Pipes and Cistern



In pipes and cistern suppose, a water tank or a cistern is connected with two types of pipes to fill and empty it.

(i) Inlet: The pipe which fills the tank up is called an inlet.

(ii) Outlet: The pipe which empties the tank is called an outlet.

If an inlet fills up the cistern in 5 hours, then in 1 hour it fills up 1/5th part of it. We say that the work done by inlet in 1 hour is 1/5.

Rule 1. Suppose a pipe fills a tank in n hours.

Then, part of the tank filled in 1 hour = 1/m, i.e., work done by the inlet in 1 hour= 1/n

Rule 2. Suppose an outlet empties a full tank in m hours.

Then, part of the tank emptied in 1 hour = 1/m, i.e., work done by the outlet in 1. hour = (-1/m)



Solved Problems on Pipes and Cistern

1. A tap A can fill a cistern in 8 hours while tap B can fill it in 4 hours. In how much times will the cistern be filled if both A and B are opened together?

Solution:

Time taken by tap A to fill the cistern = 8 hours.

Work done by tap A in 1 hour = 1/8

Time taken by tap B to fill the cistern = 4 hours.

Work done by tap B in 1 hour = 1/4

Work done by (A + B) in 1 hour = (1/8 + 1/4) = 3/8

Therefore, time taken by (A + B) to fill the cistern = 8/3 hours = 2 hours 40 min.

2. A tap A can fill a cistern in 4 hours and the tap B can empty the full cistern in 6 hours. If both the taps are opened together in the empty cistern, in how much time will the cistern be filled up?

Solution:

Time taken by tap A to fill the cistern = 4 hours.

Work done by tap A in 1 hour = 1/4th

Time taken by tap B to empty the full cistern = 6 hours.

Work done by tap B in 1 hour = -1/6th (since, tap B empties the cistern).

Work done by (A + B) in 1 hour (1/4 - 1/6) = (3 – 2)/12 = 1/12th part of the tank is filled.

Therefore, the tank will fill the cistern = 12 hours.

3. A cistern can be filled by two taps A and B in 12 hours and 16 hours respectively. The full cistern can be emptied by a third tap C in 8 hours. If all the taps are turned on at the same time, in how much time will the empty cistern be filled completely?

Solution:

Time taken by tap A to fill the cistern = 12 hours.

Time taken by tap B to fill the cistern = 16 hours.

Time taken by tap C to empty the full cistern = 8 hours.

A’s 1 hour’s work = 1/12

B’s 1 hour’s work = 1/16

C’s 1 hour’s work = -1/8 (cistern being emptied by C)

Therefore, (A + B + C)’s 1 hours net work= (1/12 + 1/16 - 1/8) = 1/48

Therefore, time taken by (A + B + C) to fill the cistern = 48 hours.

4. A tank can be filled by two taps A and B in 8 hours and 10 hours respectively. The full tank can be emptied by a third tap C in 9 hours. If all the taps are turned on at the same time, in how much time will the empty tank be filled up completely?

Solution:

Time taken by tap A to fill the tank = 8 hours.

Time taken by tap B to fill the tank = 10 hours.

Time taken by tap C to empty the full tank = 9 hours.

A’s 1 hour’s work = 1/8

B’s 1 hour’s work = 1/10

C’s 1 hour’s work = -1/9 (cistern being emptied by C)

Therefore, (A + B + C)’s 1 hours net work= (1/8 + 1/10 - 1/9) = (45 + 36 – 40)/360 = 41/360

Thus, tank will be filled completely in 360/41 hours, when all the three are opened together.

5. A cistern can be filled by one tap in 5 hours and by another in 4 hours. How long will it take to fill if both the taps are opened simultaneously?

Solution:

Time taken by first tap to fill the cistern = 5 hours.

Work done by first tap in 1 hour = 1/5

Time taken by second tap to fill the cistern = 4 hours.

Work done by second tap in 1 hour = 1/4

Work done by (first tap + second tap) in 1 hour = (1/5 + 1/4) = 9/20

Therefore, time taken by (first tap + second tap) to fill the cistern = 20/9 hours.



Time and Work
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