A takes `5(1)/(3)` days more than (A + B) together to complete a work. B takes `8(1)/(3)` days more than (A + B) together to complete a work. In how many days will A and B alone complete this work ?

Answer

To solve the problem, we need to determine how many days A and B alone will take to complete the work based on the information given about their combined work with each other.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Let \( x \) be the number of days A and B together take to complete the work.
- A takes \( x + 5\frac{1}{3} \) days to complete the work alone.
- B takes \( x + 8\frac{1}{3} \) days to complete the work alone.

2. **Convert Mixed Numbers to Improper Fractions:**
- \( 5\frac{1}{3} = \frac{16}{3} \)
- \( 8\frac{1}{3} = \frac{25}{3} \)

3. **Express A's and B's Work Rates:**
- A's work rate is \( \frac{1}{x + \frac{16}{3}} \).
- B's work rate is \( \frac{1}{x + \frac{25}{3}} \).
- The combined work rate of A and B is \( \frac{1}{x} \).

4. **Set Up the Equation:**
- The work rates of A and B together should equal their combined work rate:
\[ \frac{1}{x + \frac{16}{3}} + \frac{1}{x + \frac{25}{3}} = \frac{1}{x} \]

5. **Finding a Common Denominator:**
- The common denominator for the left side is \((x + \frac{16}{3})(x + \frac{25}{3})\):
\[ \frac{(x + \frac{25}{3}) + (x + \frac{16}{3})}{(x + \frac{16}{3})(x + \frac{25}{3})} = \frac{1}{x} \]

6. **Simplifying the Numerator:**
- Combine the numerators:
\[ 2x + \frac{41}{3} \]

7. **Cross Multiply:**
\[ (2x + \frac{41}{3})x = (x + \frac{16}{3})(x + \frac{25}{3}) \]

8. **Expanding Both Sides:**
- Left Side:
\[ 2x^2 + \frac{41}{3}x \]
- Right Side:
\[ x^2 + \frac{25}{3}x + \frac{16}{3}x + \frac{400}{9} = x^2 + \frac{41}{3}x + \frac{400}{9} \]

9. **Setting the Equation:**
\[ 2x^2 + \frac{41}{3}x = x^2 + \frac{41}{3}x + \frac{400}{9} \]

10. **Rearranging the Equation:**
\[ 2x^2 - x^2 = \frac{400}{9} \]
\[ x^2 = \frac{400}{9} \]

11. **Taking the Square Root:**
\[ x = \frac{20}{3} \]

12. **Calculating A's and B's Time Alone:**
- A's time alone:
\[ x + \frac{16}{3} = \frac{20}{3} + \frac{16}{3} = \frac{36}{3} = 12 \text{ days} \]
- B's time alone:
\[ x + \frac{25}{3} = \frac{20}{3} + \frac{25}{3} = \frac{45}{3} = 15 \text{ days} \]

### Final Answer:
- A will complete the work alone in **12 days**.
- B will complete the work alone in **15 days**.