P can complete a work in 15 days.Q can complete the same work in 10 days ,P started working and after 5 days Q joined him. How many days more will they take to complete the remaining work ?

Options

A. 1 days
B. 2 days
C. 3 days
D. 4 days

Answer

To solve the problem, we need to follow these steps:

### Step 1: Determine the work done by P and Q in one day.
- P can complete the work in 15 days. Therefore, the work done by P in one day is:
\[ \text{Work done by P in 1 day} = \frac{1}{15} \]
- Q can complete the work in 10 days. Therefore, the work done by Q in one day is:
\[ \text{Work done by Q in 1 day} = \frac{1}{10} \]

### Step 2: Calculate the work done by P in the first 5 days.
- In 5 days, the work done by P is:
\[ \text{Work done by P in 5 days} = 5 \times \frac{1}{15} = \frac{5}{15} = \frac{1}{3} \]

### Step 3: Determine the remaining work after P has worked for 5 days.
- The total work is considered as 1 unit. Therefore, the remaining work after P has completed \(\frac{1}{3}\) of the work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Calculate the combined work done by P and Q in one day.
- When Q joins P, their combined work done in one day is:
\[ \text{Combined work} = \text{Work done by P in 1 day} + \text{Work done by Q in 1 day} = \frac{1}{15} + \frac{1}{10} \]
- To add these fractions, we need a common denominator. The least common multiple of 15 and 10 is 30. Therefore:
\[ \frac{1}{15} = \frac{2}{30}, \quad \frac{1}{10} = \frac{3}{30} \]
\[ \text{Combined work} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \]

### Step 5: Calculate the number of days required to complete the remaining work.
- The remaining work is \(\frac{2}{3}\). To find out how many days it will take for P and Q to complete this remaining work, we can set up the equation:
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Combined work per day}} = \frac{\frac{2}{3}}{\frac{1}{6}} \]
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \text{Days required} = \frac{2}{3} \times 6 = 4 \]

### Conclusion
P and Q will take **4 more days** to complete the remaining work.

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