A train leaves a station A at 7 am and reaches another station B at 11 am. Another train leaves B at 8 am and reaches A at 11.30 am. The two trains cross one another at

Options

A. `8:36`am
B. `8:56`am
C. `9:00`am
D. `9:24` am

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
- Train 1 leaves station A at 7 AM and reaches station B at 11 AM.
- Thus, the travel time for Train 1 is 4 hours (from 7 AM to 11 AM).

- Train 2 leaves station B at 8 AM and reaches station A at 11:30 AM.
- Therefore, the travel time for Train 2 is 3.5 hours (from 8 AM to 11:30 AM).

### Step 2: Calculate the speeds of both trains
- Let's assume the distance between stations A and B is \( x \) kilometers.
- The speed of Train 1 is \( \frac{x}{4} \) km/h (since it covers distance \( x \) in 4 hours).
- The speed of Train 2 is \( \frac{x}{3.5} \) km/h (since it covers distance \( x \) in 3.5 hours).

### Step 3: Determine the time when both trains are in motion
- Train 1 starts at 7 AM, and Train 2 starts at 8 AM.
- By the time Train 2 starts at 8 AM, Train 1 has already traveled for 1 hour.

### Step 4: Calculate the distance covered by Train 1 by 8 AM
- In 1 hour, Train 1 covers:
\[ \text{Distance} = \text{Speed} \times \text{Time} = \frac{x}{4} \times 1 = \frac{x}{4} \text{ km} \]

### Step 5: Calculate the remaining distance between the two trains when Train 2 starts
- The remaining distance between Train 1 and Train 2 when Train 2 starts is:
\[ \text{Remaining Distance} = x - \frac{x}{4} = \frac{3x}{4} \text{ km} \]

### Step 6: Calculate the relative speed of both trains
- The relative speed of the two trains when they are moving towards each other is:
\[ \text{Relative Speed} = \frac{x}{4} + \frac{x}{3.5} \]
To add these, we need a common denominator:
\[ \frac{x}{4} = \frac{7x}{28}, \quad \frac{x}{3.5} = \frac{8x}{28} \]
Thus,
\[ \text{Relative Speed} = \frac{7x}{28} + \frac{8x}{28} = \frac{15x}{28} \text{ km/h} \]

### Step 7: Calculate the time taken to meet after Train 2 starts
- The time taken to meet after Train 2 starts is:
\[ \text{Time} = \frac{\text{Remaining Distance}}{\text{Relative Speed}} = \frac{\frac{3x}{4}}{\frac{15x}{28}} = \frac{3x}{4} \times \frac{28}{15x} = \frac{3 \times 28}{4 \times 15} = \frac{84}{60} = 1.4 \text{ hours} \]

### Step 8: Convert time into minutes
- \( 1.4 \) hours is equivalent to \( 1 \) hour and \( 24 \) minutes.

### Step 9: Determine the time of crossing
- Train 2 starts at 8 AM. Adding \( 1 \) hour and \( 24 \) minutes to this gives:
\[ 8:00 \text{ AM} + 1 \text{ hour} + 24 \text{ minutes} = 9:24 \text{ AM} \]

### Final Answer
The two trains cross each other at **9:24 AM**.