Rohan and Mohit together can build a wall in 8 days. Mohit and Vikas can build the same wall in 10 days and Vikas and Rohan can build the same wall in 12 days. In how many days can all the three complete the same wall while working together?

Options

A. `(240)/(37)`
B. `(120)/(37)`
C. `(150)/(37)`
D. `(180)/(37)`

Answer

To solve the problem step by step, we will first define the work done by each pair of individuals and then find the total work done when all three work together.

### Step 1: Define the work rates of each pair
Let:
- R = work done by Rohan in one day
- M = work done by Mohit in one day
- V = work done by Vikas in one day

From the information given:
1. Rohan and Mohit together can build a wall in 8 days.
- Therefore, their combined work rate is \( R + M = \frac{1}{8} \) (walls per day).

2. Mohit and Vikas can build the same wall in 10 days.
- Therefore, their combined work rate is \( M + V = \frac{1}{10} \) (walls per day).

3. Vikas and Rohan can build the same wall in 12 days.
- Therefore, their combined work rate is \( V + R = \frac{1}{12} \) (walls per day).

### Step 2: Set up the equations
We can summarize the work rates as:
1. \( R + M = \frac{1}{8} \) (Equation 1)
2. \( M + V = \frac{1}{10} \) (Equation 2)
3. \( V + R = \frac{1}{12} \) (Equation 3)

### Step 3: Solve the equations
To find the individual work rates, we can add all three equations:
\[ (R + M) + (M + V) + (V + R) = \frac{1}{8} + \frac{1}{10} + \frac{1}{12} \]
This simplifies to:
\[ 2R + 2M + 2V = \frac{1}{8} + \frac{1}{10} + \frac{1}{12} \]
Now, we need to find a common denominator for the right side. The LCM of 8, 10, and 12 is 120. Thus, we convert each fraction:
\[ \frac{1}{8} = \frac{15}{120}, \quad \frac{1}{10} = \frac{12}{120}, \quad \frac{1}{12} = \frac{10}{120} \]
Adding these gives:
\[ \frac{15 + 12 + 10}{120} = \frac{37}{120} \]
So we have:
\[ 2R + 2M + 2V = \frac{37}{120} \]
Dividing by 2:
\[ R + M + V = \frac{37}{240} \]

### Step 4: Find the time taken by all three together
The combined work rate of Rohan, Mohit, and Vikas is \( \frac{37}{240} \) walls per day. To find the number of days taken to complete one wall, we take the reciprocal of the work rate:
\[ \text{Days taken} = \frac{1}{\frac{37}{240}} = \frac{240}{37} \]

### Final Answer
Thus, Rohan, Mohit, and Vikas together can complete the wall in \( \frac{240}{37} \) days, which is approximately 6.49 days.