Two persons A and B can complete a task in 10 days and 15 days, respectively. They work together and A leaves 5 days before the task is completed After that, B alone completes the remaining task The number of days taken to complete the whole task is :

Options

A. 8
B. 9
C. 10
D. 12

Answer

To solve the problem, we need to determine how many days it takes for A and B to complete the task together, given that A leaves 5 days before the task is completed and B finishes the remaining work alone.

### Step-by-Step Solution:

1. **Determine the Work Rates of A and B:**
- A can complete the task in 10 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ (tasks per day)} \]
- B can complete the task in 15 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{15} \text{ (tasks per day)} \]

2. **Find the Least Common Multiple (LCM) to Determine Total Work:**
- The LCM of 10 and 15 is 30. Therefore, we can consider the total work as 30 units.

3. **Calculate the Daily Work Contribution:**
- A's daily contribution:
\[ \text{A's daily work} = \frac{30}{10} = 3 \text{ units/day} \]
- B's daily contribution:
\[ \text{B's daily work} = \frac{30}{15} = 2 \text{ units/day} \]

4. **Let the Total Time Taken to Complete the Task be X Days:**
- A works for \(X - 5\) days (since A leaves 5 days before the task is completed).
- B works for \(X\) days.

5. **Set Up the Equation for Total Work Done:**
- The total work done can be expressed as:
\[ \text{Work done by A} + \text{Work done by B} = \text{Total Work} \]
- This translates to:
\[ 3(X - 5) + 2X = 30 \]

6. **Simplify the Equation:**
- Expanding the equation:
\[ 3X - 15 + 2X = 30 \]
- Combining like terms:
\[ 5X - 15 = 30 \]

7. **Solve for X:**
- Adding 15 to both sides:
\[ 5X = 45 \]
- Dividing by 5:
\[ X = 9 \]

8. **Conclusion:**
- The total number of days taken to complete the whole task is 9 days.