Directions: Solve the following problem.

A tap can fill an empty water tank in 12 hours, while a second tap can fill it in 36 hours. A leakage outlet can drain a full tank in 60 hours. Both taps are opened together for 15 minutes. Then the second tap is closed and the leakage outlet is opened, while the first tap remains open. How long will the first tap and the leakage outlet take to fill the remaining part of the tank?

The correct answer is $14\frac{7}{12}\text{hours}$.

Step-by-Step Explanation:

Step 1: Determine the individual rates of the taps and the leakage outlet.
Let the total capacity of the water tank be 1 unit (the whole tank).

- Rate of the first tap = $\frac{1}{12}$ of the tank per hour.

- Rate of the second tap = $\frac{1}{36}$ of the tank per hour.

- Rate of the leakage outlet = $-\frac{1}{60}$ of the tank per hour (negative because it drains water).

Step 2: Calculate the part of the tank filled in the first 15 minutes.
First, convert 15 minutes into hours:

$\text{Time}=\frac{15}{60}=\frac{1}{4}\text{hour}$

Both the first and second taps are opened together for this period. Their combined filling rate per hour is:

$\text{Combined Rate}=\frac{1}{12}+\frac{1}{36}=\frac{3+1}{36}=\frac{4}{36}=\frac{1}{9}\text{of the tank per hour}$

The portion of the tank filled in $\frac{1}{4}$ hour is:

$\text{Part Filled}=\frac{1}{9}\times \frac{1}{4}=\frac{1}{36}\text{of the tank}$

Step 3: Calculate the remaining part of the tank to be filled.

$\text{Remaining Part}=1-\frac{1}{36}=\frac{35}{36}\text{of the tank}$

Step 4: Calculate the combined rate of the first tap and the leakage outlet.
After 15 minutes, the second tap is closed, and the leakage outlet is opened while the first tap remains open. Their net combined rate per hour is:

$\text{Net Rate}=\frac{1}{12}-\frac{1}{60}=\frac{5-1}{60}=\frac{4}{60}=\frac{1}{15}\text{of the tank per hour}$

Step 5: Calculate the time required to fill the remaining part of the tank.

$\text{Time Required}=\frac{\text{Remaining Part}}{\text{Net Rate}}=\frac{35/36}{1/15}=\frac{35}{36}\times 15$

Simplifying the fraction by dividing both 15 and 36 by 3:

$\text{Time Required}=\frac{35\times 5}{12}=\frac{175}{12}\text{hours}$

Converting $\frac{175}{12}$ to a mixed fraction:

$\frac{175}{12}=14\frac{7}{12}\text{hours}$

Therefore, the first tap and the leakage outlet will take $14\frac{7}{12}\text{hours}$ to fill the remaining part of the tank.