Quants 19 - Pipes and Cisterns

Key Points to Remember:

Inlet - A pipe connected with a tank or a cistern or a reservoir, that fills it, is known as an inlet.

Outlet - A pipe connected with a tank or a cistern or a reservoir, emptying it, is known as an outlet.

If a pipe can fill a tank in x hours, then part filled in 1 hour = 1/x.

If a pipe can empty a full tank in y hours, then part emptied in 1 hour = 1/y.

If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where y > x), then on opening both the pipes, the net part filled in 1 hour = 1/x - 1/y.

If a pipe can fill a tank in x hours and another pipe can empty the full tank in y hours (where x > y), then on opening both the pipes, the net part emptied in 1 hour = 1/y - 1/x.

Practice Questions:

Question 1

Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

Part filled by A in 1 hour = 1/36 

Part filled by B in 1 hour = 1/45

Part filled by A and B in 1 hour = 1/36 + 1/45 = (5+4)/180 = 9/180 = 1/20

Hence, both the pipes together will fill the tank in 20 hours. 

Question 2

Two pipes can fill a tank in 10 hours and 12 hours respectively while a third pipe empties the full tank in 20 hours. If all the three pipes operate simultaneously, in how much time will the tank be filled?

Net part filled in 1 hour = 1/10 + 1/12 - 1/20 = (6+5-3) / 60 = 8/60 = 2/15

The tank will be full in 2/15 = 7 hours 30 minutes. 

Question 3

If two pipes function simultaneously, the reservoir will be filled in 12 hours. One pipe fills the reservoir in 10 hours faster than the other. How many hours does it take the second pipe to fill the reservoir?

Let the reservoir be filled by first pipe in x hours.
Then the second pipe will fill it in x+10 hours.
So, 1/x + 1/(x+10) = 1/12
x+10+x/x(x+10) = 1/12
2x+10 / x²+10x = 1/12
24x+120 =  x²+10x
x² - 14x - 120 = 0
(x-20)(x+6) = 0
So, x = 20.
The second pipe fills the reservoir in 30 hours.


Question 4
A cistern has two taps which fill it in 12 minutes and 15 minutes respectively. There is also a waste pipe in the cistern. When all the three pipes are opened, the empty cistern is full in 20 minutes. How long will the waste pipe take to empty the full cistern?
Work done by waste pipe in 1 minute = 1/20 - (1/12 + 1/15) = 1/20 - (5+4 / 60) = 1/20 - 9/60 = 3-9 / 60 = -6/60 = -1/10 (-ve sign means emptying)
Waste pipe will empty the full cistern in 10 minutes.


Question 5
An electric pump can fill a tank in 3 hours. Because of a leak in the tank, it took 3.5 hours to fill the tank. If the tank is full, how much time will the leak take to empty it?
Work done by the leak in 1 hour = 1/3 - 1/(7/2) = 1/3 - 2/7 = 7-6 / 21 = 1/21
The leak will empty the tank in 21 hours.


Time to Think:
Two pipes can fill a cistern in 14 hours and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom it took 32 minutes more to fill the cistern. When the cistern is full, in what time will the leak empty it?