Solving Time and Work: Pipes X, Y, and Z Filling a Tank

This problem involves calculating the combined time taken by multiple pipes to fill a tank when their individual pair-wise filling times are given. We need to find the time when pipes X, Y, and Z operate simultaneously.

Understanding the Problem

We are given the time taken for pairs of pipes (X and Y, Y and Z, Z and X) to fill the tank. Our goal is to find the time taken when all three pipes work together.

Step-by-Step Solution

Step 1: Define Rates of Pipes

Let the time taken by pipe X alone to fill the tank be \( T_X \) hours, pipe Y alone be \( T_Y \) hours, and pipe Z alone be \( T_Z \) hours.

The rate at which each pipe fills the tank is the reciprocal of the time taken:

• Rate of pipe X = \( \frac{1}{T_X} \) tank per hour
• Rate of pipe Y = \( \frac{1}{T_Y} \) tank per hour
• Rate of pipe Z = \( \frac{1}{T_Z} \) tank per hour

Step 2: Formulate Equations from Given Information

We are given the time taken by pairs of pipes. We can express this in terms of their combined rates:

• X and Y together fill the tank in 45 hours:

  \( \frac{1}{T_X} + \frac{1}{T_Y} = \frac{1}{45} \) (Equation 1)

• Y and Z together fill the tank in 36 hours:

  \( \frac{1}{T_Y} + \frac{1}{T_Z} = \frac{1}{36} \) (Equation 2)

• Z and X together fill the tank in 72 hours:

  \( \frac{1}{T_Z} + \frac{1}{T_X} = \frac{1}{72} \) (Equation 3)

Step 3: Combine the Equations to Find the Total Rate

To find the combined rate of all three pipes (\( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} \)), let's add all three equations:

\( \left( \frac{1}{T_X} + \frac{1}{T_Y} \right) + \left( \frac{1}{T_Y} + \frac{1}{T_Z} \right) + \left( \frac{1}{T_Z} + \frac{1}{T_X} \right) = \frac{1}{45} + \frac{1}{36} + \frac{1}{72} \)

Combining like terms on the left side:

\( 2\left( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} \right) = \frac{1}{45} + \frac{1}{36} + \frac{1}{72} \)

Step 4: Calculate the Sum of the Fractions

To add the fractions on the right side, we need a common denominator. The Least Common Multiple (LCM) of 45, 36, and 72 is 360.

• \( \frac{1}{45} = \frac{1 \times 8}{45 \times 8} = \frac{8}{360} \)
• \( \frac{1}{36} = \frac{1 \times 10}{36 \times 10} = \frac{10}{360} \)
• \( \frac{1}{72} = \frac{1 \times 5}{72 \times 5} = \frac{5}{360} \)

Now, substitute these back into the combined equation:

\( 2\left( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} \right) = \frac{8}{360} + \frac{10}{360} + \frac{5}{360} \)

\( 2\left( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} \right) = \frac{8 + 10 + 5}{360} \)

\( 2\left( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} \right) = \frac{23}{360} \)

Step 5: Find the Combined Rate of All Three Pipes

Divide both sides by 2 to find the combined rate of X, Y, and Z working together:

\( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} = \frac{23}{360 \times 2} \)

\( \frac{1}{T_X} + \frac{1}{T_Y} + \frac{1}{T_Z} = \frac{23}{720} \) tank per hour

This is the rate at which all three pipes fill the tank simultaneously.

Step 6: Calculate the Total Time Taken

The time taken for all three pipes to fill the tank together is the reciprocal of their combined rate:

Time = \( \frac{1}{\left( \frac{23}{720} \right)} \) hours

Time = \( \frac{720}{23} \) hours

Step 7: Convert the Time to a Mixed Fraction

To express \( \frac{720}{23} \) hours as a mixed fraction, divide 720 by 23:

720 ÷ 23 = 31 with a remainder of 7.

So, \( \frac{720}{23} = 31 \frac{7}{23} \) hours.

Conclusion

Therefore, the number of hours in which all three pipes (X, Y, and Z) can fill the tank if operated simultaneously is \( 31 \frac{7}{23} \) hours.