P and Q can complete a certain in 28 days and 56 days, respectively. P works for 7 days, and then Q joins P . In how many more days, can they complete the work ?

Options

A. 7
B. 14
C. 21
D. 28

Explanation

To solve the problem step by step, we will first determine the work done by P and Q, and then calculate how much work is left after P works alone for 7 days. Finally, we will find out how many more days they will take to complete the remaining work together.

### Step 1: Determine the work done by P and Q in one day.

- P can complete the work in 28 days, so the work done by P in one day is:
\[ \text{Work done by P in one day} = \frac{1}{28} \]

- Q can complete the work in 56 days, so the work done by Q in one day is:
\[ \text{Work done by Q in one day} = \frac{1}{56} \]

### Step 2: Calculate the total work done by P in 7 days.

- In 7 days, P will complete:
\[ \text{Work done by P in 7 days} = 7 \times \frac{1}{28} = \frac{7}{28} = \frac{1}{4} \]

### Step 3: Calculate the remaining work after P has worked for 7 days.

- The total work can be considered as 1 unit (the whole work). Therefore, the remaining work after P has worked for 7 days is:
\[ \text{Remaining work} = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Calculate the combined work done by P and Q in one day.

- The combined work done by P and Q in one day is:
\[ \text{Combined work} = \frac{1}{28} + \frac{1}{56} \]

To add these fractions, we need a common denominator. The least common multiple of 28 and 56 is 56.

\[ \frac{1}{28} = \frac{2}{56} \]

Therefore,
\[ \text{Combined work} = \frac{2}{56} + \frac{1}{56} = \frac{3}{56} \]

### Step 5: Calculate how many days P and Q will take to complete the remaining work.

- To find out how many days it will take for P and Q to complete the remaining \(\frac{3}{4}\) of the work, we set up the equation:
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Combined work per day}} = \frac{\frac{3}{4}}{\frac{3}{56}} = \frac{3}{4} \times \frac{56}{3} = 14 \]

### Final Answer:
- Therefore, P and Q will take **14 more days** to complete the work after Q joins P.

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