Aman and Raman together can complete a piece of work in 30 days, Raman and Manan can complete the same work in 36 days and Manan and Aman can complete the same work in 45 days. All of the three working together can complete the same work in how many days?

Options

A. 12
B. 18
C. 24
D. 28

Answer

To solve the problem, we need to find out how many days Aman, Raman, and Manan can complete the work together. We will use the information given about the pairs of workers.

1. **Understanding the Work Rates**:
- Let the total work be represented as 1 unit.
- If Aman and Raman can complete the work in 30 days, their combined work rate is:
\[ \text{Work rate of Aman and Raman} = \frac{1}{30} \text{ units/day} \]
- If Raman and Manan can complete the work in 36 days, their combined work rate is:
\[ \text{Work rate of Raman and Manan} = \frac{1}{36} \text{ units/day} \]
- If Manan and Aman can complete the work in 45 days, their combined work rate is:
\[ \text{Work rate of Manan and Aman} = \frac{1}{45} \text{ units/day} \]

2. **Setting Up Equations**:
- Let the work rates of Aman, Raman, and Manan be \(A\), \(R\), and \(M\) respectively.
- From the above work rates, we can set up the following equations:
\[ A + R = \frac{1}{30} \quad \text{(1)} \]
\[ R + M = \frac{1}{36} \quad \text{(2)} \]
\[ M + A = \frac{1}{45} \quad \text{(3)} \]

3. **Adding the Equations**:
- Now, let's add all three equations:
\[ (A + R) + (R + M) + (M + A) = \frac{1}{30} + \frac{1}{36} + \frac{1}{45} \]
- This simplifies to:
\[ 2A + 2R + 2M = \frac{1}{30} + \frac{1}{36} + \frac{1}{45} \]
- Dividing the entire equation by 2:
\[ A + R + M = \frac{1}{2} \left( \frac{1}{30} + \frac{1}{36} + \frac{1}{45} \right) \]

4. **Finding a Common Denominator**:
- The least common multiple (LCM) of 30, 36, and 45 is 180.
- Converting each fraction:
\[ \frac{1}{30} = \frac{6}{180}, \quad \frac{1}{36} = \frac{5}{180}, \quad \frac{1}{45} = \frac{4}{180} \]
- Adding these fractions:
\[ \frac{6}{180} + \frac{5}{180} + \frac{4}{180} = \frac{15}{180} = \frac{1}{12} \]

5. **Calculating the Combined Work Rate**:
- Therefore, we have:
\[ A + R + M = \frac{1}{2} \cdot \frac{1}{12} = \frac{1}{24} \]

6. **Finding the Time Taken by All Three Together**:
- The combined work rate of Aman, Raman, and Manan is \(\frac{1}{24}\) units/day.
- Thus, the time taken to complete the work together is:
\[ \text{Time} = \frac{1 \text{ unit}}{\frac{1}{24} \text{ units/day}} = 24 \text{ days} \]

So, Aman, Raman, and Manan together can complete the work in **24 days**.