A pump can fill a tank with water in 3 hours .Because of a leak it took `3(1)/(2) `hours to fill tank .In how many hours can the leak alone drain all the water of the tank when it is full ?

Options

A. 21
B. 10
C. 30
D. 15

Answer

To solve the problem, we need to determine how long it will take for the leak alone to drain the tank when it is full. Let's break down the solution step by step.

### Step 1: Determine the rate of the pump
The pump can fill the tank in 3 hours. Therefore, the rate at which the pump fills the tank is:
\[ \text{Rate of pump} = \frac{1 \text{ tank}}{3 \text{ hours}} = \frac{1}{3} \text{ tank per hour} \]

**Hint:** To find the rate of a filling or draining device, divide the total work (in this case, filling the tank) by the time taken.

### Step 2: Determine the time taken to fill the tank with the leak
Due to the leak, it takes \(3 \frac{1}{2}\) hours (or \(3.5\) hours) to fill the tank. Therefore, the effective rate of filling the tank with the leak is:
\[ \text{Effective rate with leak} = \frac{1 \text{ tank}}{3.5 \text{ hours}} = \frac{1}{3.5} = \frac{2}{7} \text{ tank per hour} \]

**Hint:** Convert mixed numbers to improper fractions or decimals to simplify calculations.

### Step 3: Determine the rate of the leak
Let the rate at which the leak drains the tank be \(L\) (in tanks per hour). The relationship between the rates can be established as follows:
\[ \text{Rate of pump} - \text{Rate of leak} = \text{Effective rate with leak} \]
Substituting the known values:
\[ \frac{1}{3} - L = \frac{2}{7} \]

**Hint:** Set up an equation based on the relationship between the rates of filling and draining.

### Step 4: Solve for the rate of the leak
To solve for \(L\), we first need to find a common denominator for the fractions. The least common multiple of 3 and 7 is 21. Rewriting the fractions:
\[ \frac{1}{3} = \frac{7}{21} \quad \text{and} \quad \frac{2}{7} = \frac{6}{21} \]
Now substituting these values into the equation:
\[ \frac{7}{21} - L = \frac{6}{21} \]
Rearranging gives:
\[ L = \frac{7}{21} - \frac{6}{21} = \frac{1}{21} \text{ tank per hour} \]

**Hint:** When solving for a variable, isolate it on one side of the equation.

### Step 5: Determine the time taken by the leak to drain the full tank
If the leak drains at a rate of \(\frac{1}{21}\) tank per hour, the time taken by the leak to drain the full tank is the reciprocal of the rate:
\[ \text{Time taken by leak} = \frac{1 \text{ tank}}{L} = \frac{1}{\frac{1}{21}} = 21 \text{ hours} \]

**Hint:** To find the time taken to complete a task at a certain rate, take the reciprocal of the rate.

### Conclusion
The leak alone can drain all the water of the tank when it is full in **21 hours**.