Solving Pipe Filling and Emptying Problems
This problem involves calculating the time taken to fill a tank when one pipe fills it and another empties it simultaneously.
Calculating Filling Pipe Rate
The first pipe fills the tank in 5 hours. Its filling rate is the amount of the tank it fills per hour.
- Rate of filling pipe = \(\frac{1}{5}\) tank per hour.
Calculating Emptying Pipe Rate
The second pipe empties the tank in 10 hours. Its emptying rate is the amount of the tank it empties per hour.
- Rate of emptying pipe = \(\frac{1}{10}\) tank per hour.
Combined Work Rate
When both pipes work together, the net rate at which the tank fills is the filling rate minus the emptying rate.
- Net filling rate = (Rate of filling pipe) - (Rate of emptying pipe)
- Net filling rate = \(\frac{1}{5} - \frac{1}{10}\)
- To subtract, find a common denominator, which is 10:
- Net filling rate = \(\frac{2}{10} - \frac{1}{10} = \frac{1}{10}\) tank per hour.
This means that when both pipes are open, \(\frac{1}{10}\) of the tank is filled every hour.
Total Time Calculation
To find the total time required to fill the tank, divide the total work (filling 1 tank) by the net filling rate.
- Time = \(\frac{\text{Total Work}}{\text{Net Filling Rate}}\)
- Time = \(\frac{1 \text{ tank}}{\frac{1}{10} \text{ tank per hour}}\)
- Time = \(1 \times \frac{10}{1}\) hours
- Time = 10 hours.
Therefore, it will take 10 hours for both pipes to fill the tank when working together.
Similar Questions
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Important Questions from Pipe and Cistern
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Pipe J can fill a tank in 48 hours and pipe K can fill the same tank in 72 hours. If both the pipes are opened alternately for one hour each and pipe K is opened for the first hour, then in how much time the tank will be full?
