A, B, C start walking from same point, in same direction on a circular path of 12 km. Their speed are 4, 3 and 3/2 km./hr. Find time taken by them to meet at starting point ?

Options

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

Answer

To solve the problem, we need to determine the time taken by A, B, and C to meet at the starting point after walking on a circular path of 12 km. 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-by-Step Solution:

1. **Identify the Length of the Circular Path:**
- The length of the circular path is given as 12 km.

2. **Identify the Speeds of A, B, and C:**
- Speed of A = 4 km/hr
- Speed of B = 3 km/hr
- Speed of C = 3/2 km/hr (which is equal to 1.5 km/hr)

3. **Calculate the Time Taken by Each Person to Complete One Round:**
- Time taken by A to complete one round:
\[ \text{Time}_A = \frac{\text{Distance}}{\text{Speed}_A} = \frac{12 \text{ km}}{4 \text{ km/hr}} = 3 \text{ hours} \]
- Time taken by B to complete one round:
\[ \text{Time}_B = \frac{\text{Distance}}{\text{Speed}_B} = \frac{12 \text{ km}}{3 \text{ km/hr}} = 4 \text{ hours} \]
- Time taken by C to complete one round:
\[ \text{Time}_C = \frac{\text{Distance}}{\text{Speed}_C} = \frac{12 \text{ km}}{1.5 \text{ km/hr}} = 8 \text{ hours} \]

4. **Find the LCM of the Times:**
- The times taken to complete one round are:
- Time_A = 3 hours
- Time_B = 4 hours
- Time_C = 8 hours
- To find the LCM of 3, 4, and 8:
- The prime factorization of the numbers:
- 3 = 3^1
- 4 = 2^2
- 8 = 2^3
- The LCM is taken by taking the highest power of each prime:
- LCM = 2^3 * 3^1 = 8 * 3 = 24

5. **Conclusion:**
- The time taken by A, B, and C to meet at the starting point is 24 hours.

### Summary of the Solution:
The time taken by A, B, and C to meet at the starting point on a circular path of 12 km is **24 hours**.