Train Meeting Time Calculation

This problem involves calculating the time when two trains, traveling in opposite directions between two stations, will meet. We need to determine the speeds of the trains and their relative speed to find the meeting point.

Calculating Train Travel Times

First, let's find the total time each train takes for its journey:

• Train A: Departs station M at 6:40 AM and arrives at station N at 3:40 PM on the same day. The total travel time for Train A is calculated as: From 6:40 AM to 3:40 PM is 9 hours.
• Train B: Departs station N at 8:40 AM and arrives at station M at 2:40 PM on the same day. The total travel time for Train B is calculated as: From 8:40 AM to 2:40 PM is 6 hours.

Calculating Train Speeds

Let's assume the distance between station M and station N is $D$ kilometers. We can now express the speed of each train:

• Speed of Train A ($S_A$) = $\frac{\text{Distance}}{\text{Time}} = \frac{D}{9}$ km/hr.
• Speed of Train B ($S_B$) = $\frac{\text{Distance}}{\text{Time}} = \frac{D}{6}$ km/hr.

Determining the Meeting Point Calculation

Train B starts later than Train A. We need to find out how far Train A has traveled before Train B starts its journey. Train B starts at 8:40 AM, while Train A starts at 6:40 AM. The time difference is:

Time difference = 8:40 AM - 6:40 AM = 2 hours.

In these 2 hours, Train A covers a certain distance:

Distance covered by Train A by 8:40 AM = $S_A \times \text{Time} = \frac{D}{9} \times 2 = \frac{2D}{9}$.

At 8:40 AM, when Train B starts from station N, Train A is already $\frac{2D}{9}$ km away from station M. The remaining distance between the two trains is:

Remaining Distance = Total Distance - Distance covered by Train A

Remaining Distance = $D - \frac{2D}{9} = \frac{9D - 2D}{9} = \frac{7D}{9}$.

Now, both trains are moving towards each other. To find the time it takes for them to meet, we calculate their relative speed. The relative speed is the sum of their individual speeds:

Relative Speed = $S_A + S_B = \frac{D}{9} + \frac{D}{6}$.

To add these fractions, we find a common denominator, which is 18:

Relative Speed = $\frac{2D}{18} + \frac{3D}{18} = \frac{5D}{18}$ km/hr.

The time it will take for the trains to meet after 8:40 AM is calculated by dividing the remaining distance by their relative speed:

Time to meet = $\frac{\text{Remaining Distance}}{\text{Relative Speed}} = \frac{7D/9}{5D/18}$.

Time to meet = $\frac{7D}{9} \times \frac{18}{5D} = \frac{7 \times 18}{9 \times 5} = \frac{7 \times 2}{5} = \frac{14}{5}$ hours.

Now, we convert this time into hours and minutes:

$\frac{14}{5}$ hours = $2 \frac{4}{5}$ hours.

To convert the fraction of an hour to minutes:

Minutes = $\frac{4}{5} \times 60 = 4 \times 12 = 48$ minutes.

So, the trains will meet 2 hours and 48 minutes after 8:40 AM.

Final Meeting Time Calculation

To find the exact meeting time, we add this duration to Train B's departure time (8:40 AM):

Meeting Time = 8:40 AM + 2 hours 48 minutes.

Adding 2 hours to 8:40 AM gives 10:40 AM.

Adding the remaining 48 minutes to 10:40 AM gives 11:28 AM.

Therefore, the trains A and B meet at 11:28 AM.