A pipe can fili 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?
एक पाइप किसी टंकी को घंटे में भर सकता है। अपने तल में रिसाव के कारण टंकी 10 घंटे में भरती है। यदि टंकी पूरी भूरी हो तो रिसाव द्वारा वह कितने समय में खाली हो जाएगी?

Options

A. A)170 hours
B. B)80 hours
C. C)90 hours
D. D)100 hours

Explanation

To solve the problem, we need to determine how long it will take for the leak to empty a full cistern. We know the following:

1. A pipe can fill the cistern in 9 hours.
2. Due to a leak, the cistern fills up in 10 hours.

Let's break this down step by step:

### Step 1: Determine the filling rate of the pipe
The filling rate of the pipe can be calculated as follows:
- If the pipe fills the cistern in 9 hours, then its filling rate is:
\[ \text{Filling rate of the pipe} = \frac{1 \text{ cistern}}{9 \text{ hours}} = \frac{1}{9} \text{ cistern per hour} \]

### Step 2: Determine the effective filling rate with the leak
When the leak is present, the cistern fills in 10 hours. Therefore, the effective filling rate (pipe + leak) is:
- If the cistern fills in 10 hours, then the effective filling rate is:
\[ \text{Effective filling rate} = \frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10} \text{ cistern per hour} \]

### Step 3: Set up the equation for the leak's rate
Let the rate at which the leak empties the cistern be \( L \) (in cisterns per hour). The equation considering both the pipe and the leak is:
\[ \text{Filling rate of the pipe} - \text{Rate of the leak} = \text{Effective filling rate} \]
Substituting the values we have:
\[ \frac{1}{9} - L = \frac{1}{10} \]

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

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

Thus, the leak will take **90 hours** to empty the full cistern.

### Summary of Steps:
1. Calculate the filling rate of the pipe: \( \frac{1}{9} \).
2. Calculate the effective filling rate with the leak: \( \frac{1}{10} \).
3. Set up the equation for the leak's rate: \( \frac{1}{9} - L = \frac{1}{10} \).
4. Solve for \( L \): \( L = \frac{1}{90} \).
5. Calculate the time to empty the cistern: \( 90 \text{ hours} \).