P can do a piece of work in 7 days and Q can do the same work in 21 days. How many days will both take together to complete the same work?
(a)4.75
(b)5.75
(c)6
(d)5.25

Options

A. 4.75
B. 5.75
C. 6
D. 5.25

Answer

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

### Step 1: Determine the work done by P and Q in one day.
- P can complete the work in 7 days. Therefore, the work done by P in one day is:
\[ \text{Work done by P in one day} = \frac{1}{7} \text{ of the work} \]

- Q can complete the work in 21 days. Therefore, the work done by Q in one day is:
\[ \text{Work done by Q in one day} = \frac{1}{21} \text{ of the work} \]

### Step 2: Calculate the combined work done by P and Q in one day.
- To find the total work done by both P and Q in one day, we add the work done by each:
\[ \text{Total work done in one day} = \frac{1}{7} + \frac{1}{21} \]

- To add these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 21 is 21:
\[ \frac{1}{7} = \frac{3}{21} \]
\[ \frac{1}{21} = \frac{1}{21} \]
\[ \text{Total work done in one day} = \frac{3}{21} + \frac{1}{21} = \frac{4}{21} \]

### Step 3: Calculate the total time taken to complete the work together.
- If P and Q together can do \(\frac{4}{21}\) of the work in one day, we need to find out how many days it will take to complete 1 whole work:
\[ \text{Days taken} = \frac{1 \text{ (whole work)}}{\frac{4}{21}} = \frac{21}{4} = 5.25 \text{ days} \]

### Conclusion:
Thus, P and Q together will take **5.25 days** to complete the work.