A can fill a cistern in 10 hours and B can empty it in 5 hours. If they are opened together, in how many hours will cistern be empty?

Answer

To solve the problem step by step, we need to determine how long it will take for the cistern to be emptied when both A and B are working together.

### Step-by-Step Solution:

1. **Identify the Rates of A and B**:
- A can fill the cistern in 10 hours, so A's rate of work is \( \frac{1}{10} \) of the cistern per hour.
- B can empty the cistern in 5 hours, so B's rate of work is \( -\frac{1}{5} \) of the cistern per hour (negative because B is emptying the cistern).

2. **Calculate the Combined Rate**:
- The combined rate of A and B working together is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{10} - \frac{1}{5} \]
- To perform the addition, we need a common denominator. The least common multiple (LCM) of 10 and 5 is 10. Thus:
\[ \frac{1}{10} - \frac{2}{10} = -\frac{1}{10} \]
- This means that together, A and B will empty \( \frac{1}{10} \) of the cistern per hour.

3. **Determine the Total Work**:
- The total work to empty the cistern is considered as 1 full cistern.

4. **Calculate the Time to Empty the Cistern**:
- Using the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{1}{-\frac{1}{10}} = -10 \text{ hours} \]
- Since time cannot be negative, we interpret this as it taking 10 hours to completely empty the cistern.

### Final Answer:
The cistern will be empty in **10 hours** when A and B are opened together.