A, B and C start at the same time in same direction to run around a circular path of 12 km. If the speed of A, B and C is 3 km/hour, 7 km/hour and 13 km/hour respectively. After what time will they meet again at the starting point?

Options

A. 16 hr.
B. 12 hr.
C. 28 hr.
D. None of these

Answer

To solve the problem step by step, we need to determine the time taken by each runner (A, B, and C) to complete one full lap around the circular path and then find the least common multiple (LCM) of these times to find out when they will all meet at the starting point again.

### Step 1: Calculate the time taken by A
- Distance = 12 km
- Speed of A = 3 km/hour
- Time taken by A = Distance / Speed = 12 km / 3 km/hour = 4 hours

### Step 2: Calculate the time taken by B
- Speed of B = 7 km/hour
- Time taken by B = Distance / Speed = 12 km / 7 km/hour = 12/7 hours

### Step 3: Calculate the time taken by C
- Speed of C = 13 km/hour
- Time taken by C = Distance / Speed = 12 km / 13 km/hour = 12/13 hours

### Step 4: Find the LCM of the times
Now we need to find the LCM of the three times: 4 hours, 12/7 hours, and 12/13 hours.

1. Convert all times to a common format:
- 4 hours = 4/1 hours
- 12/7 hours remains as is.
- 12/13 hours remains as is.

2. To find the LCM, we can express the times with a common denominator:
- LCM(4, 12/7, 12/13) can be calculated using the LCM of the numerators divided by the GCD of the denominators.

3. The LCM of the numerators (4, 12, 12) is 12.
4. The GCD of the denominators (1, 7, 13) is 1.

Thus, the LCM of the times is:
\[ \text{LCM} = \frac{12}{1} = 12 \text{ hours} \]

### Final Answer
A, B, and C will meet again at the starting point after **12 hours**.

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