A pipe can fill a tank in 20 h. due to a leak in the bottom , it is filled in 40 h .if the tank is full how much time will the leak take to empty it ?

Options

A. 40h
B. 30h
C. 50h
D. 30h

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A pipe can fill a tank in 20 hours.
- Due to a leak, the tank is filled in 40 hours.
- We need to find out how long it will take for the leak to empty the tank.

2. **Assuming the Capacity of the Tank**:
- Let's assume the capacity of the tank is 40 units (this is chosen because it is the LCM of 20 and 40, making calculations easier).

3. **Calculating the Filling Rate of the Pipe**:
- The pipe fills the tank in 20 hours, so its rate of filling is:
\[ \text{Rate of Pipe} = \frac{40 \text{ units}}{20 \text{ hours}} = 2 \text{ units per hour} \]

4. **Calculating the Combined Rate of Pipe and Leak**:
- The pipe and the leak together fill the tank in 40 hours, so their combined rate is:
\[ \text{Rate of Pipe + Leak} = \frac{40 \text{ units}}{40 \text{ hours}} = 1 \text{ unit per hour} \]

5. **Setting Up the Equation**:
- Let \( L \) be the rate of the leak (in units per hour).
- From the above, we know:
\[ \text{Rate of Pipe} + \text{Rate of Leak} = \text{Rate of Pipe + Leak} \]
\[ 2 + L = 1 \]

6. **Solving for the Leak Rate**:
- Rearranging the equation gives:
\[ L = 1 - 2 = -1 \text{ unit per hour} \]
- The negative sign indicates that the leak is emptying the tank.

7. **Calculating the Time for the Leak to Empty the Tank**:
- The capacity of the tank is 40 units, and the leak's rate is 1 unit per hour (since it empties the tank).
- The time taken by the leak to empty the tank is:
\[ \text{Time} = \frac{\text{Capacity of Tank}}{\text{Rate of Leak}} = \frac{40 \text{ units}}{1 \text{ unit per hour}} = 40 \text{ hours} \]

### Final Answer:
The leak will take **40 hours** to empty the tank.

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