Two trains of equal length are running in opposite directions on parallel tracks. They pass a fixed pole in 18 seconds and 12 seconds respectively. In how many seconds will the two trains completely cross each other when running in opposite directions?

Introduction / Context:
This is another standard trains question dealing with two trains of equal length moving in opposite directions. It tests understanding of how to use times taken to pass a fixed point to deduce speeds, and then how to use relative speed to determine the time required for the trains to completely cross each other.

Given Data / Assumptions:

• Both trains have the same length, say L metres.
• The first train passes a pole in 18 seconds.
• The second train passes a pole in 12 seconds.
• The trains run in opposite directions on parallel tracks with uniform speeds.
• We must find the time taken for them to cross each other completely.

Concept / Approach:
When a train passes a pole, the distance covered is its own length. Therefore, for each train, speed equals length divided by time. When two trains cross each other while moving in opposite directions, the effective distance to be covered is the sum of their lengths, and the relative speed is the sum of their individual speeds. We express both speeds in terms of the common length L to find the required crossing time.

Step-by-Step Solution:
Step 1: Let the length of each train be L metres.Step 2: Speed of first train = L / 18 metres per second.Step 3: Speed of second train = L / 12 metres per second.Step 4: Relative speed when moving in opposite directions = L / 18 + L / 12.Step 5: Simplify the relative speed: L / 18 + L / 12 = (2L + 3L) / 36 = 5L / 36 metres per second.Step 6: Effective distance to be covered while crossing each other = L + L = 2L metres.Step 7: Time to cross each other = distance / relative speed = 2L / (5L / 36) = 2L * 36 / (5L) = 72 / 5 seconds = 14.4 seconds.

Verification / Alternative check:
Pick a convenient length, for example L = 180 metres.Then speed of first train = 180 / 18 = 10 metres per second, and speed of second train = 180 / 12 = 15 metres per second.Relative speed = 10 + 15 = 25 metres per second and total distance = 360 metres.Time = 360 / 25 = 14.4 seconds, confirming the result.

Why Other Options Are Wrong:
Times such as 15.5 seconds, 18.8 seconds or 20.2 seconds arise from algebraic errors or from using the difference of speeds instead of their sum. The value 16 seconds is an approximate guess and does not satisfy the correct distance and speed calculations.

Common Pitfalls:
A common mistake is to forget that the total distance during crossing is the sum of the lengths of both trains. Another error is confusing relative speed in the case of objects moving in the same direction (difference of speeds) with those moving in opposite directions (sum of speeds). Also, candidates sometimes cancel the length L incorrectly in intermediate steps, leading to wrong numerical values.

Final Answer:
The two trains will completely cross each other in 14.4 seconds.