A train leaves Manipur at 6:00 am and reaches Dispurat 10:00 am. Another train leaves Dispurat 8:00 am and reaches Manipur at 11 : 30 am. At what time do the two trains cross each other?

Options

A. `7:56` AM
B. ` 7 : 56` pm
C. `8:56` AM
D. `8:56` PM

Answer

To solve the problem of when the two trains cross each other, we can follow these steps:

### Step 1: Determine the travel times of both trains.
- The first train leaves Manipur at 6:00 am and arrives in Dispur at 10:00 am.
- **Travel time for Train 1** = 10:00 am - 6:00 am = 4 hours.

- The second train leaves Dispur at 8:00 am and arrives in Manipur at 11:30 am.
- **Travel time for Train 2** = 11:30 am - 8:00 am = 3.5 hours.

### Step 2: Calculate the speeds of both trains.
- Let the distance between Manipur and Dispur be \( D \).
- Speed of Train 1 = \( \frac{D}{4} \) (since it takes 4 hours).
- Speed of Train 2 = \( \frac{D}{3.5} \) (since it takes 3.5 hours).

### Step 3: Determine the distance covered by Train 1 before Train 2 starts.
- Train 1 starts at 6:00 am and Train 2 starts at 8:00 am.
- By 8:00 am, Train 1 has been traveling for 2 hours.
- Distance covered by Train 1 in 2 hours = Speed of Train 1 × Time = \( \frac{D}{4} \times 2 = \frac{D}{2} \).

### Step 4: Calculate the remaining distance when Train 2 starts.
- Remaining distance when Train 2 starts = Total Distance - Distance covered by Train 1 = \( D - \frac{D}{2} = \frac{D}{2} \).

### Step 5: Calculate the combined speed of both trains.
- Combined speed of both trains = Speed of Train 1 + Speed of Train 2 = \( \frac{D}{4} + \frac{D}{3.5} \).
- To add these fractions, we need a common denominator. The least common multiple of 4 and 3.5 is 14.
- Speed of Train 1 = \( \frac{D}{4} = \frac{3.5D}{14} \)
- Speed of Train 2 = \( \frac{D}{3.5} = \frac{4D}{14} \)
- Combined speed = \( \frac{3.5D}{14} + \frac{4D}{14} = \frac{7.5D}{14} = \frac{15D}{28} \).

### Step 6: Calculate the time taken to meet after Train 2 starts.
- Time taken to meet = Remaining Distance / Combined Speed = \( \frac{\frac{D}{2}}{\frac{15D}{28}} \).
- Simplifying this gives:
\[ \text{Time} = \frac{D/2}{15D/28} = \frac{28}{30} = \frac{14}{15} \text{ hours}. \]

### Step 7: Convert time to minutes.
- \( \frac{14}{15} \) hours = \( \frac{14}{15} \times 60 \) minutes = 56 minutes.

### Step 8: Determine the time of crossing.
- Train 2 starts at 8:00 am.
- Adding 56 minutes to 8:00 am gives us:
- 8:00 am + 56 minutes = 8:56 am.

### Final Answer:
The two trains cross each other at **8:56 am**.

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