A pump can fill a cistern in 2 hours. Because of a leak in the cistern it took `2(1)/(3)` hours to fill it. If the cistern is full, how much time will the leak take to empty it?

Options

A. 15 hour
B. 14 hour
C. 10 hour
D. 12 hour

Answer

To solve the problem step by step, we need to find out how long it will take for the leak to empty the cistern.

### Step 1: Determine the rate of the pump (R1)
The pump can fill the cistern in 2 hours. Therefore, the rate of the pump (R1) can be calculated as:
\[ R1 = \frac{\text{Volume}}{\text{Time}} \]
Assuming the volume of the cistern is V, we have:
\[ R1 = \frac{V}{2} \]

### Step 2: Determine the time taken to fill the cistern with the leak
The time taken to fill the cistern with the leak is given as \(2 \frac{1}{3}\) hours, which can be converted to an improper fraction:
\[ 2 \frac{1}{3} = \frac{7}{3} \text{ hours} \]

### Step 3: Determine the effective rate of filling (R1 - R2)
Let R2 be the rate of the leak. The effective rate of filling the cistern considering the leak is:
\[ R1 - R2 = \frac{V}{\frac{7}{3}} \]
This can be simplified to:
\[ R1 - R2 = \frac{3V}{7} \]

### Step 4: Set up the equation
Now we can set up the equation using the rates:
\[ R1 - R2 = \frac{3V}{7} \]
Substituting R1 from Step 1:
\[ \frac{V}{2} - R2 = \frac{3V}{7} \]

### Step 5: Solve for R2
To solve for R2, we first find a common denominator for the fractions. The common denominator of 2 and 7 is 14. We convert the fractions:
\[ \frac{V}{2} = \frac{7V}{14} \]
\[ \frac{3V}{7} = \frac{6V}{14} \]
Now, substituting back into the equation:
\[ \frac{7V}{14} - R2 = \frac{6V}{14} \]
Rearranging gives:
\[ R2 = \frac{7V}{14} - \frac{6V}{14} = \frac{1V}{14} \]

### Step 6: Calculate the time taken by the leak to empty the cistern
The rate of the leak (R2) is \(\frac{V}{14}\). To find the time (T) taken by the leak to empty the cistern, we use:
\[ R2 = \frac{V}{T} \]
Substituting R2:
\[ \frac{V}{14} = \frac{V}{T} \]
Cancelling V from both sides (assuming V ≠ 0):
\[ T = 14 \text{ hours} \]

### Final Answer
The leak will take **14 hours** to empty the cistern.

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