Abhishek, Bobby and Carlie start from the same point and travel in the same direction around 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/hr. In how many hours will they come together again?

Options

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

Answer

To solve the problem of when Abhishek, Bobby, and Carlie will come together again after starting from the same point and traveling around an island with a circumference of 6 km, we can follow these steps:

### Step 1: Determine the speeds of each person
- Abhishek's speed = 3 km/hr
- Bobby's speed = 2.5 km/hr (which is \(2 \frac{1}{2} = \frac{5}{2}\))
- Carlie's speed = 1.25 km/hr (which is \(1 \frac{1}{4} = \frac{5}{4}\))

### Step 2: Calculate the time taken by each person to complete one round (6 km)
- Time taken by Abhishek (TA) = Distance / Speed = \( \frac{6 \text{ km}}{3 \text{ km/hr}} = 2 \text{ hours} \)
- Time taken by Bobby (TB) = Distance / Speed = \( \frac{6 \text{ km}}{2.5 \text{ km/hr}} = \frac{6}{2.5} = 2.4 \text{ hours} \)
- Time taken by Carlie (TC) = Distance / Speed = \( \frac{6 \text{ km}}{1.25 \text{ km/hr}} = \frac{6}{1.25} = 4.8 \text{ hours} \)

### Step 3: Convert the times into fractions for easier calculation
- TA = 2 hours = \( \frac{2}{1} \)
- TB = 2.4 hours = \( \frac{12}{5} \)
- TC = 4.8 hours = \( \frac{24}{5} \)

### Step 4: Find the Least Common Multiple (LCM) of the times
To find the LCM of the times, we can use the formula for LCM of fractions:

\[ \text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} \]

**Numerators**: 2, 12, 24
**Denominators**: 1, 5, 5

- LCM of the numerators (2, 12, 24) = 24
- HCF of the denominators (1, 5, 5) = 1

Thus, the LCM of the times is:

\[ \text{LCM} = \frac{24}{1} = 24 \text{ hours} \]

### Conclusion
Abhishek, Bobby, and Carlie will come together again after 24 hours.

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