A tap fills a cistern in 14 hours. Another tap empties the full tank in 18 hours. How long (in hours) will it take to fill the tank completely, if the tank is empty and both the taps are open together?
This question involves calculating the combined time taken to fill a cistern when one tap is filling it and another tap is simultaneously emptying it. We need to determine the net rate at which the cistern is filled.
Understanding Tap Rates
First, let's determine the rate at which each tap works. The rate is the portion of the cistern that the tap can fill or empty in one hour.
- Filling Tap Rate: The first tap fills the cistern in 14 hours. So, in 1 hour, it fills $1/14$ of the cistern.
- Emptying Tap Rate: The second tap empties the cistern in 18 hours. So, in 1 hour, it empties $1/18$ of the cistern.
Calculating Net Filling Rate
When both taps are open, the filling tap adds water, and the emptying tap removes water. The net rate of filling is the difference between the filling rate and the emptying rate.
Net Filling Rate = (Rate of Filling Tap) - (Rate of Emptying Tap)
Net Filling Rate = $\frac{1}{14} - \frac{1}{18}$
To subtract these fractions, we find a common denominator. The least common multiple (LCM) of 14 and 18 is 126.
Net Filling Rate = $\frac{1 \times 9}{14 \times 9} - \frac{1 \times 7}{18 \times 7}$
Net Filling Rate = $\frac{9}{126} - \frac{7}{126}$
Net Filling Rate = $\frac{9 - 7}{126}$
Net Filling Rate = $\frac{2}{126}$
Simplifying the fraction:
Net Filling Rate = $\frac{1}{63}$
This means that when both taps are open, $\frac{1}{63}$ of the cistern is filled every hour.
Calculating Total Time to Fill
To find the total time it takes to fill the entire cistern (1 whole cistern), we take the reciprocal of the net filling rate.
Time = $\frac{\text{Total Work}}{\text{Net Filling Rate}}$
Time = $\frac{1}{\frac{1}{63}}$
Time = $63$ hours
Therefore, it will take 63 hours to fill the tank completely when both taps are open.
Important Questions from Pipe and Cistern
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:
‘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?
