Calculating Train Meeting Time Between Stations

This problem asks us to find the exact time when two trains, Train A and Train B, meet while traveling between station M and station N. We are given their departure and arrival times.

Analyzing Train Schedules and Durations

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

• Train A departs station M at 8:00 a.m. and arrives at station N at 3:00 p.m.
• Train B departs station N at 10:00 a.m. and arrives at station M at 3:00 p.m.

We can calculate the duration of their journeys:

• Train A's total travel time ($T_A$): From 8:00 a.m. to 3:00 p.m. is 7 hours.
• Train B's total travel time ($T_B$): From 10:00 a.m. to 3:00 p.m. is 5 hours.

Determining Speeds and Positions

Let's assume the distance between station M and station N is represented by $D$. We can calculate the speed of each train.

• Speed of Train A ($S_A$) = Distance / Time = $D / T_A = D / 7$ (distance units per hour).
• Speed of Train B ($S_B$) = Distance / Time = $D / T_B = D / 5$ (distance units per hour).

Train B starts later than Train A. Train B starts at 10:00 a.m., while Train A started at 8:00 a.m. By the time Train B starts its journey (at 10:00 a.m.), Train A has already been traveling for 2 hours (from 8:00 a.m. to 10:00 a.m.).

Let's calculate the distance Train A has covered by 10:00 a.m.:

Distance covered by A = Speed of A $\times$ Time traveled

Distance covered by A = $S_A \times 2 \text{ hours} = (D/7) \times 2 = \frac{2D}{7}$

At 10:00 a.m., the remaining distance between Train A and Train B is:

Remaining Distance = Total Distance - Distance covered by A

Remaining Distance = $D - \frac{2D}{7} = \frac{7D - 2D}{7} = \frac{5D}{7}$

Calculating the Meeting Time

From 10:00 a.m. onwards, both trains are moving towards each other. To find when they meet, we need to consider their relative speed.

Relative speed ($S_{rel}$) = Speed of A + Speed of B

$S_{rel} = S_A + S_B = \frac{D}{7} + \frac{D}{5}$

To add these fractions, we find a common denominator (35):

$S_{rel} = \frac{5D}{35} + \frac{7D}{35} = \frac{12D}{35}$ (distance units per hour).

Now, we can calculate the time it takes for the trains to cover the remaining distance between them, starting from 10:00 a.m.

Time to meet (after 10:00 a.m.) = Remaining Distance / Relative Speed

$t_{meet} = \frac{5D/7}{12D/35}$

$t_{meet} = \frac{5D}{7} \times \frac{35}{12D}$

The $D$ cancels out:

$t_{meet} = \frac{5}{7} \times \frac{35}{12} = \frac{5 \times 5}{12} = \frac{25}{12}$ hours.

Let's convert this time duration into hours and minutes:

$\frac{25}{12}$ hours = $2$ whole hours and $\frac{1}{12}$ of an hour.

To convert $\frac{1}{12}$ hours to minutes:

Minutes = $\frac{1}{12} \times 60 = 5$ minutes.

So, the time taken for the trains to meet after 10:00 a.m. is 2 hours and 5 minutes.

Finally, we add this time duration to Train B's departure time (10:00 a.m.) to find the meeting time:

Meeting Time = 10:00 a.m. + 2 hours 5 minutes

Meeting Time = 12:05 p.m.