Vijay and Sahil together complete a piece of work in 40 days, Sahil and Ranjit can complete the same work in 48 days and Ranjit and Vijay can complete the same work in 60 days. In how many days can all the three complete the same work while working together?

Options

A. 16
B. 24
C. 32
D. 38

Answer

To solve the problem, we need to find out how many days Vijay, Sahil, and Ranjit can complete the work together. We will use the concept of work rates.

1. **Define Work Rates**:
Let the work rates of Vijay, Sahil, and Ranjit be \( V \), \( S \), and \( R \) respectively. The work done is inversely proportional to the time taken.

2. **Set Up Equations**:
From the information given:
- Vijay and Sahil together complete the work in 40 days:
\[ V + S = \frac{1}{40} \quad \text{(Equation 1)} \]
- Sahil and Ranjit together complete the work in 48 days:
\[ S + R = \frac{1}{48} \quad \text{(Equation 2)} \]
- Ranjit and Vijay together complete the work in 60 days:
\[ R + V = \frac{1}{60} \quad \text{(Equation 3)} \]

3. **Add the Equations**:
Now, we will add all three equations:
\[ (V + S) + (S + R) + (R + V) = \frac{1}{40} + \frac{1}{48} + \frac{1}{60} \]
This simplifies to:
\[ 2V + 2S + 2R = \frac{1}{40} + \frac{1}{48} + \frac{1}{60} \]
Dividing the entire equation by 2:
\[ V + S + R = \frac{1}{2} \left( \frac{1}{40} + \frac{1}{48} + \frac{1}{60} \right) \]

4. **Calculate the Right Side**:
We need to find a common denominator for \( 40, 48, \) and \( 60 \). The LCM of these numbers is \( 240 \).
\[ \frac{1}{40} = \frac{6}{240}, \quad \frac{1}{48} = \frac{5}{240}, \quad \frac{1}{60} = \frac{4}{240} \]
Adding these fractions:
\[ \frac{6}{240} + \frac{5}{240} + \frac{4}{240} = \frac{15}{240} = \frac{1}{16} \]

5. **Substituting Back**:
Now substituting back into our equation:
\[ V + S + R = \frac{1}{2} \left( \frac{1}{16} \right) = \frac{1}{32} \]

6. **Conclusion**:
Therefore, the time taken by all three working together to complete the work is:
\[ \text{Time} = \frac{1}{V + S + R} = \frac{1}{\frac{1}{32}} = 32 \text{ days} \]

**Final Answer**: Vijay, Sahil, and Ranjit together can complete the work in **32 days**.