A can do a piece of work in 6 days and A alone can do it in 10 days.In how many days can B alone do it?.

Answer

To solve the problem step by step, we need to find out how many days B alone can complete the work given that A and B together can complete it in 6 days, and A alone can complete it in 10 days.

### Step-by-Step Solution:

1. **Determine the total work in terms of units**:
- Let’s assume the total work is 30 units (this is the least common multiple of the days A and B take to complete the work).

2. **Calculate the work done by A and B together in one day**:
- Since A and B together can complete the work in 6 days, the work done by A and B together in one day is:
\[ \text{Work done by A and B together in one day} = \frac{30 \text{ units}}{6 \text{ days}} = 5 \text{ units/day} \]

3. **Calculate the work done by A alone in one day**:
- A alone can complete the work in 10 days, so the work done by A alone in one day is:
\[ \text{Work done by A alone in one day} = \frac{30 \text{ units}}{10 \text{ days}} = 3 \text{ units/day} \]

4. **Calculate the work done by B alone in one day**:
- To find the work done by B alone in one day, we subtract the work done by A from the work done by A and B together:
\[ \text{Work done by B alone in one day} = \text{Work done by A and B together} - \text{Work done by A alone} \]
\[ \text{Work done by B alone in one day} = 5 \text{ units/day} - 3 \text{ units/day} = 2 \text{ units/day} \]

5. **Calculate the number of days B alone will take to complete the work**:
- Now, to find out how many days B alone will take to complete the total work, we use the formula:
\[ \text{Days taken by B} = \frac{\text{Total work}}{\text{Work done by B alone in one day}} \]
\[ \text{Days taken by B} = \frac{30 \text{ units}}{2 \text{ units/day}} = 15 \text{ days} \]

### Final Answer:
B alone can complete the work in **15 days**.