P,Q and R can complete a work in 12,24 and 6 days respectively. The work will be completed in __ days if all of them are working together.

Options

A. 7
B. 24/7
C. 8/7
D. 9

Answer

To solve the problem of how many days P, Q, and R will take to complete the work together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Individual Work Rates**:
- P can complete the work in 12 days, so P's work rate is \( \frac{1}{12} \) of the work per day.
- Q can complete the work in 24 days, so Q's work rate is \( \frac{1}{24} \) of the work per day.
- R can complete the work in 6 days, so R's work rate is \( \frac{1}{6} \) of the work per day.

2. **Calculate Combined Work Rate**:
- To find the combined work rate of P, Q, and R, we add their individual work rates:
\[ \text{Combined Work Rate} = \frac{1}{12} + \frac{1}{24} + \frac{1}{6} \]

3. **Finding a Common Denominator**:
- The least common multiple (LCM) of the denominators (12, 24, 6) is 24.
- Convert each fraction to have a denominator of 24:
\[ \frac{1}{12} = \frac{2}{24}, \quad \frac{1}{24} = \frac{1}{24}, \quad \frac{1}{6} = \frac{4}{24} \]

4. **Add the Converted Fractions**:
- Now, add the fractions:
\[ \text{Combined Work Rate} = \frac{2}{24} + \frac{1}{24} + \frac{4}{24} = \frac{7}{24} \]

5. **Calculate Time Taken to Complete the Work Together**:
- The total work can be considered as 1 (the whole work).
- Time taken to complete the work together is given by:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{1}{\frac{7}{24}} = \frac{24}{7} \text{ days} \]

### Final Answer:
The work will be completed in \( \frac{24}{7} \) days if all of them are working together.