To determine the time when both trains meet, we can follow these steps:

1. Determine the total time taken by each train for its journey:

• Train 1 (leaves station A at 5:00 AM and reaches station B at 9:00 AM):

  • Time taken = 9:00 AM - 5:00 AM = 4 hours.

• Train 2 (leaves station B at 7:00 AM and reaches station A at 10:30 AM):

  • Time taken = 10:30 AM - 7:00 AM = 3.5 hours.

2. Calculate the relative positions of both trains when Train 2 departs:

At 7:00 AM (when Train 2 departs), Train 1 has been traveling for 2 hours (since it started at 5:00 AM). Since Train 1 takes 4 hours to complete the journey, it has covered half the distance (2 hours out of 4 hours).

3. Calculate the speed ratios of the trains:

Since both trains are traveling toward each other, we need to consider their relative speeds. Let's assume that the distance between Station A and Station B is $D$.

• Speed of Train 1 = $\frac{D}{4}$ (distance covered per hour).
• Speed of Train 2 = $\frac{D}{3.5}$.

4. Set up the equation for the meeting time:

Let the time after 7:00 AM when both trains meet be $t$ hours. In that time:

• Train 1 will cover a distance of $\frac{D}{4} \times t$,
• Train 2 will cover a distance of $\frac{D}{3.5} \times t$.

Since Train 1 had already covered half the distance before 7:00 AM, the remaining distance is $\frac{D}{2}$. Therefore, the sum of the distances covered by both trains should equal $\frac{D}{2}$:

$\frac{D}{4} \times t + \frac{D}{3.5} \times t = \frac{D}{2}$

5. Solve for $t$:

Dividing the entire equation by $D$:

$\frac{t}{4} + \frac{t}{3.5} = \frac{1}{2}$

To simplify the fractions, we convert 3.5 to $\frac{7}{2}$:

$\frac{t}{4} + \frac{2t}{7} = \frac{1}{2}$

Find the least common denominator (28) and rewrite the equation:

$\frac{7t}{28} + \frac{8t}{28} = \frac{14}{28}$

Simplifying:

$\frac{15t}{28} = \frac{14}{28}$

Multiply both sides by 28:

$15t = 14$

Solve for $t$:

$t = \frac{14}{15} \text{ hours} \approx 0.933 \text{ hours}.$

This is approximately 56 minutes.

6. Determine the time:

Since both trains meet 56 minutes after 7:00 AM, they will meet at:

$7:00 \, \text{AM} + 0:56 \, \text{minutes} = 7:56 \, \text{AM}.$

Answer: The two trains will meet at 7:56 AM.

---

Would you like any further details, or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the meeting time change if Train 2 started earlier?
2. What if Train 1 had a different travel time, say 5 hours instead of 4?
3. How can we calculate the distance between the stations if we knew the speeds of the trains?
4. What would happen if both trains started at the same time but from different stations?
5. How would the solution change if the trains were moving at constant but different accelerations?

Tip: When solving problems with relative motion, focus on the combined speed or distance between the objects moving towards or away from each other.