Solving Pipe and Tank Filling Problems

This problem involves calculating the time taken to fill a tank when different pipes are working together, some filling and some emptying.

Let's break down the problem step-by-step by finding the work rate of each pipe per hour.

• Pipe X fills the tank in 16 hours.
• Pipe Y fills the tank in 20 hours.
• Pipe Z empties the tank in 25 hours.

Step 1: Calculate the hourly rates of each pipe

The rate of a pipe is the fraction of the tank it can fill or empty in one hour.

• Rate of pipe X = \( \frac{1}{\text{Time taken by X}} = \frac{1}{16} \) of the tank per hour (filling).
• Rate of pipe Y = \( \frac{1}{\text{Time taken by Y}} = \frac{1}{20} \) of the tank per hour (filling).
• Rate of pipe Z = \( \frac{1}{\text{Time taken by Z}} = \frac{1}{25} \) of the tank per hour (emptying). Since it empties, its rate is negative for combined work.

Step 2: Calculate the combined rate of pipes X and Y when opened together

Initially, only pipes X and Y are open. Their combined filling rate is the sum of their individual rates.

Combined rate of X and Y = Rate of X + Rate of Y

\( \frac{1}{16} + \frac{1}{20} \)

To add these fractions, we find a common denominator, which is the LCM of 16 and 20. The LCM is 80.

\( \frac{1 \times 5}{16 \times 5} + \frac{1 \times 4}{20 \times 4} = \frac{5}{80} + \frac{4}{80} = \frac{5+4}{80} = \frac{9}{80} \) of the tank per hour.

Step 3: Calculate the amount of tank filled in the first 6 hours

Pipes X and Y work together for the first 6 hours. The amount filled is their combined rate multiplied by the time.

Amount filled in 6 hours = Combined rate of X and Y × Time

\( \frac{9}{80} \times 6 = \frac{54}{80} = \frac{27}{40} \) of the tank.

Step 4: Calculate the remaining capacity of the tank

The tank is not fully filled after 6 hours. The remaining capacity is the total capacity (1 whole tank) minus the amount filled.

Remaining capacity = \( 1 - \frac{27}{40} = \frac{40}{40} - \frac{27}{40} = \frac{40-27}{40} = \frac{13}{40} \) of the tank.

Step 5: Calculate the combined rate of pipes X, Y, and Z when all are open

After 6 hours, pipe Z is also opened. Now, pipes X and Y are filling, and pipe Z is emptying. The combined rate is the sum of the filling rates minus the emptying rate.

Combined rate of X, Y, and Z = Rate of X + Rate of Y - Rate of Z

\( \frac{1}{16} + \frac{1}{20} - \frac{1}{25} \)

To combine these fractions, find the LCM of 16, 20, and 25. The LCM is 400.

\( \frac{1 \times 25}{16 \times 25} + \frac{1 \times 20}{20 \times 20} - \frac{1 \times 16}{25 \times 16} = \frac{25}{400} + \frac{20}{400} - \frac{16}{400} = \frac{25+20-16}{400} = \frac{45-16}{400} = \frac{29}{400} \) of the tank per hour.

Since the combined rate is positive (\( \frac{29}{400} \)), the tank will continue to fill.

Step 6: Calculate the time taken to fill the remaining capacity with all three pipes open

The time taken to fill the remaining \( \frac{13}{40} \) of the tank at a combined rate of \( \frac{29}{400} \) per hour is:

Time = \( \frac{\text{Remaining Capacity}}{\text{Combined Rate of X, Y, Z}} \)

\( \frac{\frac{13}{40}}{\frac{29}{400}} = \frac{13}{40} \times \frac{400}{29} \)

Simplify the expression:

\( \frac{13}{\cancel{40}} \times \frac{\cancel{400}^{10}}{29} = 13 \times \frac{10}{29} = \frac{130}{29} \) hours.

Step 7: Calculate the total time taken to fill the tank

The total time is the sum of the time pipes X and Y worked alone and the time all three pipes worked together.

Total time = Time in first 6 hours + Time with X, Y, and Z open

Total time = \( 6 + \frac{130}{29} \)

Convert 6 to a fraction with denominator 29:

\( 6 = \frac{6 \times 29}{29} = \frac{174}{29} \)

Total time = \( \frac{174}{29} + \frac{130}{29} = \frac{174+130}{29} = \frac{304}{29} \) hours.

Therefore, the total time taken to completely fill the tank is \( \frac{304}{29} \) hours.

This matches one of the given options.