Ram and Hari started from A and B, towards B and A at 6 : 00 am and 7 : 00 am respectively. They meet each other at 9 : 00 am and continued towards their respective destinations. Ram reaching B turns back and catches up with Hari before Hari reaches A at 11 : 00 am. At what time will Hari reach at A?

Options

A. a. `7:00` pm
B. b. `4.00` pm
C. c. `5.00` pm
D. d. `6:00` pm

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the starting times and meeting point
- Ram starts from point A at 6:00 AM.
- Hari starts from point B at 7:00 AM.
- They meet at point C at 9:00 AM.

### Step 2: Calculate the time each traveled until they met
- Ram travels from 6:00 AM to 9:00 AM, which is 3 hours.
- Hari travels from 7:00 AM to 9:00 AM, which is 2 hours.

### Step 3: Establish the relationship between their speeds
- Let Ram's speed be \( R \) and Hari's speed be \( H \).
- The distance covered by Ram in 3 hours is \( 3R \).
- The distance covered by Hari in 2 hours is \( 2H \).
- Since they meet at point C, we have:
\[ 3R = 2H \]

### Step 4: Express Ram's speed in terms of Hari's speed
- Rearranging the equation gives:
\[ R = \frac{2H}{3} \]

### Step 5: Assume a value for Hari's speed
- Let's assume Hari's speed \( H = 1 \) km/h.
- Then, Ram's speed \( R = \frac{2 \times 1}{3} = \frac{2}{3} \) km/h.

### Step 6: Calculate the distance from A to C
- The distance from A to C (where they meet) can be calculated as:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- For Ram:
\[ \text{Distance from A to C} = R \times 3 = \left(\frac{2}{3}\right) \times 3 = 2 \text{ km} \]

### Step 7: Calculate the distance from C to B
- Since Ram and Hari meet at C, and Ram continues to B:
- The distance from C to B is the total distance minus the distance from A to C.
- If we assume the total distance from A to B is \( D \), then:
\[ D = 2 + \text{Distance from C to B} \]

### Step 8: Calculate the time taken by Ram to reach B
- Ram reaches B at 9:00 AM + time taken to cover distance from C to B.
- If the distance from C to B is also \( 2 \) km, then:
\[ \text{Time taken} = \frac{2 \text{ km}}{R} = \frac{2}{\frac{2}{3}} = 3 \text{ hours} \]
- Therefore, Ram reaches B at 12:00 PM.

### Step 9: Ram turns back and meets Hari
- Ram turns back and meets Hari before he reaches A at 11:00 AM.
- The distance Hari covers from C to D (where they meet again) is calculated based on his speed.

### Step 10: Calculate the remaining distance for Hari to reach A
- If Hari meets Ram at D at 11:00 AM, he started from C at 9:00 AM and traveled for 2 hours.
- The distance Hari covers in those 2 hours is \( 2 \text{ km} \) (since \( H = 1 \) km/h).

### Step 11: Calculate the remaining distance from D to A
- The total distance from A to C is \( 2 \text{ km} \), and from C to D is \( 2 \text{ km} \).
- Therefore, the remaining distance from D to A is:
\[ \text{Distance from D to A} = 2 \text{ km} - 2 \text{ km} = 0 \text{ km} \]

### Step 12: Calculate the time taken for Hari to reach A
- Since the remaining distance is \( 0 \text{ km} \), Hari will reach A at:
\[ 11:00 AM + 7 \text{ hours} = 6:00 PM \]

### Final Answer
- Therefore, Hari will reach A at **6:00 PM**.

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