A and B together can do a certain work in x days. Working alone, A and B can do the same workin (x + 8) and (x + 18) days, respectively. A and B together will complete of the same work in:

Options

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

Answer

To solve the problem, we need to find out how many days A and B together will take to complete the work given that A can do the work alone in (x + 8) days and B can do it alone in (x + 18) days.

### Step-by-Step Solution:

1. **Understanding the Work Rates:**
- Let the total work be denoted as 1 unit of work.
- The rate of work done by A alone is \( \frac{1}{x + 8} \) (work per day).
- The rate of work done by B alone is \( \frac{1}{x + 18} \) (work per day).
- The rate of work done by A and B together is \( \frac{1}{x} \) (work per day).

2. **Setting Up the Equation:**
- The combined work rate of A and B can also be expressed as the sum of their individual work rates:
\[ \frac{1}{x} = \frac{1}{x + 8} + \frac{1}{x + 18} \]

3. **Finding a Common Denominator:**
- The common denominator for the right side is \( (x + 8)(x + 18) \). Thus, we rewrite the equation:
\[ \frac{1}{x} = \frac{(x + 18) + (x + 8)}{(x + 8)(x + 18)} \]
- Simplifying the numerator:
\[ (x + 18) + (x + 8) = 2x + 26 \]
- Therefore, the equation becomes:
\[ \frac{1}{x} = \frac{2x + 26}{(x + 8)(x + 18)} \]

4. **Cross-Multiplying:**
- Cross-multiplying gives us:
\[ (x + 8)(x + 18) = x(2x + 26) \]

5. **Expanding Both Sides:**
- Expanding the left side:
\[ x^2 + 26x + 144 \]
- Expanding the right side:
\[ 2x^2 + 26x \]

6. **Setting Up the Quadratic Equation:**
- Setting both sides equal:
\[ x^2 + 26x + 144 = 2x^2 + 26x \]
- Rearranging gives:
\[ 0 = 2x^2 - x^2 + 26x - 26x - 144 \]
\[ x^2 - 144 = 0 \]

7. **Solving for x:**
- This simplifies to:
\[ x^2 = 144 \]
- Taking the square root gives:
\[ x = 12 \]

8. **Conclusion:**
- Therefore, A and B together will complete the work in \( x = 12 \) days.

### Final Answer:
A and B together will complete the work in **12 days**.