# Pipe and Cistern

Ankur Kulhari

First let us understand frequently used and important keywords of this topic.
Pipe: It is used to either fill the tank or empty the the tank.
Tap: It is also user for, either to fill the tank or to empty the tank.
Tank or Cistern or Reservoir: It is used to store the water.
Inlet: A pipe or a tap connected with a tank (or a cistern or a reservoir), if it fills it.
Outlet: A pipe or a tap connected with a tank (or a cistern or a reservoir), if it empties it.
Most of the rules are similar to time and work concepts:
Here we can assume, inlet is a person doing positive work and outlet is a person doing negative work.

1. If a pipe can fill a tank in ‘a’ hour, then the part filled in 1 hr. = 1/a
2. If two pipes can fill a tank in a and b hours respectively, then both together in 1 hr. can fill = 1/a + 1/b part of the tank.
They together can fill the tank in = (ab)/(a+b) time
3. If a pipe can empty a tank in b hour, then the part of the tank emptied in 1 hr. = 1/b
4. If both pipes (one can fill in a hr. and another can empty in b hrs) are opened simultaneously (and 1/a>1/b), then, the net part filled in 1 hour = (part filled in 1 hr.) – (part emptied in 1 hr.)
= (1/a – 1/b)
Together they can fill the tank in = ab/(a-b) hours
5. If both pipes are opened simultaneously (and 1/b>1/a), then

the net part emptied in 1 hour = (part emptied in 1 hr.) – (part filled in 1 hr.)
= (1/b – 1/a)
Together they can fill the tank in = ab/(b-a) hours
6. 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
= 1/(1/x+1/y+1/z) = xyz/xy+yz+xz hours
7. Two pipes can fill a cistern in x, y hours while working alone and a third pipe can empty the cistern in z hours. If all the three pipes are opened together, the time taken to fill (or empty) the cistern is given by
= 1/[(1/x)+(1/y)-(1/z)] = xyz/yz+xz-xy hours
{we can remember the –ve sign by as 1/x+1/y-1/z, so with z numerator will be xy, so with xy it will be –ve}

Example 1. A pipe takes 12 min. to fill 3/4th of a tank. How much time it will take to fill the half of the tank?

Explanation:
3/4th of pipe is filled in = 12 min
Full tank can be filled in = 12/(3/4) = 12*4/3 16 min
half of the tank will be filled in = 16*(1/2) = 8 min.

Example 2. Two pipes can fill a reservoir in 20 and 30 mins working alone respectively. If both pipes opened at a time, find the time when the reservoir will be filled fully.

Explanation:
In 20 min pipe#1 fill = 1 reservoir
In 1 min pipe#1 fill = 1/20 part
In 30 min pipe#2 empty = 1 reservoir
In 1 min pipe#2 empty = 1/30 part
In 1 min overall filled (both pipes opened) = 1/20 – 1/30 = 1/60 part
=> 1/60 part is filled in = 1 min
1 full reservoir can be filled in = 60 min

Example 3. Two Pipes A and B can fill a tank in 15 and 18 min alone respectively. A hole in the tank takes 90 min to empty it. In how much time the tank will be emptied, if both pipes and hole are opened simultaneously?

Explanation:
In 1 min = (1/15 + 1/18 – 1/ 90) = 1/9 part is filled
=> The tank can be emptied in 9 min.

Example 4. There are two tapes in a reservoir, named A and B. Tab A takes 20 min and B 40 min to fill the tank fully alone. Both the taps are opened simultaneously. Tap A was closed after 10 min. In how much more time will the reservoir be full?

Explanation:
Part filled by both A and B in 1 min = 1/20 + 1/40 = 3/40
Part filled by both in 10 min = 10*3/40 = 3/4
Part left empty = 1-3/4 = 1/4
Now 1/4 part has to fill by pipe B alone,
B fill 1 reservoir in 40 min
1/4 part reservoir can be filled in = 40*1/4 = 10 min
=> Total time taken to fill the tank = 10 min together + 10 min B alone
= 20 min

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