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
Pipes A, B and C can fill a tank in 20, 30 and 60 hours, respectively. Pipes A, B and C are opened at 7 a.m., 8 a.m., and 9 a.m., respectively, on the same day. When will the tank be full?
There are two water taps in a tank which can fill the empty tank in 12 hours and 18 hours respectively. It is seen that there is a leakage point at the bottom of the tank which can empty the completely filled tank in 36 hours. If both the water taps are opened at the same time to fill the empty tank and leakage point was repaired after 1 hour, then in how much time the empty tank will be completely filled?
Two pipes A and B can fill a tank in 12 minutes and 24 minutes, respectively, while a third pipe C can empty the full tank in 32 minutes. All the three pipes are opened simultaneously. However, pipe C is closed 2 minutes before the tank is filled. In how much time (in minutes) will the tank be full?
Pipes A and B can fill a tank in 12 hours and 16 hours respectively and pipe C can empty the full tank in 24 hours. All three pipes are opened together, but after 4 hours pipe B is closed. In how many hours, the empty tank will be completely filled?
Pipes A and B can fill a tank in 43.2 minutes and 108 minutes, respectively. Pipe C can empty it at 3 litres/minute. When all the three pipes are opened together, they fill the tank in 54 minutes. The capacity (in litres) of the tank is:
