A and B together can do a piece of work in 12 days and A alone can do it in 18 days. In how many days canBalone do it?

Answer

To solve the problem step by step, we will use the concept of work done in terms of days.

### Step-by-Step Solution:

1. **Understand the Work Rates**:
- Let the total work be represented as 1 unit of work.
- A and B together can complete the work in 12 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1 \text{ unit of work}}{12 \text{ days}} = \frac{1}{12} \text{ units/day} \]

2. **Calculate A's Work Rate**:
- A alone can complete the work in 18 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ unit of work}}{18 \text{ days}} = \frac{1}{18} \text{ units/day} \]

3. **Set Up the Equation for B's Work Rate**:
- Let B's work rate be \( \frac{1}{x} \) units/day, where \( x \) is the number of days B takes to complete the work alone.
- According to the work rates, we can set up the equation:
\[ \text{Work rate of A} + \text{Work rate of B} = \text{Work rate of A and B} \]
This translates to:
\[ \frac{1}{18} + \frac{1}{x} = \frac{1}{12} \]

4. **Solve for B's Work Rate**:
- Rearranging the equation gives:
\[ \frac{1}{x} = \frac{1}{12} - \frac{1}{18} \]
- To subtract these fractions, find a common denominator, which is 36:
\[ \frac{1}{12} = \frac{3}{36}, \quad \frac{1}{18} = \frac{2}{36} \]
- Thus:
\[ \frac{1}{x} = \frac{3}{36} - \frac{2}{36} = \frac{1}{36} \]

5. **Calculate x**:
- Taking the reciprocal gives:
\[ x = 36 \]
- Therefore, B alone can complete the work in 36 days.

### Final Answer:
B alone can do the work in **36 days**.