Two pipes A and B can fill a cistern in 12 hours and 8 hours respectively. The pipes are opened simultaneously, and it is found that due to leakage in bottom, 12 min extra are taken to the cistern to be filled. If the cistern is full, in how much time the leak empties the cistern alone?

Options

A. A)120 hours
B. B)112 hours
C. C)108 hours
D. D)132 hours

Answer

To solve the problem step by step, we will follow the outlined procedure:

### Step 1: Determine the rates of pipes A and B
- Pipe A can fill the cistern in 12 hours, so its rate is \( \frac{1}{12} \) of the cistern per hour.
- Pipe B can fill the cistern in 8 hours, so its rate is \( \frac{1}{8} \) of the cistern per hour.

**Hint:** To find the rate of work, use the formula \( \text{Rate} = \frac{1}{\text{Time}} \).

### Step 2: Calculate the combined rate of pipes A and B
- The combined rate when both pipes are opened together is:
\[ \text{Combined Rate} = \frac{1}{12} + \frac{1}{8} \]
To add these fractions, find a common denominator (which is 24):
\[ \text{Combined Rate} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24} \]
- Thus, together they can fill the cistern at a rate of \( \frac{5}{24} \) of the cistern per hour.

**Hint:** When adding fractions, always find a common denominator.

### Step 3: Calculate the time taken by both pipes to fill the cistern without leakage
- The time taken to fill the cistern without leakage is the reciprocal of the combined rate:
\[ \text{Time without leakage} = \frac{1}{\frac{5}{24}} = \frac{24}{5} \text{ hours} \]

**Hint:** To find the time taken, use the formula \( \text{Time} = \frac{1}{\text{Rate}} \).

### Step 4: Account for the extra time due to leakage
- It is given that due to leakage, it takes an extra 12 minutes to fill the cistern. Convert 12 minutes to hours:
\[ 12 \text{ minutes} = \frac{12}{60} = \frac{1}{5} \text{ hours} \]
- Therefore, the total time taken with leakage is:
\[ \text{Total time with leakage} = \frac{24}{5} + \frac{1}{5} = \frac{25}{5} = 5 \text{ hours} \]

**Hint:** When converting minutes to hours, divide by 60.

### Step 5: Set up the equation to find the rate of the leak
- Let the rate of the leak be \( L \) (in terms of cisterns per hour). The equation when both pipes and the leak are working together is:
\[ \text{Combined Rate} - L = \text{Rate with leakage} \]
Thus:
\[ \frac{5}{24} - L = \frac{1}{5} \]

**Hint:** To isolate \( L \), rearrange the equation.

### Step 6: Solve for the leak's rate
- Rearranging gives:
\[ L = \frac{5}{24} - \frac{1}{5} \]
To subtract these fractions, find a common denominator (which is 120):
\[ L = \frac{25}{120} - \frac{24}{120} = \frac{1}{120} \]

**Hint:** When subtracting fractions, ensure you have a common denominator.

### Step 7: Calculate the time taken by the leak to empty the cistern alone
- If the leak empties the cistern at a rate of \( \frac{1}{120} \) of the cistern per hour, the time taken by the leak to empty the entire cistern is:
\[ \text{Time for leak} = \frac{1}{L} = \frac{1}{\frac{1}{120}} = 120 \text{ hours} \]

**Hint:** To find the time taken for a single task, take the reciprocal of the rate.

### Final Answer
The leak will empty the cistern alone in **120 hours**.