\(7\frac{17}{31} \) 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
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:
