This problem involves calculating the time taken by an emptying pipe when combined with two filling pipes. We need to determine the rate at which each pipe works and then use these rates to find the time the emptying pipe takes to empty the cistern alone.
Understanding Pipe Rates
The rate at which a pipe works is the fraction of the cistern it can fill or empty in one minute. If a pipe can fill a cistern in T minutes, its filling rate is $1/T$ of the cistern per minute.
- Pipe 1 fills the cistern in 40 minutes. Its filling rate is $R_1 = \frac{1}{40}$ cistern/minute.
- Pipe 2 fills the cistern in 14 minutes. Its filling rate is $R_2 = \frac{1}{14}$ cistern/minute.
- Let the pipe at the bottom, which empties the cistern, take E minutes to empty it. Its emptying rate is $R_E = \frac{1}{E}$ cistern/minute.
- When all three pipes are opened simultaneously, the cistern gets filled in 20 minutes. The combined rate of filling is $R_{combined} = \frac{1}{20}$ cistern/minute.
Combined Rate Calculation
The combined rate is the sum of the filling rates minus the emptying rate (since it removes water):
Combined Rate = (Rate of Pipe 1) + (Rate of Pipe 2) - (Rate of Emptying Pipe)
Using the rates calculated above:
$ R_{combined} = R_1 + R_2 - R_E $ $ \frac{1}{20} = \frac{1}{40} + \frac{1}{14} - \frac{1}{E} $Solving for the Emptying Pipe Time (E)
To find E, we first rearrange the equation:
$ \frac{1}{E} = \frac{1}{40} + \frac{1}{14} - \frac{1}{20} $Now, we find a common denominator for 40, 14, and 20. The least common multiple (LCM) of these numbers is 280.
Convert the fractions to have the denominator 280:
- $\frac{1}{40} = \frac{1 \times 7}{40 \times 7} = \frac{7}{280}$
- $\frac{1}{14} = \frac{1 \times 20}{14 \times 20} = \frac{20}{280}$
- $\frac{1}{20} = \frac{1 \times 14}{20 \times 14} = \frac{14}{280}$
Substitute these back into the equation for $\frac{1}{E}$:
$ \frac{1}{E} = \frac{7}{280} + \frac{20}{280} - \frac{14}{280} $ $ \frac{1}{E} = \frac{7 + 20 - 14}{280} $ $ \frac{1}{E} = \frac{27 - 14}{280} $ $ \frac{1}{E} = \frac{13}{280} $Now, to find E, we take the reciprocal of both sides:
$ E = \frac{280}{13} $Converting to Mixed Fraction
To express the time in a more understandable format, we convert the improper fraction $\frac{280}{13}$ into a mixed number:
Divide 280 by 13:
$ 280 \div 13 = 21 \text{ with a remainder of } 7 $So, the time taken is:
$ E = 21\frac{7}{13} \text{ minutes} $Therefore, the pipe at the bottom of the tank will take $21\frac{7}{13}$ minutes to empty the completely filled cistern if no other pipe is open.
Important Questions from Pipe and Cistern
Pipe P can fill $\frac{2}{3}$ part of a tank in 16 hours, and pipe Q can fill $\frac{5}{6}$ part of the same tank in 20 hours. Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 8 hours. Pipes P, Q and R together can fill the empty tank in:
- Pipe A can fill a tank in 18 hours, pipe B can fill the same tank in 30 hours and pipe C can fill the same tank in 18 hours. The time taken by them to fill the same tank if they operate together is::
Pipe P can fill $\frac{2}{3}$ part of a tank in 24 hours, and pipe Q can fill $\frac{2}{3}$ part of the same tank in 16 hours. Both P and Q were kept open for 4 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 5 hours. Pipes P, Q and R together can fill the empty tank in:
- Pipe P can fill $\frac{5}{7}$ part of a tank in 30 hours, and pipe Q can fill $\frac{2}{5}$ part of the same tank in 28 hours. Both P and Q were kept open for 7 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 8 hours. Pipes P, Q and R together can fill the empty tank in:
Pipe P can fill $\frac{3}{8}$ part of a tank in 21 hours and pipe Q can fill $\frac{2}{7}$ part of the same tank in 16 hours. Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 3 hours. Pipes P, Q and R together can fill the empty tank in:
