From a point on a 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 step by step, we need to calculate the time taken by each runner to complete one lap around the circular track and then find the least common multiple (LCM) of those times.

### Step 1: Calculate the time taken by A to complete the track
- **Speed of A** = \(2 \frac{1}{2}\) km/h = \( \frac{5}{2} \) km/h
- **Circumference of the track** = 5 km
- **Time taken by A** = \( \frac{\text{Circumference}}{\text{Speed}} = \frac{5 \text{ km}}{\frac{5}{2} \text{ km/h}} = 2 \text{ hours} \)

**Hint:** To find the time taken to complete a circular track, divide the length of the track by the speed of the runner.

### Step 2: Calculate the time taken by B to complete the track
- **Speed of B** = 3 km/h
- **Time taken by B** = \( \frac{5 \text{ km}}{3 \text{ km/h}} = \frac{5}{3} \text{ hours} \)

**Hint:** Use the same formula: Time = Distance / Speed for each runner.

### Step 3: Calculate the time taken by C to complete the track
- **Speed of C** = 2 km/h
- **Time taken by C** = \( \frac{5 \text{ km}}{2 \text{ km/h}} = \frac{5}{2} \text{ hours} \)

**Hint:** Remember to convert the speeds into a consistent unit (km/h) before calculating the time.

### Step 4: Find the LCM of the times taken by A, B, and C
- Times:
- A: 2 hours = \( \frac{2}{1} \)
- B: \( \frac{5}{3} \) hours
- C: \( \frac{5}{2} \) hours

To find the LCM of these fractions, we can convert them to a common denominator:
- The LCM of the numerators (2, 5, 5) is 10.
- The GCD of the denominators (1, 3, 2) is 1.

Thus, the LCM of the times is:
\[ \text{LCM} = \frac{\text{LCM of numerators}}{\text{GCD of denominators}} = \frac{10}{1} = 10 \text{ hours} \]

**Hint:** To find the LCM of fractions, find the LCM of the numerators and the GCD of the denominators.

### Conclusion
A, B, and C will meet again at the starting point after **10 hours**.

**Final Answer:** 10 hours