By working together, A and B can finish a work in 15 days. If B alone can finish the work in 20 days, in how many days can A alone finish the work?

Answer

To solve the problem step by step, we will use the concept of work done and the formula for combined work.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A and B together can complete the work in 15 days.
- B alone can complete the work in 20 days.
- We need to find out how many days A alone would take to complete the work.

2. **Setting Up the Variables**:
- Let the number of days A alone takes to finish the work be \( X \).
- The work done by A in one day is \( \frac{1}{X} \).
- The work done by B in one day is \( \frac{1}{20} \) (since B takes 20 days).

3. **Combined Work**:
- When A and B work together, they complete the work in 15 days, so their combined work in one day is \( \frac{1}{15} \).

4. **Setting Up the Equation**:
- According to the formula for combined work:
\[ \text{Work done by A in one day} + \text{Work done by B in one day} = \text{Combined work done in one day} \]
- This can be expressed as:
\[ \frac{1}{X} + \frac{1}{20} = \frac{1}{15} \]

5. **Finding a Common Denominator**:
- The common denominator for \( X \), \( 20 \), and \( 15 \) is \( 60X \).
- Rewriting the equation:
\[ \frac{60}{60X} + \frac{3X}{60X} = \frac{4X}{60X} \]
- This simplifies to:
\[ 60 + 3X = 4X \]

6. **Solving for X**:
- Rearranging the equation gives:
\[ 4X - 3X = 60 \]
\[ X = 60 \]

7. **Conclusion**:
- A alone can finish the work in **60 days**.