A and B can do a work in 8 days and A alone can do it in 12 days. In how many days can B alone do it?

Answer

To solve the problem, we need to find out how many days B alone can do the work, given that A and B together can complete the work in 8 days, and A alone can do it in 12 days.

### Step-by-Step Solution:

1. **Determine the total work done**:
- If A and B together can complete the work in 8 days, we can express the total work in terms of work units. Let's assume the total work is 24 units (this is a convenient number since both 8 and 12 are factors of 24).
- Therefore, the total work = 24 units.

2. **Calculate the work done by A in one day**:
- A can complete the work alone in 12 days. Thus, the amount of work A does in one day is:
\[ \text{Work done by A in one day} = \frac{24 \text{ units}}{12 \text{ days}} = 2 \text{ units/day} \]

3. **Calculate the combined work done by A and B in one day**:
- A and B together can complete the work in 8 days. Therefore, the amount of work they do together in one day is:
\[ \text{Work done by A and B in one day} = \frac{24 \text{ units}}{8 \text{ days}} = 3 \text{ units/day} \]

4. **Determine the work done by B in one day**:
- We know the combined work done by A and B in one day is 3 units, and A alone does 2 units. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \text{Work done by A and B} - \text{Work done by A} = 3 \text{ units/day} - 2 \text{ units/day} = 1 \text{ unit/day} \]

5. **Calculate the number of days B needs to complete the work alone**:
- Since B can do 1 unit of work in one day, the total number of days B will take to complete the 24 units of work is:
\[ \text{Days for B} = \frac{24 \text{ units}}{1 \text{ unit/day}} = 24 \text{ days} \]

### Final Answer:
B alone can complete the work in **24 days**.

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