\(7\frac{17}{31} \) hours
This problem involves calculating the time it takes for two pipes, Pipe A and Pipe B, working together to fill half of an empty cistern. We are given the individual times each pipe takes to fill the cistern.
Understanding Pipe Rates
First, let's determine the rate at which each pipe fills the cistern. The rate is the fraction of the cistern that can be filled in one hour.
- Pipe A fills the cistern in 26 hours. So, the rate of Pipe A is $ \frac{1}{26} $ of the cistern per hour.
- Pipe B fills the cistern in 36 hours. So, the rate of Pipe B is $ \frac{1}{36} $ of the cistern per hour.
Calculating Combined Filling Rate
When both pipes operate together, their rates add up. We need to find the combined rate.
Combined Rate = Rate of Pipe A + Rate of Pipe B
Combined Rate = $ \frac{1}{26} + \frac{1}{36} $
To add these fractions, we find a common denominator. The Least Common Multiple (LCM) of 26 and 36 is 468.
- $26 = 2 \times 13$
- $36 = 2^2 \times 3^2$
- LCM(26, 36) = $2^2 \times 3^2 \times 13 = 4 \times 9 \times 13 = 468$
Now, we convert the fractions:
Combined Rate = $ \frac{1 \times 18}{26 \times 18} + \frac{1 \times 13}{36 \times 13} = \frac{18}{468} + \frac{13}{468} $
Combined Rate = $ \frac{18 + 13}{468} = \frac{31}{468} $ cistern per hour.
Determining Time to Fill Full Cistern
The time taken to fill the entire cistern together is the reciprocal of their combined rate.
Time to fill the full cistern = $ \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{31}{468}} $ hours
Time to fill the full cistern = $ \frac{468}{31} $ hours.
Calculating Time for Half Cistern
The question asks for the time taken to fill only half of the cistern. Therefore, we need half the time it takes to fill the full cistern.
Time to fill half cistern = $ \frac{1}{2} \times (\text{Time to fill the full cistern}) $
Time to fill half cistern = $ \frac{1}{2} \times \frac{468}{31} $ hours
Time to fill half cistern = $ \frac{468}{2 \times 31} = \frac{234}{31} $ hours.
Converting to Mixed Fraction
To express the answer in the format given in the options, we convert the improper fraction $ \frac{234}{31} $ into a mixed fraction.
Divide 234 by 31:
$ 234 \div 31 = 7 $ with a remainder of $ 234 - (7 \times 31) = 234 - 217 = 17 $.
So, the time taken is $ 7 \frac{17}{31} $ hours.
Thus, the time taken by Pipe A and Pipe B to fill half of the cistern together is $ 7 \frac{17}{31} $ hours.
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?
