Abhishek, Bobby and Charlie start from the same point and travel in the same direction round an Island 6 km in circumference. Abhishek travels at the rate of 3, Bobby at the rate of `2 1/2`and Charlie at the rate of `1 1/4` km/hour. In how many hours will they come together at the starting point again?

Options

A. 6 hrs
B. 12 hrs
C. 24 hrs
D. 15 hrs

Answer

To solve the problem, we need to determine how long it will take for Abhishek, Bobby, and Charlie to meet again at the starting point after traveling around the island. We will calculate the time taken by each person to complete one full round and then find the least common multiple (LCM) of these times.

### Step 1: Calculate the time taken by each person to complete one round.

1. **Abhishek's speed**: 3 km/h
- Time taken by Abhishek (A) = Distance / Speed = 6 km / 3 km/h = 2 hours

2. **Bobby's speed**: 2 1/2 km/h = 5/2 km/h
- Time taken by Bobby (B) = Distance / Speed = 6 km / (5/2) km/h = 6 * (2/5) = 12/5 hours

3. **Charlie's speed**: 1 1/4 km/h = 5/4 km/h
- Time taken by Charlie (C) = Distance / Speed = 6 km / (5/4) km/h = 6 * (4/5) = 24/5 hours

### Step 2: List the times taken by each person.
- Abhishek: 2 hours
- Bobby: 12/5 hours
- Charlie: 24/5 hours

### Step 3: Convert Abhishek's time to a fraction for easier calculation.
- Abhishek's time: 2 hours = 2/1 hours

### Step 4: Find the LCM of the times.
We need to find the LCM of the three times: 2/1, 12/5, and 24/5.

#### Step 4.1: Find LCM of the numerators.
- Numerators: 2, 12, 24
- Prime factorization:
- 2 = 2
- 12 = 2^2 * 3
- 24 = 2^3 * 3

The LCM of the numerators is:
- Highest power of 2: 2^3 = 8
- Highest power of 3: 3^1 = 3
- LCM = 8 * 3 = 24

#### Step 4.2: Find GCD of the denominators.
- Denominators: 1, 5, 5
- GCD = 1 (since 1 is the only common factor)

#### Step 5: Calculate the LCM of the times.
- LCM of the times = LCM of numerators / GCD of denominators = 24 / 1 = 24 hours.

### Conclusion
Abhishek, Bobby, and Charlie will all meet at the starting point again after **24 hours**.

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