Calculating Time to Fill Tank Fraction
The problem involves two pipes, one filling a tank and the other emptying it. We need to find the time it takes to fill two-thirds of the tank when both pipes operate simultaneously.
Pipe Rates
- Filling Pipe Rate: This pipe fills the tank in 18 minutes. Its rate is $\frac{1}{18}$ of the tank per minute.
- Emptying Pipe Rate: This pipe empties the full tank in 72 minutes. Its rate is $\frac{1}{72}$ of the tank per minute.
Net Filling Rate
When both pipes are open, the net rate at which the tank fills is the difference between the filling rate and the emptying rate.
Net filling rate = (Filling Pipe Rate) - (Emptying Pipe Rate)
Net filling rate = $\frac{1}{18} - \frac{1}{72}$
To subtract these fractions, find a common denominator, which is 72:
Net filling rate = $\frac{4}{72} - \frac{1}{72} = \frac{3}{72}$ tank/min
Simplifying the net rate:
Net filling rate = $\frac{1}{24}$ tank/min
Time to Fill Two-Thirds of the Tank
The net rate tells us that $\frac{1}{24}$ of the tank is filled every minute. Therefore, the entire tank would be filled in 24 minutes (since Time = Total Work / Rate = 1 / (1/24) = 24 minutes).
We need to find the time required to fill only two-thirds ($\frac{2}{3}$) of the tank.
Time to fill $\frac{2}{3}$ of the tank = (Total time to fill the tank) $\times$ (Fraction of the tank to fill)
Time = $24 \text{ min} \times \frac{2}{3}$
Time = $\frac{24 \times 2}{3} \text{ min}$
Time = $8 \times 2 \text{ min}$
Time = $16 \text{ min}$
Conclusion
It will take 16 minutes to fill two-thirds of the tank when both pipes are operated together.
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?
