Understanding the Train Meeting Problem

This problem involves calculating the time when two trains, traveling towards each other from different stations, will meet. We are given the departure and arrival times for both trains.

Analyzing Train Schedules

First, let's determine the total travel time for each train:

• Train A: Departs Station M at 7:55 a.m. and arrives at Station N at 2:55 p.m.
• Train B: Departs Station N at 9:55 a.m. and arrives at Station M at 2:55 p.m.

Calculating Travel Durations

We calculate the duration of the journey for each train:

• Train A's Duration: From 7:55 a.m. to 2:55 p.m. is exactly 7 hours.
• Train B's Duration: From 9:55 a.m. to 2:55 p.m. is exactly 5 hours.

Calculating Meeting Time Using Relative Speed

Let the distance between Station M and Station N be $D$. Let the speeds of Train A and Train B be $V_A$ and $V_B$, respectively.

Using the formula: Speed = Distance / Time

• Speed of Train A: $V_A = \frac{D}{7}$ (units of distance per hour)
• Speed of Train B: $V_B = \frac{D}{5}$ (units of distance per hour)

Adjusting for Different Departure Times

Train A starts at 7:55 a.m., and Train B starts at 9:55 a.m. Train B starts 2 hours later than Train A.

Let's find out where Train A is when Train B starts its journey at 9:55 a.m.

• In the 2 hours from 7:55 a.m. to 9:55 a.m., Train A travels a distance of: Distance = Speed × Time = $V_A \times 2 = \frac{D}{7} \times 2 = \frac{2D}{7}$
• At 9:55 a.m., the remaining distance between Train A and Train B is: Remaining Distance = Total Distance - Distance covered by A = $D - \frac{2D}{7} = \frac{7D - 2D}{7} = \frac{5D}{7}$

Calculating Time to Meet

From 9:55 a.m. onwards, both trains are moving towards each other. We can use the concept of relative speed.

• Relative speed of Train A and Train B = $V_A + V_B$ Relative Speed = $\frac{D}{7} + \frac{D}{5} = \frac{5D + 7D}{35} = \frac{12D}{35}$
• The time it takes for them to meet after 9:55 a.m. is: Time to Meet = Remaining Distance / Relative Speed Time = $\frac{\frac{5D}{7}}{\frac{12D}{35}} = \frac{5D}{7} \times \frac{35}{12D}$
• Canceling $D$ and simplifying: Time = $\frac{5}{7} \times \frac{35}{12} = \frac{5 \times 5}{12} = \frac{25}{12}$ hours

Converting Time and Finding the Meeting Point

Now, we convert $\frac{25}{12}$ hours into hours and minutes:

• $\frac{25}{12}$ hours = $2$ whole hours and $\frac{1}{12}$ of an hour.
• $\frac{1}{12}$ of an hour = $\frac{1}{12} \times 60$ minutes = $5$ minutes.
• So, the time taken to meet after 9:55 a.m. is 2 hours and 5 minutes.

To find the actual meeting time, we add this duration to Train B's departure time (9:55 a.m.):

Meeting Time = 9:55 a.m. + 2 hours 5 minutes

9:55 a.m. + 2 hours = 11:55 a.m.

11:55 a.m. + 5 minutes = 12:00 p.m.

Summary of Calculation

At 9:55 a.m.:

• Train A has traveled $\frac{2D}{7}$.
• Remaining distance = $\frac{5D}{7}$.
• Relative speed = $\frac{12D}{35}$.
• Time to meet after 9:55 a.m. = $\frac{25}{12}$ hours = 2 hours 5 minutes.
• Meeting Time = 9:55 a.m. + 2 hours 5 minutes = 12:00 p.m.

Therefore, Trains A and B meet at 12:00 p.m.