A and B are two railway stations 90 km apart. A train leaves A at 9:00 am, heading towards B at a speed of 40 km/hr. Another train leaves B at 10:30 am, heading towards A at a speed of 20 km/hr. The trains meet each other at

The correct option is 11 : 00 am.

Let us break down the solution step-by-step to understand why this is the correct meeting time:

Step 1: Understand the initial positions and times
The distance between station A and station B is 90 km.
Train 1 leaves station A at 9:00 am traveling towards B at a speed of 40 km/hr.
Train 2 leaves station B at 10:30 am traveling towards A at a speed of 20 km/hr.

Step 2: Calculate the distance covered by the first train before the second train starts
The second train starts at 10:30 am. Between 9:00 am and 10:30 am, Train 1 has already been traveling for 1.5 hours (or 1 hour and 30 minutes).
Using the formula:
$$\text{Distance}=\text{Speed}\times \text{Time}$$
The distance covered by Train 1 in 1.5 hours is:
$$40\text{ km/hr}\times 1.5\text{ hr}=60\text{ km}$$

Step 3: Calculate the remaining distance between the two trains at 10:30 am
At 10:30 am, Train 1 is 60 km away from station A, moving towards B.
The total distance between A and B is 90 km, so the remaining distance separating the two trains is:
$$90\text{ km}-60\text{ km}=30\text{ km}$$

Step 4: Calculate the relative speed of the two trains
Since the two trains are moving towards each other, their relative speed is the sum of their individual speeds:
$$\text{Relative Speed}=40\text{ km/hr}+20\text{ km/hr}=60\text{ km/hr}$$

Step 5: Calculate the time taken to meet after 10:30 am
The time required to cover the remaining 30 km at the relative speed of 60 km/hr is:
$$\text{Time}=\frac{\text{Remaining Distance}}{\text{Relative Speed}}=\frac{30}{60}\text{ hours}=0.5\text{ hours}$$
Since 0.5 hours is equal to 30 minutes, the trains will meet 30 minutes after the second train starts.

Step 6: Determine the exact meeting time
Adding 30 minutes to the start time of the second train (10:30 am):
$$10:30\text{ am}+30\text{ minutes}=11:00\text{ am}$$
Thus, the trains meet each other at 11:00 am.