Four runers started running the race in the same direction around a circular path of 7 km. Their speeds are 4,3,9 and 3.5km/hr individually. If they have started their race at 6 o'clock in the morning, then at what time they will be at the starting point?

Answer

To solve the problem of when the four runners will meet again at the starting point after starting their race at 6 o'clock in the morning, we need to determine how long each runner takes to complete one lap around the circular track of 7 km. Then, we will find the least common multiple (LCM) of these times to determine when they will all be together at the starting point.

### Step-by-Step Solution:

1. **Calculate the time taken by each runner to complete one lap:**
- **Runner 1:** Speed = 4 km/hr
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{7 \text{ km}}{4 \text{ km/hr}} = \frac{7}{4} \text{ hours} = 1.75 \text{ hours} \]
- **Runner 2:** Speed = 3 km/hr
\[ \text{Time} = \frac{7 \text{ km}}{3 \text{ km/hr}} = \frac{7}{3} \text{ hours} \approx 2.33 \text{ hours} \]
- **Runner 3:** Speed = 9 km/hr
\[ \text{Time} = \frac{7 \text{ km}}{9 \text{ km/hr}} = \frac{7}{9} \text{ hours} \approx 0.78 \text{ hours} \]
- **Runner 4:** Speed = 3.5 km/hr
\[ \text{Time} = \frac{7 \text{ km}}{3.5 \text{ km/hr}} = 2 \text{ hours} \]

2. **Convert times into a common format (in hours):**
- Runner 1: \(1.75\) hours
- Runner 2: \(\frac{7}{3}\) hours
- Runner 3: \(\frac{7}{9}\) hours
- Runner 4: \(2\) hours

3. **Find the LCM of the times:**
- To find the LCM of the fractions, we take the LCM of the numerators and the GCD (Greatest Common Divisor) of the denominators.
- Numerators: \(7, 7, 7, 2\) → LCM = \(14\)
- Denominators: \(4, 3, 9, 1\) → GCD = \(1\)
- Thus, the LCM of the times is:
\[ \text{LCM} = \frac{\text{LCM of numerators}}{\text{GCD of denominators}} = \frac{14}{1} = 14 \text{ hours} \]

4. **Determine the time they will meet at the starting point:**
- They started at 6:00 AM. Adding 14 hours to this time:
\[ 6:00 \text{ AM} + 14 \text{ hours} = 8:00 \text{ PM} \]

### Final Answer:
The four runners will meet at the starting point at **8:00 PM**.

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