A, B and C start together from the same place to walk round a circular path of length 12 km. A walks at the rate of 4 km/h. B 3 km/h and C `(3)/(2)` km / h. They will meet together at the starting place at the end of.

Options

A. 10 hours
B. 12 hours
C. 15 hours
D. 24 hours

Answer

To find out when A, B, and C will meet together at the starting place after walking around a circular path of length 12 km, we need to calculate the time taken by each person to complete one full revolution around the circular path.

### Step 1: Calculate the time taken by A to complete one revolution.
- Speed of A = 4 km/h
- Length of the circular path = 12 km

Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For A:
\[ \text{Time taken by A} = \frac{12 \text{ km}}{4 \text{ km/h}} = 3 \text{ hours} \]

### Step 2: Calculate the time taken by B to complete one revolution.
- Speed of B = 3 km/h

For B:
\[ \text{Time taken by B} = \frac{12 \text{ km}}{3 \text{ km/h}} = 4 \text{ hours} \]

### Step 3: Calculate the time taken by C to complete one revolution.
- Speed of C = \( \frac{3}{2} \) km/h = 1.5 km/h

For C:
\[ \text{Time taken by C} = \frac{12 \text{ km}}{1.5 \text{ km/h}} = 8 \text{ hours} \]

### Step 4: Find the least common multiple (LCM) of the times taken by A, B, and C.
We need to find the LCM of 3 hours, 4 hours, and 8 hours.

- The multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
- The multiples of 4: 4, 8, 12, 16, 20, 24, ...
- The multiples of 8: 8, 16, 24, ...

The smallest common multiple is 24 hours.

### Conclusion:
A, B, and C will meet together at the starting place at the end of **24 hours**.

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