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P fills the tank in 5 hours, Q fills it in 6 hours, and R empties it in 12 hours. If all three pipes are opened together, how long will it take to fill the tank?

Mathematics

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Answer

6017 hours\text{} \frac{60}{17} \text{ hours} \text{}

Step-by-step solution

4 steps

Here's how to solve the problem:

  1. Determine the individual rates of each pipe.

    Pipe P fills the tank in 5 hours, so its filling rate is 15\frac{1}{5} of the tank per hour.
    Pipe Q fills the tank in 6 hours, so its filling rate is 16\frac{1}{6} of the tank per hour.
    Pipe R empties the tank in 12 hours, so its emptying rate is −112-\frac{1}{12} of the tank per hour.

  2. Calculate the combined rate of all three pipes working together.

    The combined rate is the sum of the individual rates, with emptying rates being negative.

    Combined Rate=15+16−112\text{Combined Rate} = \frac{1}{5} + \frac{1}{6} - \frac{1}{12}

  3. Find a common denominator and sum the fractions.

    The least common multiple (LCM) of 5, 6, and 12 is 60.

    Combined Rate=1×125×12+1×106×10−1×512×5\text{Combined Rate} = \frac{1 \times 12}{5 \times 12} + \frac{1 \times 10}{6 \times 10} - \frac{1 \times 5}{12 \times 5} Combined Rate=1260+1060−560\text{Combined Rate} = \frac{12}{60} + \frac{10}{60} - \frac{5}{60} Combined Rate=12+10−560\text{Combined Rate} = \frac{12 + 10 - 5}{60} Combined Rate=22−560\text{Combined Rate} = \frac{22 - 5}{60} Combined Rate=1760 tank/hour\text{Combined Rate} = \frac{17}{60} \text{ tank/hour}

  4. Calculate the total time taken to fill the tank.

    The time taken is the reciprocal of the combined rate.

    Time=1Combined Rate\text{Time} = \frac{1}{\text{Combined Rate}} Time=11760\text{Time} = \frac{1}{\frac{17}{60}} Time=6017 hours\text{Time} = \frac{60}{17} \text{ hours}

    The tank will be filled in 6017\frac{60}{17} hours.

    The final answer is 6017 hours\text{} \frac{60}{17} \text{ hours} \text{}

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