Ramesh, Dinesh and Mahesh start to run in a circular ground. Ramesh completes a circle in `1(2)/(7)` seconds, Dinesh complets a circle in `1(1)/(14)` seconds and Mahesh complete a circle in `1(3)/(7)` second. If they start running at the same time then after how much time they will meet ? (a) `12(6)/(7)` Second (b) `7(6)/(15)` Second (c) `(7)/(90)` Second (d) None of these

Options

A. `12(6)/(7)` Second
B. `7(6)/(15)` Second
C. `(7)/(90)` Second
D. None of these

Answer

To solve the problem of when Ramesh, Dinesh, and Mahesh will meet while running on a circular ground, we need to determine the time each of them takes to complete one full circle and then find the least common multiple (LCM) of these times.

### Step-by-Step Solution:

1. **Convert the times into improper fractions:**
- Ramesh completes a circle in \(1 \frac{2}{7}\) seconds:
\[ 1 \frac{2}{7} = \frac{7 \times 1 + 2}{7} = \frac{9}{7} \text{ seconds} \]
- Dinesh completes a circle in \(1 \frac{1}{14}\) seconds:
\[ 1 \frac{1}{14} = \frac{14 \times 1 + 1}{14} = \frac{15}{14} \text{ seconds} \]
- Mahesh completes a circle in \(1 \frac{3}{7}\) seconds:
\[ 1 \frac{3}{7} = \frac{7 \times 1 + 3}{7} = \frac{10}{7} \text{ seconds} \]

2. **Identify the times taken to complete one circle:**
- Ramesh: \(\frac{9}{7}\) seconds
- Dinesh: \(\frac{15}{14}\) seconds
- Mahesh: \(\frac{10}{7}\) seconds

3. **Find the LCM of the times:**
- To find the LCM, we need to express all times in a common format. The denominators are 7 and 14. The least common multiple of 7 and 14 is 14.
- Convert each time to have a denominator of 14:
- Ramesh:
\[ \frac{9}{7} = \frac{9 \times 2}{7 \times 2} = \frac{18}{14} \]
- Dinesh:
\[ \frac{15}{14} = \frac{15}{14} \text{ (already in the correct form)} \]
- Mahesh:
\[ \frac{10}{7} = \frac{10 \times 2}{7 \times 2} = \frac{20}{14} \]

4. **Calculate the LCM of the numerators:**
- The numerators are 18, 15, and 20.
- Prime factorization:
- \(18 = 2 \times 3^2\)
- \(15 = 3 \times 5\)
- \(20 = 2^2 \times 5\)
- The LCM is found by taking the highest power of each prime:
- \(2^2\) from 20
- \(3^2\) from 18
- \(5\) from 15
- Therefore,
\[ \text{LCM} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 \]

5. **Convert the LCM back to the time format:**
- Since we calculated the LCM with a common denominator of 14, we need to divide the LCM of the numerators by the common denominator:
\[ \text{Time} = \frac{180}{14} = \frac{90}{7} \text{ seconds} \]

6. **Convert \(\frac{90}{7}\) into a mixed number:**
- Dividing 90 by 7 gives:
\[ 90 \div 7 = 12 \text{ remainder } 6 \]
- Thus,
\[ \frac{90}{7} = 12 \frac{6}{7} \text{ seconds} \]

### Final Answer:
The time after which Ramesh, Dinesh, and Mahesh will meet is \(12 \frac{6}{7}\) seconds.