Train A crosses a man standing on a platform in 8 seconds and crosses a 180 m long platform P in 17 seconds. If Train A crosses Train B, which is coming from the opposite direction at 108 km/h, in 8 seconds, then how much time (in seconds) will Train B take to cross platform P?

Introduction / Context:
This is a multi step trains question involving the length and speed of one train, the length of a platform, and the relative speed when another train comes from the opposite direction. We must first determine the speed and length of Train A, then use the crossing time with Train B to find the length of Train B, and finally calculate how long Train B will take to cross the same platform P.

Given Data / Assumptions:

• Train A crosses a man in 8 seconds.
• Train A crosses a platform of length 180 m in 17 seconds.
• Train A crosses Train B in 8 seconds when Train B runs in the opposite direction at 108 km/h.
• We assume constant speeds for both trains.
• Train B must cross the same 180 m platform completely.

Concept / Approach:
When a train crosses a man, the distance is just the length of the train. When it crosses a platform, the distance is the sum of the train length and platform length. From these, we can find Train A speed and length. Then we use relative speed for opposite directions to find Train B length based on the time taken to cross each other. Finally, we use Train B speed and its length plus platform length to compute the time required to cross platform P.

Step-by-Step Solution:
Step 1: Let length of Train A be LA metres and its speed be vA m/s. Step 2: From crossing a man, LA / vA = 8, so LA = 8 * vA. Step 3: From crossing a 180 m platform in 17 seconds, (LA + 180) / vA = 17. Step 4: Substitute LA = 8 * vA into LA + 180 = 17 * vA, giving 8 * vA + 180 = 17 * vA. Step 5: Simplify to 180 = 9 * vA, so vA = 20 m/s. Step 6: Then length of Train A = LA = 8 * 20 = 160 m. Step 7: Speed of Train B = 108 km/h = 108 * 5 / 18 = 30 m/s. Step 8: Relative speed when trains move in opposite directions = 20 + 30 = 50 m/s. Step 9: Let length of Train B be LB. Time to cross each other is 8 seconds, so (LA + LB) / 50 = 8. Step 10: This gives LA + LB = 400, so LB = 400 - 160 = 240 m. Step 11: For Train B crossing platform P, distance = LB + 180 = 240 + 180 = 420 m. Step 12: Time = distance / speed = 420 / 30 = 14 seconds.

Verification / Alternative check:
We can quickly check each piece. Train A speed 20 m/s gives length 160 m from LA = vA * 8. Crossing a 180 m platform gives total distance 340 m, and 340 / 20 = 17 seconds, matching the data. For Train B, length 240 m with Train A in opposite direction: total length 400 m, relative speed 50 m/s, time 400 / 50 = 8 seconds, also correct. Finally, Train B crossing 180 m platform means 420 m at 30 m/s, confirming 14 seconds.

Why Other Options Are Wrong:

• 16 seconds: Would imply total distance 480 m, not 420 m, for the given speed of Train B.
• 11 seconds: Leads to unrealistically short distance and contradicts the derived length of Train B.
• 12 seconds: Gives distance 360 m which is less than LB + platform length.

Common Pitfalls:
Learners may mix up which train length is used in which step or forget that crossing a man only involves the train length. Some also incorrectly use the difference instead of the sum of speeds when trains move in opposite directions. It is also easy to misinterpret times or skip the step of converting units from km/h to m/s when needed, although here Train A speed was directly found in m/s from the time relationships.

Final Answer:
Train B will take 14 seconds to cross platform P.