Train Meeting Time Solution

This guide provides a step-by-step solution to calculate the exact time when two trains, Train A and Train B, meet while traveling between stations M and N.

Train A Travel Details

First, we analyze the journey of Train A:

• Departure Station: M
• Departure Time: 8:25 AM
• Arrival Station: N
• Arrival Time: 3:25 PM
• Total Travel Duration for Train A: 7 hours (from 8:25 AM to 3:25 PM).

Train B Travel Details

Next, we look at the details for Train B:

• Departure Station: N
• Departure Time: 10:25 AM
• Arrival Station: M
• Arrival Time: 3:25 PM
• Total Travel Duration for Train B: 5 hours (from 10:25 AM to 3:25 PM).

Speeds of Train A and B

Let the total distance between Station M and Station N be denoted by '$D$'. The speed of each train is calculated using the formula: Speed = Distance / Time.

• Speed of Train A ($S_A$) = $\frac{D}{7}$ distance units per hour.
• Speed of Train B ($S_B$) = $\frac{D}{5}$ distance units per hour.

Distance Before Both Trains Meet

Train A starts 2 hours earlier than Train B. Train A departs at 8:25 AM, while Train B departs at 10:25 AM. We need to calculate the distance Train A covers in these 2 hours.

• Time difference = 10:25 AM - 8:25 AM = 2 hours.
• Distance covered by Train A by 10:25 AM = Speed of Train A $\times$ Time = $S_A \times 2 \text{ hours}$.
• Distance = $\frac{D}{7} \times 2 = \frac{2D}{7}$.
• At 10:25 AM, when Train B starts its journey from Station N, Train A has already traveled $\frac{2D}{7}$ of the total distance from Station M.
• The remaining distance between Train A and Train B at 10:25 AM = Total Distance - Distance covered by A = $D - \frac{2D}{7} = \frac{7D - 2D}{7} = \frac{5D}{7}$.

Relative Speed Calculation

From 10:25 AM onwards, both trains are moving towards each other. To find out how quickly they cover the remaining distance, we calculate their relative speed, which is the sum of their individual speeds.

• Relative Speed = Speed of Train A + Speed of Train B = $S_A + S_B$.
• Relative Speed = $\frac{D}{7} + \frac{D}{5}$.
• To add these fractions, we find a common denominator, which is 35.
• Relative Speed = $\frac{5D}{35} + \frac{7D}{35} = \frac{12D}{35}$ distance units per hour.

Meeting Time Calculation

Now, we calculate the time it takes for the trains to cover the remaining distance ($\frac{5D}{7}$) at their calculated relative speed ($\frac{12D}{35}$). The formula is Time = Distance / Speed.

• Time to meet (after 10:25 AM) = $\frac{\text{Remaining Distance}}{\text{Relative Speed}}$.
• Time = $\frac{5D/7}{12D/35}$.
• To divide by a fraction, we multiply by its reciprocal: Time = $\frac{5D}{7} \times \frac{35}{12D}$.
• Simplifying the expression by canceling out $D$ and common factors: Time = $\frac{5 \times 35}{7 \times 12} = \frac{5 \times 5}{12} = \frac{25}{12}$ hours.
• Next, we convert this time duration from hours into hours and minutes.
• $\frac{25}{12}$ hours can be written as $2$ full hours and $\frac{1}{12}$ of an hour ($25 \div 12 = 2$ with a remainder of 1).
• To convert the fraction of an hour ($\frac{1}{12}$) into minutes, we multiply by 60:
• Minutes = $\frac{1}{12} \times 60 = 5$ minutes.
• So, the trains will meet 2 hours and 5 minutes after Train B started its journey.
• Finally, we add this duration to Train B's departure time (10:25 AM):
• Meeting Time = 10:25 AM + 2 hours 5 minutes = 12:30 PM.

Therefore, the trains A and B will meet at 12:30 PM.