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
Two pipes A and B can independently fill a tank completely in 20 and 30 minutes respectively. If both the pipes are opened simultaneously, how much time will they take to fill the tank completely?
The compound interest on Rs. 64,000 for 3 years, compounded annually at 7.5% p.a. is
A water tank can be emptied in 40 minutes by a pipe of d 'diameter, so how long will it take for a 2d diameter pipe to be emptied?
A pipe can fill a tank in 4 hours, while a leak which is at one-fourth of the height of the tank from bottom can empty upto that part in 2 hours. If both are operated simultaneously and initially the tank is full, then when it will be one-fourth full?
Two pipes X and Y can fill an empty tank in 16 hours and 20 hours respectively. Pipe Z alone can empty the completely filled tank in 25 hours. Firstly both pipes X and Y are opened and after 6 hours pipe Z is also opened. What will be the total time (in hours) taken to completely fill the tank?
