Two trains are moving in opposite directions at speeds of 30 km/h and 45 km/h respectively. Their lengths are 450 metres and 550 metres respectively. How many seconds does the slower train take to completely cross the faster train?

Introduction / Context:
This question involves two trains moving in opposite directions, which means their relative speed is the sum of their individual speeds. To find how long one train takes to cross the other, we must consider the total distance (sum of their lengths) and the relative speed. The result will be the time during which they completely pass each other.

Given Data / Assumptions:

• Speed of the slower train = 30 km/h.
• Speed of the faster train = 45 km/h.
• Length of the slower train = 450 m.
• Length of the faster train = 550 m.
• The trains move in opposite directions.
• Speeds are constant and track is straight.

Concept / Approach:
When two bodies move in opposite directions, their relative speed equals the sum of their individual speeds. For one train to fully cross the other, it must cover a distance equal to the sum of both lengths. We convert speeds from km/h to m/s, add them to get relative speed, and then use time = distance / speed to find the crossing time in seconds.

Step-by-Step Solution:
Step 1: Total distance to be covered = 450 + 550 = 1000 m. Step 2: Convert speeds to m/s using 1 km/h = 5/18 m/s. Step 3: 30 km/h = 30 * 5/18 = 150/18 = 8.333... m/s. Step 4: 45 km/h = 45 * 5/18 = 225/18 = 12.5 m/s. Step 5: Relative speed = 8.333... + 12.5 = 20.833... m/s. Step 6: Time to cross = distance / relative speed = 1000 / 20.833... seconds. Step 7: Calculate: 1000 / (75 * 5/18) can be simplified as 1000 / (75 * 1000 / 3600) = 3600 / 150 = 48 seconds.

Verification / Alternative check:
Using the exact relation, relative speed = (30 + 45) km/h = 75 km/h. Time in hours = distance in km / speed in km/h = 1 km / 75 km/h = 1/75 hours. Converting to seconds: (1/75) * 3600 = 3600 / 75 = 48 seconds, which matches the earlier result and confirms correctness.

Why Other Options Are Wrong:
Values such as 54, 62, or 72 seconds would correspond to smaller relative speeds than 75 km/h for the same 1 km distance. The distractor 52 seconds is close but cannot be obtained from the exact speed and distance relationship. Only 48 seconds satisfies the mathematical relationship precisely.

Common Pitfalls:
Some learners forget to convert metres to kilometres or mix units, which leads to incorrect answers. Others may mistakenly subtract the train speeds even though the trains move in opposite directions. Remember to add speeds for opposite directions and to include the sum of lengths when calculating the distance for crossing.

Final Answer:
The slower train takes 48 seconds to completely cross the faster train.