Cistern Filling and Emptying Rate Calculation
This problem involves calculating the net time taken to fill a cistern when one inlet (tap) and one outlet (pipe) are operating simultaneously. We determine the rate at which each device works and then find the combined rate.
Work Rate Analysis
- Tap Filling Rate: The tap fills the cistern in 5 hours. So, its filling rate is \( \frac{1}{5} \) of the cistern per hour.
- Outlet Pipe Emptying Rate: The outlet pipe empties the cistern in 9 hours. So, its emptying rate is \( \frac{1}{9} \) of the cistern per hour.
Combined Operation Rate
When both the tap and the outlet pipe are open, the tap adds water, and the pipe removes water. The net rate of filling is the difference between the filling rate and the emptying rate.
- Net Filling Rate = (Tap Filling Rate) - (Outlet Pipe Emptying Rate)
- Net Filling Rate = \( \frac{1}{5} - \frac{1}{9} \)
To subtract these fractions, we find a common denominator, which is 45.
- Net Filling Rate = \( \frac{9}{45} - \frac{5}{45} \)
- Net Filling Rate = \( \frac{4}{45} \) of the cistern per hour.
Time Calculation
The time required to fill the cistern is the reciprocal of the net filling rate.
- Time = \( \frac{1}{\text{Net Filling Rate}} \)
- Time = \( \frac{1}{\frac{4}{45}} \) hours
- Time = \( \frac{45}{4} \) hours
Convert to Hours and Minutes
To express \( \frac{45}{4} \) hours in hours and minutes:
- \( \frac{45}{4} = 11 \frac{1}{4} \) hours
- This means 11 full hours and \( \frac{1}{4} \) of an hour.
- Convert \( \frac{1}{4} \) hour to minutes: \( \frac{1}{4} \times 60 \text{ minutes} = 15 \text{ minutes} \).
- Therefore, the total time taken is 11 hours and 15 minutes.
Similar Questions
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- Two pipes A and B can fill an empty cistern in 18 and 27 hours, respectively. Pipe C can drain the entire cistern in 45 hours when no other pipe is in operation. Initially, when the cistern was empty Pipe A and Pipe C were turned on. After a few hours Pipe A was turned off and Pipe B was turned on instantly. In all, it took 55 hours to fill the cistern. For how many hours was Pipe B turned on?
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Important Questions from Pipe and Cistern
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