A train that is 270 metres long is running at a speed of 120 km/h. It crosses another train running in the opposite direction at a speed of 80 km/h in 9 seconds. What is the length of the other train in metres?

Introduction / Context:
This question focuses on relative speed and the total effective distance when two trains running in opposite directions cross each other. It tests whether you can combine speeds given in km/h, convert them to metres per second, and use the crossing time to find the unknown length of the second train.

Given Data / Assumptions:

• Length of first train = 270 metres.
• Speed of first train = 120 km/h.
• Speed of second train = 80 km/h.
• The trains move in opposite directions on parallel tracks.
• Time taken to cross each other completely = 9 seconds.
• We need the length of the second train.

Concept / Approach:
When two trains run in opposite directions, their relative speed is the sum of their individual speeds. The distance that must be covered for them to cross each other completely is the sum of their lengths. We first convert both speeds from km/h to metres per second, add them to obtain the relative speed, multiply by the given time to get total distance, and finally subtract the known train length from this total to find the unknown length.

Step-by-Step Solution:
Step 1: Convert speeds to metres per second using 1 km/h = 5 / 18 metres per second.Step 2: Speed of first train = 120 * 5 / 18 = 600 / 18 = 100 / 3 metres per second.Step 3: Speed of second train = 80 * 5 / 18 = 400 / 18 = 200 / 9 metres per second.Step 4: Relative speed = 100 / 3 + 200 / 9 = (300 + 200) / 9 = 500 / 9 metres per second.Step 5: Total distance covered while crossing = relative speed * time = (500 / 9) * 9 = 500 metres.Step 6: Let the length of the second train be L metres. Then 270 + L = 500, so L = 500 - 270 = 230 metres.

Verification / Alternative check:
If the second train is 230 metres long, total length is 270 + 230 = 500 metres.With relative speed 500 / 9 metres per second, time = distance / speed = 500 / (500 / 9) = 9 seconds.This equals the given crossing time, confirming the correctness of the length.

Why Other Options Are Wrong:
Values such as 240 metres, 260 metres or 320 metres do not satisfy the distance = speed * time relationship when checked back with the same data. For example, a longer train such as 320 metres would yield a total length larger than 500 metres and hence a longer crossing time than 9 seconds. The value 200 metres is too small and leads to a shorter crossing time than stated.

Common Pitfalls:
Common mistakes include forgetting to convert speeds from km/h to metres per second, adding lengths incorrectly, or failing to use the sum of speeds in opposite directions. Some candidates also mix up the roles of the two trains and subtract lengths instead of adding them. Keeping the relationships clear and checking units at each stage helps avoid such errors.

Final Answer:
The length of the other train is 230 metres.