A train leaves station X at 5 a.m. and reaches station Y at 9 a.m. Another train leaves station Y at 7 a.m. and reaches station X at 10:30 a.m. At what time do the two trains cross each other ?

Options

A. `7:36` am
B. `7:56` am
C. `8:36` am
D. `8:56` am

Explanation

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 A leaves station X at 5 a.m. and arrives at station Y at 9 a.m.
- The travel time for Train A is 4 hours (from 5 a.m. to 9 a.m.).

- Train B leaves station Y at 7 a.m. and arrives at station X at 10:30 a.m.
- The travel time for Train B is 3.5 hours (from 7 a.m. to 10:30 a.m.).

**Hint:** Calculate the total travel time for each train by subtracting the departure time from the arrival time.

### Step 2: Calculate the speeds of both trains.
- Let the distance between stations X and Y be \( D \) km.
- Speed of Train A = Distance / Time = \( D / 4 \) km/h.
- Speed of Train B = Distance / Time = \( D / 3.5 \) km/h.

**Hint:** Use the formula for speed, which is distance divided by time.

### Step 3: Set up the equation for the meeting point.
- Train A starts at 5 a.m. and Train B starts at 7 a.m.
- Let \( t \) be the time in hours after 7 a.m. when the two trains meet.
- By the time Train B starts (7 a.m.), Train A has already traveled for 2 hours.
- Distance traveled by Train A by 7 a.m. = Speed of Train A × Time = \( (D / 4) \times 2 = D / 2 \) km.
- Distance traveled by Train B after \( t \) hours = Speed of Train B × Time = \( (D / 3.5) \times t \) km.

**Hint:** Remember that the total distance covered by both trains when they meet should equal the total distance \( D \).

### Step 4: Write the equation for the distances.
- The sum of the distances traveled by both trains when they meet should equal the total distance \( D \):
\[ \frac{D}{2} + \left(\frac{D}{3.5}\right) t = D \]

**Hint:** Rearrange the equation to isolate \( t \).

### Step 5: Solve for \( t \).
- Multiply through by 14 (the LCM of 2 and 3.5) to eliminate the denominators:
\[ 14 \left(\frac{D}{2}\right) + 14 \left(\frac{D}{3.5}\right) t = 14D \]
- This simplifies to:
\[ 7D + 4Dt = 14D \]
- Rearranging gives:
\[ 4Dt = 14D - 7D \]
\[ 4Dt = 7D \]
- Dividing both sides by \( D \) (assuming \( D \neq 0 \)):
\[ 4t = 7 \]
\[ t = \frac{7}{4} = 1.75 \text{ hours} \]

**Hint:** Convert the decimal to hours and minutes for clarity.

### Step 6: Convert \( t \) to hours and minutes.
- \( 1.75 \) hours is \( 1 \) hour and \( 0.75 \times 60 = 45 \) minutes.
- Thus, \( t = 1 \) hour and \( 45 \) minutes.

**Hint:** Remember to add this time to the starting time of Train B (7 a.m.).

### Step 7: Find the meeting time.
- Adding \( 1 \) hour and \( 45 \) minutes to \( 7 \) a.m. gives:
- \( 7:00 + 1:45 = 8:45 \) a.m.

**Final Answer:** The two trains cross each other at **8:45 a.m.**