A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern fills up in 10 hours. If the cistern is full, in how much time will it be emptied by the leak?

Answer

To solve the problem, we need to determine how long it will take for the leak to empty the cistern when it is full. We will use the concept of work done per hour.

### Step-by-step Solution:

1. **Determine the rate of the pipe filling the cistern:**
The pipe can fill the cistern in 9 hours. Therefore, the rate of the pipe is:
\[ \text{Rate of pipe} = \frac{1 \text{ cistern}}{9 \text{ hours}} = \frac{1}{9} \text{ cistern per hour} \]

**Hint:** To find the rate of work done, divide 1 (whole job) by the time taken to complete the job.

2. **Determine the rate of the cistern filling with the leak:**
The cistern fills up in 10 hours due to the leak. Therefore, the combined rate of the pipe and the leak is:
\[ \text{Rate with leak} = \frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10} \text{ cistern per hour} \]

**Hint:** Again, divide 1 (whole job) by the time taken to complete the job to find the rate.

3. **Set up the equation for the leak's rate:**
Let the rate of the leak be \( L \) (in cisterns per hour). The equation representing the situation is:
\[ \text{Rate of pipe} - \text{Rate of leak} = \text{Rate with leak} \]
Substituting the known values:
\[ \frac{1}{9} - L = \frac{1}{10} \]

**Hint:** Use the rates you calculated to set up an equation that reflects the relationship between the pipe and the leak.

4. **Solve for the leak's rate \( L \):**
Rearranging the equation gives:
\[ L = \frac{1}{9} - \frac{1}{10} \]
To subtract these fractions, find a common denominator (which is 90):
\[ L = \frac{10}{90} - \frac{9}{90} = \frac{1}{90} \text{ cistern per hour} \]

**Hint:** When subtracting fractions, make sure to convert them to have a common denominator.

5. **Determine the time taken by the leak to empty the cistern:**
If the leak empties at a rate of \( \frac{1}{90} \) cisterns per hour, then the time taken to empty 1 full cistern is:
\[ \text{Time} = \frac{1 \text{ cistern}}{L} = \frac{1}{\frac{1}{90}} = 90 \text{ hours} \]

**Hint:** To find the time taken to complete a job, divide the total work (1 job) by the rate of work done.

### Final Answer:
The leak will take **90 hours** to empty the full cistern.