A pipe can fill a tank in 24 hrs. Due to a leakage in the bottom, it is filled in 36 hrs. If the tank is half full, how much time will the leak take to empty the tank?

Options

A. 48 hrs
B. 72 hrs
C. 36 hrs
D. 24 hrs

Answer

To solve the problem, we need to determine the rate at which the pipe fills the tank and the rate at which the leak empties the tank.

### Step-by-Step Solution:

1. **Determine the filling rate of the pipe:**
- The pipe can fill the tank in 24 hours.
- Therefore, the filling rate of the pipe is:
\[ \text{Filling rate of the pipe} = \frac{1 \text{ tank}}{24 \text{ hours}} = \frac{1}{24} \text{ tanks per hour} \]

2. **Determine the effective filling rate with leakage:**
- Due to the leakage, the tank is filled in 36 hours.
- Thus, the effective filling rate (pipe filling minus leakage) is:
\[ \text{Effective filling rate} = \frac{1 \text{ tank}}{36 \text{ hours}} = \frac{1}{36} \text{ tanks per hour} \]

3. **Set up the equation to find the leakage rate:**
- Let the rate at which the leak empties the tank be \( L \) (in tanks per hour).
- The effective filling rate can be expressed as:
\[ \text{Filling rate of the pipe} - \text{Leak rate} = \text{Effective filling rate} \]
- Substituting the known values:
\[ \frac{1}{24} - L = \frac{1}{36} \]

4. **Solve for \( L \):**
- Rearranging the equation gives:
\[ L = \frac{1}{24} - \frac{1}{36} \]
- To subtract these fractions, find a common denominator. The least common multiple of 24 and 36 is 72.
- Convert the fractions:
\[ L = \frac{3}{72} - \frac{2}{72} = \frac{1}{72} \text{ tanks per hour} \]

5. **Determine the time taken by the leak to empty half the tank:**
- If the tank is half full, the leak will take time \( T \) to empty it:
\[ \text{Time} = \frac{\text{Amount of tank}}{\text{Leak rate}} = \frac{0.5 \text{ tank}}{L} \]
- Substituting the value of \( L \):
\[ T = \frac{0.5}{\frac{1}{72}} = 0.5 \times 72 = 36 \text{ hours} \]

### Final Answer:
The leak will take **36 hours** to empty the tank when it is half full.