A takes 4 hours more than that of A and B working together to finish certain piece of work. While B takes 9 hours more than that of A and B working together. Then find in how many hours A and B working together can finish the work?

Answer

To solve the problem, we need to find out how many hours A and B together can finish the work. Let's denote the time taken by A and B working together as \( x \) hours.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( x \) be the time taken by A and B working together to complete the work.
- Then, A takes \( x + 4 \) hours to finish the work alone.
- B takes \( x + 9 \) hours to finish the work alone.

2. **Calculate the Work Done**:
- The work done by A in one hour is \( \frac{1}{x + 4} \).
- The work done by B in one hour is \( \frac{1}{x + 9} \).
- The work done by A and B together in one hour is \( \frac{1}{x} \).

3. **Set Up the Equation**:
- According to the work rates, the combined work done by A and B in one hour can be expressed as:
\[ \frac{1}{x} = \frac{1}{x + 4} + \frac{1}{x + 9} \]

4. **Find a Common Denominator**:
- The common denominator for the right side of the equation is \( (x + 4)(x + 9) \).
- Rewrite the equation:
\[ \frac{1}{x} = \frac{(x + 9) + (x + 4)}{(x + 4)(x + 9)} \]
- Simplifying the numerator:
\[ \frac{1}{x} = \frac{2x + 13}{(x + 4)(x + 9)} \]

5. **Cross-Multiply**:
- Cross-multiplying gives:
\[ (2x + 13)x = (x + 4)(x + 9) \]

6. **Expand Both Sides**:
- Left side:
\[ 2x^2 + 13x \]
- Right side:
\[ x^2 + 13x + 36 \]

7. **Set Up the Quadratic Equation**:
- Now, equate both sides:
\[ 2x^2 + 13x = x^2 + 13x + 36 \]
- Simplifying gives:
\[ 2x^2 - x^2 + 13x - 13x - 36 = 0 \]
\[ x^2 - 36 = 0 \]

8. **Solve the Quadratic Equation**:
- Factoring gives:
\[ (x - 6)(x + 6) = 0 \]
- Thus, \( x = 6 \) or \( x = -6 \) (we discard the negative solution since time cannot be negative).

9. **Conclusion**:
- Therefore, A and B working together can finish the work in **6 hours**.