Pipe Filling Rates Calculation

This problem involves calculating the time it takes for multiple pipes, P, Q, and R, to fill a tank together. We need to determine the filling/emptying rate of each pipe and then find their combined rate to solve the question: Pipes P, Q and R together can fill the empty tank in:

Step 1: Determine the filling rate of Pipe P

Pipe P fills $\frac{3}{8}$ of the tank in 21 hours.

The rate of filling for Pipe P is calculated as:

$ \text{Rate P} = \frac{\text{Work Done}}{\text{Time Taken}} $

$ \text{Rate P} = \frac{3/8 \text{ tank}}{21 \text{ hours}} = \frac{3}{8 \times 21} = \frac{1}{8 \times 7} = \frac{1}{56} \text{ tank per hour.} $

This means Pipe P fills $\frac{1}{56}$ of the tank every hour.

Step 2: Determine the filling rate of Pipe Q

Pipe Q fills $\frac{2}{7}$ of the tank in 16 hours.

The rate of filling for Pipe Q is calculated similarly:

$ \text{Rate Q} = \frac{\text{Work Done}}{\text{Time Taken}} $

$ \text{Rate Q} = \frac{2/7 \text{ tank}}{16 \text{ hours}} = \frac{2}{7 \times 16} = \frac{1}{7 \times 8} = \frac{1}{56} \text{ tank per hour.} $

Pipe Q also fills $\frac{1}{56}$ of the tank every hour.

Step 3: Determine the emptying rate of Pipe R

The problem provides information about Pipe R's action in a specific context: "Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 3 hours." We use this scenario to find R's rate.

First, let's calculate the amount of water filled by pipes P and Q together during the 2 hours they were open:

$ \text{Work done by P and Q in 2 hours} = (\text{Rate P} + \text{Rate Q}) \times \text{Time} $

$ \text{Work} = \left( \frac{1}{56} + \frac{1}{56} \right) \times 2 \text{ hours} $

$ \text{Work} = \left( \frac{2}{56} \right) \times 2 = \frac{1}{28} \times 2 = \frac{1}{14} \text{ tank.} $

The problem states that Pipe R emptied *this* specific amount of water ($\frac{1}{14}$ tank) in 3 hours. From this, we can calculate R's emptying rate:

$ \text{Rate R} = \frac{\text{Work Done (emptied)}}{\text{Time Taken}} $

$ \text{Rate R} = \frac{1/14 \text{ tank}}{3 \text{ hours}} = \frac{1}{14 \times 3} = \frac{1}{42} \text{ tank per hour.} $

Since Pipe R's function is to empty the tank, its rate is considered negative when calculating the combined effect of filling pipes:

$ \text{Rate R (effective)} = -\frac{1}{42} \text{ tank per hour.} $

Step 4: Calculate the combined filling rate of P, Q, and R

The final question asks how long pipes P, Q, and R *together* can fill an *empty* tank. To find this, we sum their individual rates, remembering that R's rate is negative because it empties:

$ \text{Combined Rate} = \text{Rate P} + \text{Rate Q} + \text{Rate R (effective)} $

$ \text{Combined Rate} = \frac{1}{56} + \frac{1}{56} + \left(-\frac{1}{42}\right) $

First, combine the rates of P and Q:

$ \text{Combined Rate} = \frac{2}{56} - \frac{1}{42} = \frac{1}{28} - \frac{1}{42} $

To subtract these fractions, we find a common denominator. The Least Common Multiple (LCM) of 28 and 42 is 84.

$ \text{Combined Rate} = \frac{1 \times 3}{28 \times 3} - \frac{1 \times 2}{42 \times 2} = \frac{3}{84} - \frac{2}{84} $

$ \text{Combined Rate} = \frac{3 - 2}{84} = \frac{1}{84} \text{ tank per hour.} $

Step 5: Calculate the time taken to fill the empty tank

The combined rate calculated ($\frac{1}{84}$ tank per hour) is positive, indicating that pipes P, Q, and R working together can indeed fill the empty tank.

The time required to fill the entire tank (which represents 1 unit of work) is determined by dividing the total work by the combined rate:

$ \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} $

$ \text{Time} = \frac{1 \text{ tank}}{1/84 \text{ tank per hour}} = 1 \times \frac{84}{1} = 84 \text{ hours.} $

Final Answer

Therefore, pipes P, Q, and R working together can fill the empty tank in 84 hours.