Train Meeting Time Calculation

This problem involves calculating the time when two trains, traveling towards each other from different stations, will meet. We need to determine their speeds and relative positions.

1. Calculate Travel Times

• Train A: Leaves M at 6:25 AM, reaches N at 3:25 PM.
  • Travel duration = 3:25 PM - 6:25 AM = 9 hours.
• Train B: Leaves N at 8:25 AM, reaches M at 2:25 PM.
  • Travel duration = 2:25 PM - 8:25 AM = 6 hours.

2. Calculate Speeds

Let the distance between Station M and Station N be D.

• 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.

3. Determine Relative Positions at 8:25 AM

We use 8:25 AM, the departure time of Train B, as a reference point.

• Train A departs at 6:25 AM. By 8:25 AM, it has traveled for 2 hours (8:25 AM - 6:25 AM = 2 hours).
• Distance covered by Train A by 8:25 AM = $S_A \times 2 = \frac{D}{9} \times 2 = \frac{2D}{9}$.
• At 8:25 AM, Train A is at a distance of $\frac{2D}{9}$ from Station M.
• At 8:25 AM, Train B is at Station N (starting its journey).
• The distance between Train A and Train B at 8:25 AM is $D - \frac{2D}{9} = \frac{9D - 2D}{9} = \frac{7D}{9}$.

4. Calculate Relative Speed

Since the trains are traveling towards each other, their speeds add up.

• Relative Speed = $S_A + S_B = \frac{D}{9} + \frac{D}{6}$.
• To add these fractions, find a common denominator (18):
• Relative Speed = $\frac{2D}{18} + \frac{3D}{18} = \frac{5D}{18}$ km/hr.

5. Calculate Time to Meet After 8:25 AM

Time = $\frac{\text{Distance between trains}}{\text{Relative Speed}}$

• Time = $\frac{7D/9}{5D/18} = \frac{7D}{9} \times \frac{18}{5D}$.
• Cancel out D and simplify: Time = $\frac{7 \times 18}{9 \times 5} = \frac{7 \times 2}{5} = \frac{14}{5}$ hours.

6. Convert Time and Find Final Meeting Time

• Convert $\frac{14}{5}$ hours into hours and minutes:
• $\frac{14}{5}$ hours = $2 \frac{4}{5}$ hours.
• $\frac{4}{5}$ hours = $\frac{4}{5} \times 60$ minutes = 48 minutes.
• So, the trains meet 2 hours and 48 minutes after 8:25 AM.
• Meeting Time = 8:25 AM + 2 hours 48 minutes.
• Meeting Time = 10:25 AM + 48 minutes = 11:13 AM.

Therefore, the trains A and B meet at 11:13 AM.