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 step by step, let's break it down:

### Step 1: Determine the filling rate of Tap A
Tap A can fill the tank in 12 hours. Therefore, the rate of Tap A (in terms of tank per hour) 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 filling, divide 1 tank by the number of hours it takes to fill it.

### Step 2: Determine the effective filling rate with the leak
With the leak, the tank is filled in 15 hours. Thus, the effective rate of filling (Tap A minus the leak) is:
\[ \text{Effective Rate} = \frac{1 \text{ tank}}{15 \text{ hours}} = \frac{1}{15} \text{ tanks per hour} \]

**Hint:** Similar to step 1, divide 1 tank by the total time taken with the leak to find the effective rate.

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

**Hint:** Use the equation to find the leak's rate by rearranging it.

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

**Hint:** Subtract the fractions carefully, ensuring you have a common denominator.

### Step 5: Determine the time taken by the leak to empty the tank
The leak's rate is \( \frac{1}{60} \) tanks per hour, meaning it takes:
\[ \text{Time taken by leak} = \frac{1 \text{ tank}}{L} = \frac{1}{\frac{1}{60}} = 60 \text{ hours} \]

**Hint:** To find the time to empty the tank, take the reciprocal of the leak's rate.

### Conclusion
The leak can empty the full tank in **60 hours**.

**Final Answer:** 60 hours