Two pipes P and Q can fill a tank in 6 hours and 9 hours respectively, while a third pipe R can empty the tank in 12 hours. Initially, P and R are open for 4 hours. Then P is close... Two pipes P and Q can fill a tank in 6 hours and 9 hours respectively, while a third pipe R can empty the tank in 12 hours. Initially, P and R are open for 4 hours. Then P is closed and Q is opened. After 6 more hours R is closed. The total time taken to fill the tank (in hours) is _____ Read more

Steps to Solve

1. Calculate the filling rates of pipes P, Q, and R

The rate of pipe P is $\frac{1}{6}$ of the tank per hour,
the rate of pipe Q is $\frac{1}{9}$ of the tank per hour,
and the rate of pipe R is $\frac{1}{12}$ of the tank per hour.

2. Determine the amount filled by P and R in 4 hours

When both P and R are open, the net rate is: $$ \text{Net rate} = \frac{1}{6} - \frac{1}{12} $$

Calculating the common denominator: $$ \frac{1}{6} = \frac{2}{12} $$ So, $$ \text{Net rate} = \frac{2}{12} - \frac{1}{12} = \frac{1}{12} $$

In 4 hours, the amount filled is: $$ \text{Amount filled} = \text{Net rate} \times \text{Time} = \frac{1}{12} \times 4 = \frac{4}{12} = \frac{1}{3} $$

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

After 4 hours, the tank has $\frac{1}{3}$ filled, so the remaining portion is: $$ 1 - \frac{1}{3} = \frac{2}{3} $$

4. Determine the amount filled by P and Q in 6 hours

Next, pipe P is closed and pipe Q is opened with R closed after 6 hours of operation. The amount filled by P and Q is: $$ \text{Net rate} = \frac{1}{9} - \frac{1}{12} $$

Calculating this: $$ \text{Let the common denominator be } 36: $$ $$ \frac{1}{9} = \frac{4}{36}, \quad \frac{1}{12} = \frac{3}{36} $$ Thus, $$ \text{Net rate} = \left(\frac{4}{36} - \frac{3}{36}\right) = \frac{1}{36} $$

In 6 hours, the amount filled is: $$ \text{Amount filled} = \frac{1}{36} \times 6 = \frac{6}{36} = \frac{1}{6} $$

5. Calculate the total amount completed to fill the tank

Adding the amount filled in the previous steps gives: $$ \text{Total amount filled} = \frac{1}{3} + \frac{1}{6} $$

Finding a common denominator for addition (6): $$ \frac{1}{3} = \frac{2}{6} $$ So, $$ \text{Total amount filled} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} $$

6. Calculate the remaining time to fill the tank

Now, $\frac{1}{2}$ of the tank is left, and now only Q is open: $$ \text{Remaining time} = \frac{\text{Remaining amount}}{\text{rate of Q}} = \frac{\frac{1}{2}}{\frac{1}{9}} $$ Calculating this: $$ \text{Remaining time} = \frac{1}{2} \times 9 = 4.5 \text{ hours} $$

7. Calculate total time taken to fill the tank

Adding the hours used: $$ \text{Total time} = 4 + 6 + 4.5 = 14.5 \text{ hours} $$