From a point on circular track 5 km long A, B and C started running in the same direction at the same time with speed of `2(1)/(2)` km per hour, 3 km per hour and 2 km per hour respectively . Then on the starting point all three will meet again after

Options

A. 30 hours
B. 6 hours
C. 10 hours
D. 15 hours

Answer

To solve the problem of when A, B, and C will meet again at the starting point on a circular track of 5 km, we can follow these steps:

### Step 1: Determine the speeds of A, B, and C
- A's speed = 2.5 km/h
- B's speed = 3 km/h
- C's speed = 2 km/h

### Step 2: Calculate the time taken by each runner to complete one lap
Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

For A:
\[ T_A = \frac{5 \text{ km}}{2.5 \text{ km/h}} = 2 \text{ hours} \]

For B:
\[ T_B = \frac{5 \text{ km}}{3 \text{ km/h}} = \frac{5}{3} \text{ hours} \]

For C:
\[ T_C = \frac{5 \text{ km}}{2 \text{ km/h}} = 2.5 \text{ hours} \]

### Step 3: Find the least common multiple (LCM) of the times
The times we have are:
- \( T_A = 2 \) hours
- \( T_B = \frac{5}{3} \) hours
- \( T_C = 2.5 \) hours

To find the LCM, we first convert all times to a common fraction:
- \( T_A = 2 = \frac{6}{3} \)
- \( T_B = \frac{5}{3} \)
- \( T_C = 2.5 = \frac{5}{2} = \frac{15}{6} \)

Now, we need to find the LCM of the numerators (6, 5, 15) and the GCD of the denominators (3, 3, 2).

### Step 4: Calculate the LCM of the numerators
- LCM of 6, 5, and 15 = 30

### Step 5: Calculate the GCD of the denominators
- GCD of 3, 3, and 2 = 1

### Step 6: Calculate the overall LCM
The overall LCM is given by:
\[ \text{LCM} = \frac{\text{LCM of numerators}}{\text{GCD of denominators}} = \frac{30}{1} = 30 \text{ hours} \]

### Conclusion
All three runners will meet again at the starting point after **30 hours**.

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