Calculating Pipe Rates
The rate at which the inlet pipe fills the tank is the reciprocal of the time it takes to fill it.
- Inlet pipe rate: $\frac{1}{4}$ of the tank per hour.
- Outlet pipe rate: $\frac{1}{6}$ of the tank per hour.
Determining Net Filling Rate
When both pipes are open, the net rate is the difference between the filling rate and the emptying rate.
Net filling rate = Inlet rate - Outlet rate
Net filling rate = $\frac{1}{4} - \frac{1}{6}$
To subtract, find a common denominator, which is 12:
Net filling rate = $\frac{3}{12} - \frac{2}{12} = \frac{1}{12}$ of the tank per hour.
Finding Time to Half-Fill the Tank
The net rate of $\frac{1}{12}$ means the tank fills completely in 12 hours (since Time = Total Work / Rate = $1 \div \frac{1}{12} = 12$ hours).
We need to find the time to fill half the tank (i.e., $\frac{1}{2}$ of the tank).
Time to half-fill = (Amount to fill) / (Net filling rate)
Time to half-fill = $\frac{1}{2} \div \frac{1}{12}$
Time to half-fill = $\frac{1}{2} \times 12 = 6$ hours.
Therefore, the tank will be half-full in 6 hours.
Similar Questions
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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
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Pipe J can fill a tank in 48 hours and pipe K can fill the same tank in 72 hours. If both the pipes are opened alternately for one hour each and pipe K is opened for the first hour, then in how much time the tank will be full?
