A pump can fill a tank with water in 2 hours. Because of a leak in the tank it was taking `2(1)/(3)`, hours to fill the tank. The leak can drain all the water off the tank in :

Options

A. 8 hours
B. 7 hours
C. `4(1)/(3)` hours
D. 14 hours

Answer

To solve the problem step by step, we need to find out how long the leak can drain all the water from the tank.

### Step 1: Determine the rate of the pump
The pump can fill the tank in 2 hours. Therefore, the rate of the pump is:
\[ \text{Rate of pump} = \frac{1 \text{ tank}}{2 \text{ hours}} = \frac{1}{2} \text{ tanks per hour} \]

**Hint:** To find the rate of a pump, divide 1 (the whole tank) by the time it takes to fill the tank.

### Step 2: Convert the time taken with the leak into an improper fraction
The time taken to fill the tank with the leak is \(2 \frac{1}{3}\) hours. We convert this to an improper fraction:
\[ 2 \frac{1}{3} = \frac{7}{3} \text{ hours} \]

**Hint:** To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.

### Step 3: Determine the rate of filling with the leak
The rate of filling the tank with the leak is:
\[ \text{Rate with leak} = \frac{1 \text{ tank}}{\frac{7}{3} \text{ hours}} = \frac{3}{7} \text{ tanks per hour} \]

**Hint:** To find the rate when given time in hours, divide 1 by the time in hours (which may need to be converted to an improper fraction).

### Step 4: Determine the rate of the leak
The rate of the leak can be found by subtracting the rate of filling with the leak from the rate of the pump:
\[ \text{Rate of leak} = \text{Rate of pump} - \text{Rate with leak} \]
\[ \text{Rate of leak} = \frac{1}{2} - \frac{3}{7} \]

To perform this subtraction, we need a common denominator. The least common multiple of 2 and 7 is 14.
\[ \frac{1}{2} = \frac{7}{14}, \quad \frac{3}{7} = \frac{6}{14} \]
So,
\[ \text{Rate of leak} = \frac{7}{14} - \frac{6}{14} = \frac{1}{14} \text{ tanks per hour} \]

**Hint:** To subtract fractions, convert them to have a common denominator.

### Step 5: Determine the time taken by the leak to drain the tank
If the leak drains at a rate of \(\frac{1}{14}\) tanks per hour, then the time taken to drain the entire tank is the reciprocal of the rate:
\[ \text{Time taken by leak} = \frac{1 \text{ tank}}{\frac{1}{14} \text{ tanks per hour}} = 14 \text{ hours} \]

**Hint:** To find the time taken to drain the tank, take the reciprocal of the rate of the leak.

### Final Answer
The leak can drain all the water off the tank in **14 hours**.