A pipe can fill a cistern in 20 minutes where as the cistern when full can be emptied by a leak in 28 minutes. When both are opened, The time taken to fill the cistern is:
70 min
Cistern Filling Time Calculation
This problem involves calculating the time taken to fill a cistern when a filling pipe and an emptying leak are operating simultaneously.
Determine Individual Rates
- Filling Pipe Rate: The pipe fills the cistern in 20 minutes. So, in 1 minute, it fills $\frac{1}{20}$ of the cistern.
- Leak Emptying Rate: The leak empties the full cistern in 28 minutes. So, in 1 minute, it empties $\frac{1}{28}$ of the cistern.
Calculate Net Filling Rate
When both the pipe and the leak are open, the net rate at which the cistern is filled is the difference between the filling rate and the emptying rate.
- Net Filling Rate per minute = (Rate of Filling) - (Rate of Emptying)
- Net Filling Rate = $\frac{1}{20} - \frac{1}{28}$
- To subtract these fractions, find a common denominator. The Least Common Multiple (LCM) of 20 and 28 is 140.
- Net Filling Rate = $\frac{7}{140} - \frac{5}{140} = \frac{7 - 5}{140} = \frac{2}{140}$
- Simplify the fraction: Net Filling Rate = $\frac{1}{70}$ of the cistern per minute.
Calculate Total Time to Fill
The time taken to fill the cistern is the reciprocal of the net filling rate.
- Time = $\frac{1}{\text{Net Filling Rate per minute}}$
- Time = $\frac{1}{\frac{1}{70}}$ minutes
- Time = 70 minutes
Therefore, when both the pipe and the leak are open, it takes 70 minutes to fill the cistern.
Important Questions from Pipe and Cistern
‘A’ pipe can empty a tank in 20 minutes. The second pipe ‘B’ has a diameter twice as that of ‘A’. If both A & B pipe are attached to the tank how much time will be required to empty the tank?
Pipes A and B can empty a full tank in 16 hours and 24 hours, respectively. Pipe C alone can fill the empty tank in 4 hours. If A, B and C are opened together, the tank will be 35% full after :
Pipes A and B can fill a tank in 36 minutes and 45 minutes, respectively. Both these pipes were opened simultaneously. After 20 minutes, a leak at the bottom of the tank was spotted which was immediately sealed. The tank was full in another 15 minutes. The leak alone can empty the full tank in:
A cistern has a leak which would empty it in 6 hours. A tap is turned on which admits 10 litres of water per minute into the cistern. When it is full it is now emptied in 10 hours. What is the capacity (in litres) of the cistern?
A tank has two inlets, A and B, which can fill it in 15 hours and 20 hours, respectively. An outlet C can empty the full tank in 12 hours. If A, B and C are opened together when the tank is empty, then in how much time will the tank be filled?
