Shortcut -Pipes and Cisterns
One pipe fills the tank and the other empties the tank.
$\dfrac{1}{T}=\dfrac{1}{A}-\dfrac{1}{B}$
The pipe which empties the tank is doing negative work. So we have to assign ‘-‘symbol for the work done by that pipe.
T – Total time taken to fill the tank.
Question:
Pipe A can fill a tank in 4 hours and pipe B can empty the tank in 5 hours. If both pipes are opened together, how long will they take to fill a complete tank?
Answer: A = 4;
B=5
Substitute the values in the above equation, we get
Time taken to fill the tank = (4 x 5)/(-4 + 5)
= 20 hours
Shortcut -Pipes and Cisterns
Two pipes fill a tank and another pipes empties the tank. $\dfrac{1}{T}=\dfrac{1}{A}+\dfrac{1}{B}-\dfrac{1}{C}$
Question:
Pipes A and B can fill a tank in 10 hours and 12 hours respectively. Pipe C can empty a tank in 20 hours. If all the pipes are opened together, how long will they take to fill a complete tank?
Answer: A = 10;
B = 12;
C = 20
Substitute the values in the above equation, we get
Time taken to fill the tank
= (10 x 12 x 20 )/[(-10 x 12) + (12 x 20) + (10 x 20)]
= 7. 5 hours
= 7 hours 30 minutes
A pipe is used to fill or
empty the tank or cistern.
Inlet Pipe: A pipe used to fill the tank or cistern is known as Inlet Pipe.
Outlet Pipe: A pipe used to empty the tank or cistern is known as Outlet
Pipe.
Basic Formulas of Pipes and Cisterns
1.If an inlet pipe can fill the tank in x hours, then the part filled in 1 hour = $\dfrac{1}
{x}$
2.If an outlet pipe can empty the tank in y hours, then the part of the tank emptied in 1 hour =
$\dfrac{1}{y}$
3.If both inlet and outlet valves are kept open, then the net part of the tank filled in 1 hour is
$\dfrac{1}{x}-\dfrac{1}{y}$
Shortcut Methods for Pipes and Cisterns
Rule 1:Two pipes can fill (or empty) a cistern in x and y hours while working alone. If both pipes are opened together, then the time taken to fill (or empty) the cistern is given by $(\dfrac{xy}{x+y})$hours.
Example:
Two pipes A and B can fill a cistern in 20 and 30 minutes respectively. If both the pipes are opened together, how long will it take to fill the cistern?
Solution:Let’s say x = 20;
y = 30
$(\dfrac{xy}{x+y})$=$(\dfrac{20 \times 30}{20+30})$
=12 minutes.
So it willl take 12 minutes for both the pipes to full the cistern.
Three pipes can fill (or empty) a cistern in x, y and z hours while working alone. If all the three pipes are opened together, the time taken to fill (or empty) the cistern is given by $\dfrac{xyz}{xy+yz+zx}$hours
Example:
Three pipes can fill a tank in 20 minutes, 30 minutes and 40 minutes respectively while working alone. If, all the pipes are opened together, how long will it take to fill the tank full?
Solution:Let’s say x = 20 minutes;
y = 30 minutes;
z = 40 minutes
$\dfrac{xyz}{xy+yz+zx}$=$\dfrac{20 \times 30 \times 40 }{20 \times 30+30 \times 40 +40
\times 20 }$
=9.23
So it will take 9.23 minutes to fill the tank full.
If a pipe can fill a cistern in x hours and another can fill the same cistern
in y hours, but a third one can empty the full tank in z hours, and all of them are opened
together, then Net part filled in 1 hour = $\dfrac{1}{x}+\dfrac{1}{y}-\dfrac{1}{z}$
Time taken to fill the full cistern=$\dfrac{xyz}{yz+xz-xy}$
Example:
Two pipes can fill a cistern in 20 minutes and 30 minutes respectively. Third pipe can empty the tank in 40 minutes. If all the three pipes are opened together, how long it will take to fill the tank full?
Solution:Let’s say x = 20;
y = 30 ;
z = 40;
$\dfrac{xyz}{yz+xz-xy}$=$\dfrac{20 \times 30 \times 40 }{30 \times 40 +20 \times 40 - 20
\times 30}$
=17.14
So it will take 17.14 minutes to fill the tank full.
A pipe can fill a cistern in x hours. Because of a leak in the bottom, it is filled in y hours. If it is full, the time taken by the leak to empty the cistern is $\dfrac{xy}{y- x}$ hours.
Example:
A pipe can fill a tank in 3 hours. Because of leak in the bottom, it is filled in 4 hours. If the tank is full, how much time will the leak take to empty it?
Solution:Work done by leak in one hour=$\dfrac{1}{3}$-$\dfrac{1}{4}$=$\dfrac{1}{12}$
So leak will empty the tank in 12 hours.
By formula
Let’s say x = 3 and y = 4
$\dfrac{xy}{y-x}$=$\dfrac{3 \times 4}{4-3}$
=12 hours