Three person A, B, C rum along a circular path 12 km long.They start their race from the same point and at the same time with a speed of `3 km//hr. 7 km//hr` and `13 km//hr`. respectively. After what time will they meet again ?

Options

A. 12 hrs
B. 9 hrs
C. 24 hrs
D. 16 hrs

Answer

To find out after what time A, B, and C will meet again while running along a circular path of 12 km, we can follow these steps:

### Step 1: Determine the time taken by each person to complete one round.
- **For A**:
- Speed = 3 km/hr
- Time = Distance / Speed = 12 km / 3 km/hr = 4 hours

- **For B**:
- Speed = 7 km/hr
- Time = Distance / Speed = 12 km / 7 km/hr = 12/7 hours

- **For C**:
- Speed = 13 km/hr
- Time = Distance / Speed = 12 km / 13 km/hr = 12/13 hours

### Step 2: List the times taken for one complete round.
- A takes 4 hours
- B takes 12/7 hours
- C takes 12/13 hours

### Step 3: Find the least common multiple (LCM) of the times taken.
To find the LCM, we need to convert all times to a common format (fractions):
- A: 4 hours = 4/1 hours
- B: 12/7 hours
- C: 12/13 hours

Now, we need to find the LCM of the numerators (4, 12, 12) and the highest common factor (HCF) of the denominators (1, 7, 13).

### Step 4: Calculate the LCM of the numerators.
- The LCM of 4, 12, and 12 is 12.

### Step 5: Calculate the HCF of the denominators.
- The HCF of 1, 7, and 13 is 1 (since they have no common factors other than 1).

### Step 6: Calculate the time after which they will meet again.
Using the formula:
\[ \text{Time} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{12}{1} = 12 \text{ hours} \]

### Final Answer:
A, B, and C will meet again after **12 hours**.

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