A train of length 100 m crosses another train of length 150 m, running on a parallel track in the opposite direction, in 9 seconds. If the speed of the 150 m long train is 40 km/hr, what is the speed of the other 100 m long train in km/hr?

Introduction / Context:
This question uses the concept of relative speed for two trains moving in opposite directions on parallel tracks. When they cross each other, the effective distance is the sum of their lengths, and the relative speed is the sum of their individual speeds. We are given the speed of one train and the total time to cross, and must find the speed of the other train.

Given Data / Assumptions:
- Length of first train (unknown speed) = 100 m.
- Length of second train (known speed) = 150 m.
- Time taken for the trains to completely cross each other = 9 seconds.
- Speed of the 150 m train = 40 km/hr.
- Trains are moving in opposite directions at constant speeds.

Concept / Approach:
When two trains move in opposite directions, their relative speed is the sum of their speeds. The distance covered during the crossing is the sum of their lengths. Using distance = speed * time, we can compute the relative speed in meters per second, then subtract the known train's speed (converted to m/s) to get the unknown speed. Finally, convert that unknown speed back to km/hr.

Step-by-Step Solution:
Step 1: Total distance to be covered while crossing = 100 m + 150 m = 250 m.Step 2: Time taken to cross = 9 seconds.Step 3: Relative speed in m/s = distance / time = 250 / 9 m/s.Step 4: Convert speed of 150 m train to m/s: 40 km/hr = 40 * (5 / 18) = 200 / 18 = 100 / 9 m/s.Step 5: Let the speed of the 100 m train be v m/s. Since directions are opposite, relative speed = v + 100 / 9.Step 6: Set v + 100 / 9 = 250 / 9. So v = (250 / 9) - (100 / 9) = 150 / 9 = 50 / 3 m/s.Step 7: Convert v to km/hr by multiplying by 18 / 5: v = (50 / 3) * (18 / 5) = (50 * 18) / 15 = 900 / 15 = 60 km/hr.

Verification / Alternative check:
Check the combined speed. 60 km/hr + 40 km/hr = 100 km/hr. In m/s, 100 km/hr = 100 * (5 / 18) = 500 / 18 = 250 / 9 m/s. Time to cross 250 m at 250 / 9 m/s is distance / speed = 250 / (250 / 9) = 9 seconds, which matches the given value. This confirms that 60 km/hr is correct.

Why Other Options Are Wrong:
30 km/hr or 42 km/hr would make the relative speed too small, leading to a larger crossing time than 9 seconds. 48 km/hr or 50 km/hr similarly fail to produce the required 250 / 9 m/s relative speed when added to 40 km/hr. Only 60 km/hr gives the correct combination.

Common Pitfalls:
Common mistakes include forgetting to convert from km/hr to m/s, subtracting the speeds instead of adding them for opposite directions, or using only one train's length instead of the sum of both lengths. Consistently applying the relative speed concept and unit conversions avoids these errors.

Final Answer:
The speed of the 100 m long train is 60 km/hr.