A tap can fill a cistern in 32 hours, while another tap can empty the full cistern in 64 hours. If the cistern is initially empty and both taps are opened together, find the time (in hours) required to fill one-fourth of the cistern.
Calculating Cistern Fill Time with Combined Taps
This problem involves calculating the time taken to fill a portion of a cistern when one tap fills it and another empties it simultaneously.
Step 1: Determine Individual Tap Rates
We first find the rate at which each tap works:
- Rate of the filling tap (Tap 1): Fills $\frac{1}{32}$ of the cistern per hour.
- Rate of the emptying tap (Tap 2): Empties $\frac{1}{64}$ of the cistern per hour.
Step 2: Calculate the Net Filling Rate
When both taps are open, the net rate is the filling rate minus the emptying rate:
Net Rate = Rate (Fill) - Rate (Empty)
Net Rate = $\frac{1}{32} - \frac{1}{64}$
To subtract, find a common denominator (64):
Net Rate = $\frac{2}{64} - \frac{1}{64} = \frac{1}{64}$
So, the net rate of filling is $\frac{1}{64}$ of the cistern per hour.
Step 3: Calculate Time to Fill the Entire Cistern
The time required to fill the entire cistern (1 whole unit) is the reciprocal of the net filling rate:
Time (Full Cistern) = $\frac{1}{\text{Net Rate}} = \frac{1}{1/64} = 64$ hours.
Step 4: Calculate Time to Fill One-Fourth of the Cistern
The question asks for the time to fill only one-fourth ($\frac{1}{4}$) of the cistern. We multiply the time to fill the full cistern by the required fraction:
Time (1/4 Cistern) = Time (Full Cistern) $\times \frac{1}{4}$
Time (1/4 Cistern) = $64 \text{ hours} \times \frac{1}{4}$
Time (1/4 Cistern) = $16$ hours.
Therefore, it takes 16 hours to fill one-fourth of the cistern when both taps are opened 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?
