A and B together can do a piece of work in 20 days and A alone can do it in 30 days. B alone can do the work in how many days?

Options

A. a)45
B. b)60
C. c)75
D. d)90

Answer

To solve the problem, we need to determine how many days B alone can complete the work given the information about A and B working together and A working alone.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- A and B together can complete the work in 20 days.
- A alone can complete the work in 30 days.

2. **Calculating Work Rates**:
- The work done in one day by A and B together (combined work rate) is:
\[ \text{Work Rate of A and B} = \frac{1}{20} \text{ (work per day)} \]
- The work done in one day by A alone is:
\[ \text{Work Rate of A} = \frac{1}{30} \text{ (work per day)} \]

3. **Finding Work Rate of B**:
- Let the work rate of B be \( \frac{1}{x} \), where \( x \) is the number of days B takes to complete the work alone.
- According to the work rates, we have:
\[ \frac{1}{30} + \frac{1}{x} = \frac{1}{20} \]

4. **Solving for B's Work Rate**:
- Rearranging the equation:
\[ \frac{1}{x} = \frac{1}{20} - \frac{1}{30} \]
- To subtract the fractions, find a common denominator (which is 60):
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
- Thus,
\[ \frac{1}{x} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

5. **Finding x**:
- Taking the reciprocal gives:
\[ x = 60 \]

### Conclusion:
B alone can complete the work in **60 days**.