errors.

Final Answer:
The length of the slower train is 150 metres.