Understanding Pipe Filling and Emptying Rates

This problem deals with the concept of 'Time and Work', specifically applied to pipes filling and emptying a tank. To solve it, we need to calculate the rate at which each pipe works and then find the combined effect when they operate together.

Key Concepts:

• Filling Rate: If a pipe can fill a tank in a certain number of hours, its filling rate is the fraction of the tank it fills in one hour. For instance, a pipe filling a tank in 10 hours has a filling rate of $\frac{1}{10}$ of the tank per hour.
• Emptying Rate: Similarly, if a pipe empties a tank in a certain number of hours, its emptying rate is the fraction of the tank it empties in one hour. A pipe emptying a tank in 20 hours has an emptying rate of $\frac{1}{20}$ of the tank per hour.
• Net Rate: When multiple pipes work simultaneously, the net rate of filling is calculated by adding the rates of all filling pipes and subtracting the rates of all emptying pipes.

Calculating Combined Time for Tank Filling

Let's follow these steps to find the total time required to fill the tank:

1. Calculate the rate of the first pipe (filling):

   The first pipe fills the tank in 10 hours. Its filling rate is:

   Rate$_1 = \frac{1}{10}$ tank/hour.

2. Calculate the rate of the second pipe (filling):

   The second pipe fills the tank in 12 hours. Its filling rate is:

   Rate$_2 = \frac{1}{12}$ tank/hour.

3. Calculate the rate of the third pipe (emptying):

   The third pipe empties the tank in 20 hours. Its emptying rate is:

   Rate$_3 = \frac{1}{20}$ tank/hour.

4. Determine the combined net filling rate:

   When all pipes are opened together, the net rate is the sum of the filling rates minus the emptying rate:

   Net Rate = Rate$_1 +$ Rate$_2 -$ Rate$_3

   Substituting the values:

   Net Rate = $\frac{1}{10} + \frac{1}{12} - \frac{1}{20}$

   To combine these fractions, we find the Least Common Multiple (LCM) of the denominators 10, 12, and 20. The LCM is 60.

   Convert each fraction to have the denominator 60:

   Net Rate = $\frac{1 \times 6}{10 \times 6} + \frac{1 \times 5}{12 \times 5} - \frac{1 \times 3}{20 \times 3}$

   Net Rate = $\frac{6}{60} + \frac{5}{60} - \frac{3}{60}$

   Combine the numerators:

   Net Rate = $\frac{6 + 5 - 3}{60} = \frac{8}{60}$ tank/hour.

   Simplify the net rate:

   Net Rate = $\frac{2}{15}$ tank/hour.

5. Calculate the total time to fill the tank:

   The time taken to fill the tank is the reciprocal of the net filling rate:

   Time = $\frac{1}{\text{Net Rate}}$

   Time = $\frac{1}{2/15}$ hours

   Time = $\frac{15}{2}$ hours

   Converting this improper fraction to a decimal gives:

   Time = $7.5$ hours

Therefore, it will take 7.5 hours for the tank to be completely filled when all three pipes are opened simultaneously.