Pipe A usually fills a tank in 2 hours. On account of a leak at the bottom of the tank, it takes pipe A 30 more minutes to fill the tank. How long will the leak take to empty a full tank if pipe A is shut?

Options

A. 2 hours 30 minutes
B. 5 hours
C. 4 hours
D. 10 hours

Answer

To solve the problem, let's break it down step by step.

### Step 1: Determine the filling rate of Pipe A
Pipe A fills the tank in 2 hours. Therefore, its filling rate is:
\[ \text{Filling Rate of Pipe A} = \frac{1 \text{ tank}}{2 \text{ hours}} = \frac{1}{2} \text{ tanks per hour} \]

**Hint:** To find the filling rate, divide 1 tank by the time taken in hours.

### Step 2: Calculate the new filling time with the leak
Due to the leak, Pipe A takes an additional 30 minutes to fill the tank. This means it now takes:
\[ 2 \text{ hours} + 30 \text{ minutes} = 2 \text{ hours} + 0.5 \text{ hours} = 2.5 \text{ hours} \]

**Hint:** Convert minutes to hours by dividing by 60.

### Step 3: Determine the effective filling rate with the leak
Now, we calculate the effective filling rate when the leak is present:
\[ \text{Effective Filling Rate} = \frac{1 \text{ tank}}{2.5 \text{ hours}} = \frac{1}{2.5} \text{ tanks per hour} = \frac{2}{5} \text{ tanks per hour} \]

**Hint:** To find the effective filling rate, divide 1 tank by the new filling time in hours.

### Step 4: Find the rate of the leak
The difference in rates between Pipe A and the effective filling rate gives us the rate of the leak:
\[ \text{Rate of Leak} = \text{Filling Rate of Pipe A} - \text{Effective Filling Rate} \]
\[ \text{Rate of Leak} = \frac{1}{2} - \frac{2}{5} \]

To subtract these fractions, we need a common denominator. The least common multiple of 2 and 5 is 10:
\[ \frac{1}{2} = \frac{5}{10}, \quad \frac{2}{5} = \frac{4}{10} \]
So,
\[ \text{Rate of Leak} = \frac{5}{10} - \frac{4}{10} = \frac{1}{10} \text{ tanks per hour} \]

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

### Step 5: Calculate how long the leak will take to empty the tank
If the leak empties the tank at a rate of \(\frac{1}{10}\) tanks per hour, the time taken to empty 1 full tank is:
\[ \text{Time to empty the tank} = \frac{1 \text{ tank}}{\frac{1}{10} \text{ tanks per hour}} = 10 \text{ hours} \]

**Hint:** To find the time taken to empty a tank, divide 1 tank by the rate of the leak.

### Final Answer
The leak will take **10 hours** to empty a full tank if Pipe A is shut.