A person starts to walk at 7 am from A and reaches B at 11 am and another person starts to walk from B at 8 am and reaches A at 11:30 am. Find the time at which they will meet each other.

Answer

To solve the problem, we need to determine when two people walking towards each other will meet. Here's a step-by-step breakdown of the solution:

### Step 1: Determine the time taken by each person to travel from A to B and B to A.
- The first person starts walking from A at 7 AM and reaches B at 11 AM.
- This means the first person takes 4 hours to cover the distance from A to B.

**Time taken by Person 1 = 11 AM - 7 AM = 4 hours**

- The second person starts walking from B at 8 AM and reaches A at 11:30 AM.
- This means the second person takes 3.5 hours to cover the distance from B to A.

**Time taken by Person 2 = 11:30 AM - 8 AM = 3.5 hours**

### Step 2: Calculate the speeds of both persons.
- Let’s assume the distance between A and B is D km.
- The speed of the first person (P1) is calculated as:

\[ \text{Speed of P1} = \frac{D}{\text{Time taken by P1}} = \frac{D}{4} \text{ km/h} \]

- The speed of the second person (P2) is calculated as:

\[ \text{Speed of P2} = \frac{D}{\text{Time taken by P2}} = \frac{D}{3.5} \text{ km/h} \]

### Step 3: Find the relative speed when both persons are walking towards each other.
- The relative speed when both persons are walking towards each other is the sum of their speeds:

\[ \text{Relative Speed} = \text{Speed of P1} + \text{Speed of P2} = \frac{D}{4} + \frac{D}{3.5} \]

### Step 4: Calculate the distance covered by the first person before the second person starts walking.
- The first person starts walking at 7 AM, and the second person starts at 8 AM. Therefore, the first person walks for 1 hour before the second person starts.

- In that 1 hour, the distance covered by the first person is:

\[ \text{Distance covered by P1 in 1 hour} = \text{Speed of P1} \times 1 = \frac{D}{4} \text{ km} \]

### Step 5: Calculate the remaining distance when the second person starts walking.
- The remaining distance between the two persons when the second person starts walking is:

\[ \text{Remaining Distance} = D - \frac{D}{4} = \frac{3D}{4} \text{ km} \]

### Step 6: Set up the equation to find the time taken to meet after the second person starts walking.
- Let t be the time taken (in hours) for them to meet after 8 AM. The equation can be set up as:

\[ \text{Remaining Distance} = \text{Relative Speed} \times t \]

Substituting the values:

\[ \frac{3D}{4} = \left(\frac{D}{4} + \frac{D}{3.5}\right) \times t \]

### Step 7: Solve for t.
- Simplifying the relative speed:

\[ \frac{D}{4} + \frac{D}{3.5} = \frac{7D + 4D}{28} = \frac{11D}{28} \]

- Now substituting back into the equation:

\[ \frac{3D}{4} = \frac{11D}{28} \times t \]

- Dividing both sides by D (assuming D ≠ 0):

\[ \frac{3}{4} = \frac{11}{28} \times t \]

- Cross-multiplying to solve for t:

\[ 3 \times 28 = 11 \times 4t \implies 84 = 44t \implies t = \frac{84}{44} = \frac{21}{11} \text{ hours} \]

### Step 8: Convert t into hours and minutes.
- Converting \(\frac{21}{11}\) hours into hours and minutes:

\[ \frac{21}{11} \text{ hours} \approx 1.909 \text{ hours} \approx 1 \text{ hour and } 54 \text{ minutes} \]

### Step 9: Determine the meeting time.
- Since the second person starts walking at 8 AM, we add 1 hour and 54 minutes to 8 AM:

\[ 8:00 AM + 1:54 = 9:54 AM \]

### Final Answer:
The two persons will meet at **9:54 AM**.

---