Tap A can fill the empty tank in 12 hours, but due to a leak in the bottom it is filled in 15 hours. If the tank is full and then tap A is closed then in how many hours the leak can empty it?

Options

A. 45 hours
B. 48 hours
C. 52 hours
D. 60 hours

Answer

To solve the problem, we need to determine how long it takes for the leak to empty the tank after it has been filled by tap A. Here’s a step-by-step solution:

### Step 1: Determine the filling rate of Tap A
Tap A can fill the tank in 12 hours. Therefore, the rate at which Tap A fills the tank is:
\[ \text{Rate of Tap A} = \frac{1 \text{ tank}}{12 \text{ hours}} = \frac{1}{12} \text{ tanks per hour} \]

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

### Step 2: Determine the combined filling rate of Tap A and the leak
Due to the leak, the tank is filled in 15 hours. Therefore, the combined rate of Tap A and the leak is:
\[ \text{Rate of Tap A + Leak} = \frac{1 \text{ tank}}{15 \text{ hours}} = \frac{1}{15} \text{ tanks per hour} \]

**Hint:** Similar to the previous step, divide 1 by the time it takes to fill the tank with both the tap and the leak.

### Step 3: Set up the equation for the leak's rate
Let the rate of the leak be \( L \) (in tanks per hour). The equation for the combined rates is:
\[ \text{Rate of Tap A} + \text{Rate of Leak} = \text{Rate of Tap A + Leak} \]
Substituting the known values:
\[ \frac{1}{12} + L = \frac{1}{15} \]

**Hint:** Use the equation of rates to find the unknown rate of the leak.

### Step 4: Solve for the leak's rate
To solve for \( L \), we first find a common denominator for the fractions. The least common multiple of 12 and 15 is 60. Rewriting the equation:
\[ \frac{5}{60} + L = \frac{4}{60} \]
Now, isolate \( L \):
\[ L = \frac{4}{60} - \frac{5}{60} = -\frac{1}{60} \]

**Hint:** When you isolate the variable, make sure to subtract correctly.

### Step 5: Interpret the leak's rate
The negative sign indicates that the leak is emptying the tank at a rate of \( \frac{1}{60} \) tanks per hour. This means the leak empties 1 tank in 60 hours.

**Hint:** A negative rate indicates removal or emptying, while a positive rate indicates filling.

### Step 6: Conclusion
Thus, the time taken by the leak to empty the full tank is:
\[ \text{Time} = \frac{1 \text{ tank}}{L} = \frac{1}{-\frac{1}{60}} = 60 \text{ hours} \]

The leak can empty the tank in **60 hours**.

**Final Answer:** The leak can empty the tank in **60 hours**.