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
Pipes A and B can fill a rectangular tank in 40 minutes and 120 minutes, respectively. Pipe C can empty the completely filled same tank in 240 minutes. If pipes A, B and C are opened at the same time, then how many minutes will it take to fill the tank?
- Tap A can empty a tank in 30 minutes. Tap B can empty it in 15 minutes. If both the taps operate simultaneously, how much time is needed to empty the tank?
- A tank has three pipes A, B and C. A and B are inlet pipes while C is an outlet pipe. Pipe A can fill the task in 15 minutes. Pipe B can fill it in 20 minutes. Pipe C can empty it in 30 minutes. The pipes are opened in the following cycle.
Minute 1: Only A
Minute 2: Only B
Minute 3: Only C
The cycle repeats till the tank is full. How much time would it take to completely fill the tank? - Three pipes of diameters 2 cm, 3 cm and 4 cm are running together to fill a cistern. The smallest pipe alone can fill the cistern in 232 minutes. The amount of water flowing in each pipe is proportional to the square of its diameter. The time (in minutes) taken by all pipes running together to completely fill the cistern is:
- Pipe A can fill a tank three times as fast as another pipe B. If the two pipes working together can fill the tank in 36 minutes, then in what time will the slower pipe alone fill the tank?
- Two pipes, A and B, can fill a water tank in 1 hour and \(1\frac{1}{2}\) hours, respectively. Pipe A is opened. After 15 minutes without closing it, pipe B is also opened. How much more time will both the pipes take to fill the tank?
- A pipe can fill a tank in 5 hours, and another pipe can empty the same tank in 10 hours. How long will it take for both pipes to fill the tank when working together?
- Two taps, A and B, can fill a cistern in 6 hours and 7.5 hours, respectively. Both the taps are opened for 2 hours and then B is turned off. How much time will A take to fill the remaining cistern?
- A cistern has two inlets \(I_1\) and \(I_2\) which can fill it in 16 hours and 20 hours, respectively. An outlet can empty the full cistern in 12 hours. If all the three pipes are opened together in the empty cistern, how much time will they take to fill the cistern completely?
- A pipe can fill a sump with water in 2 hours. Because of a leak, it took $2 \frac{1}{3}$ hours to fill the sump. The leak can drain all the water of the sump in:
Important Questions from Pipe and Cistern
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The compound interest on Rs. 64,000 for 3 years, compounded annually at 7.5% p.a. is
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Two pipes can fill a tank in 10 hrs and 12 hrs, respectively, while the third can empty it in 20 hrs. If all the pipes are opened together, how much time will it take for the tank to be filled up?
