Pipes and Cistern
 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 cistern or reservoir, emptying it, is known as an outlet.
 If a pipe can fill a tank in xhours, then:
part filled in 1 hour =  1  . 
x 
 If a pipe can empty a tank in yhours, then:
part emptied in 1 hour =  1  . 
y 
 If a pipe can fill a tank in xhours and another pipe can empty the full tank in y hours (where y > x), then on opening both the pipes, then
the net part filled in 1 hour =  1  –  1  .  
x  y 
 If a pipe can fill a tank in xhours and another pipe can empty the full tank in y hours (where x > y), then on opening both the pipes, then
the net part emptied in 1 hour =  1  –  1  .  
y  x 
Questions:
LevelI:
1.  Three pipes A, B and C can fill a tank from empty to full in 30 minutes, 20 minutes, and 10 minutes respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P,Q and R respectively. What is the proportion of the solution R in the liquid in the tank after 3 minutes?  

2.  Pipes A and B can fill a tank in 5 and 6 hours respectively. Pipe C can empty it in 12 hours. If all the three pipes are opened together, then the tank will be filled in:  

3.  A pump can fill a tank with water in 2 hours. Because of a leak, it took 2 hours to fill the tank. The leak can drain all the water of the tank in:  


4. 
Two pipes A and B can fill a cistern in 37 minutes and 45 minutes respectively. Both pipes are opened. The cistern will be filled in just half an hour, if the B is turned off after:  

5.  A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is:  


6. 
Two pipes can fill a tank in 20 and 24 minutes respectively and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is: 


7.  A tank is filled in 5 hours by three pipes A, B and C. The pipe C is twice as fast as B and B is twice as fast as A. How much time will pipe A alone take to fill the tank?  

8.  Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?  

9.  Two pipes A and B can fill a tank in 20 and 30 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?  

10.  Two pipes A and B can fill a tank in 15 minutes and 20 minutes respectively. Both the pipes are opened together but after 4 minutes, pipe A is turned off. What is the total time required to fill the tank?  


11. 
LevelII:
One pipe can fill a tank three times as fast as another pipe. If together the two pipes can fill the tank in 36 minutes, then the slower pipe alone will be able to fill the tank in: 


12.  A large tanker can be filled by two pipes A and B in 60 minutes and 40 minutes respectively. How many minutes will it take to fill the tanker from empty state if B is used for half the time and A and B fill it together for the other half?  

13.  A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?  

14.  Three taps A, B and C can fill a tank in 12, 15 and 20 hours respectively. If A is open all the time and B and C are open for one hour each alternately, the tank will be full in:  

15.  Three pipes A, B and C can fill a tank in 6 hours. After working at it together for 2 hours, C is closed and A and B can fill the remaining part in 7 hours. The number of hours taken by C alone to fill the tank is:  


16. 



17. 



18. 



Answers:
LevelI:
Answer:1 Option B
Explanation:
Part filled by (A + B + C) in 3 minutes = 3  1  +  1  +  1  =  3 x  11  =  11  .  
30  20  10  60  20 
Part filled by C in 3 minutes =  3  . 
10 
Required ratio =  3  x  20  =  6  .  
10  11  11 
Answer:2 Option C
Explanation:
Net part filled in 1 hour  1  +  1  –  1  =  17  .  
5  6  12  60 
The tank will be full in  60  hours i.e., 3  9  hours. 
17  17 
Answer:3 Option D
Explanation:
Work done by the leak in 1 hour =  1  –  3  =  1  .  
2  7  14 
Leak will empty the tank in 14 hrs.
Answer:4 Option B
Explanation:
Let B be turned off after x minutes. Then,
Part filled by (A + B) in x min. + Part filled by A in (30 –x) min. = 1.
x  2  +  1  + (30 – x).  2  = 1  
75  45  75 
11x  +  (60 2x)  = 1  
225  75 
11x + 180 – 6x = 225.
x = 9.
Answer:5 Option C
Explanation:
Suppose, first pipe alone takes x hours to fill the tank .
Then, second and third pipes will take (x 5) and (x – 9) hours respectively to fill the tank.
1  +  1  =  1  
x  (x – 5)  (x – 9) 
x – 5 + x  =  1  
x(x – 5)  (x – 9) 
(2x – 5)(x – 9) = x(x – 5)
x^{2} – 18x + 45 = 0
(x – 15)(x – 3) = 0
x = 15. [neglecting x = 3]
Answer:6 Option C
Explanation:
Work done by the waste pipe in 1 minute =  1  –  1  +  1  
15  20  24 
=  1  –  11  
15  120 
= –  1  . [ve sign means emptying] 
40 
Volume of  1  part = 3 gallons. 
40 
Volume of whole = (3 x 40) gallons = 120 gallon
Answer:7 Option C
Explanation:
Suppose pipe A alone takes x hours to fill the tank.
Then, pipes B and C will take  x  and  x  hours respectively to fill the tank. 
2  4 
1  +  2  +  4  =  1  
x  x  x  5 
7  =  1  
x  5 
x = 35 hrs.
Answer:8 Option C
Explanation:
Let the cistern be filled by pipe A alone in x hours.
Then, pipe B will fill it in (x + 6) hours.
1  +  1  =  1  
x  (x + 6)  4 
x + 6 + x  =  1  
x(x + 6)  4 
x^{2} – 2x – 24 = 0
(x 6)(x + 4) = 0
x = 6. [neglecting the negative value of x]
Answer:9 Option A
Explanation:
Part filled by A in 1 min =  1  . 
20 
Part filled by B in 1 min =  1  . 
30 
Part filled by (A + B) in 1 min =  1  +  1  =  1  .  
20  30  12 
Both pipes can fill the tank in 12 minutes.
Answer:10 Option D
Explanation:
Part filled in 4 minutes = 4  1  +  1  =  7  .  
15  20  15 
Remaining part =  1 –  7  =  8  .  
15  15 
Part filled by B in 1 minute =  1 
20 
1  :  8  :: 1 : x  
20  15 
x =  8  x 1 x 20  = 10  2  min = 10 min. 40 sec.  
15  3 
The tank will be full in (4 min. + 10 min. + 40 sec.) = 14 min. 40 sec.
LevelII:
Answer:11 Option C
Explanation:
Let the slower pipe alone fill the tank in x minutes.
Then, faster pipe will fill it in  x  minutes. 
3 
1  +  3  =  1  
x  x  36 
4  =  1  
x  36 
x = 144 min.
Answer:12 Option D
Explanation:
Part filled by (A + B) in 1 minute =  1  +  1  =  1  .  
60  40  24 
Suppose the tank is filled in x minutes.
Then,  x  1  +  1  = 1  
2  24  40 
x  x  1  = 1  
2  15 
x = 30 min.
Answer:13 Option B
Explanation:
Time taken by one tap to fill half of the tank = 3 hrs.
Part filled by the four taps in 1 hour =  4 x  1  =  2  .  
6  3 
Remaining part =  1 –  1  =  1  .  
2  2 
2  :  1  :: 1 : x  
3  2 
x =  1  x 1 x  3  =  3  hours i.e., 45 mins.  
2  2  4 
So, total time taken = 3 hrs. 45 mins.
Answer:14 Option C
Explanation:
(A + B)’s 1 hour’s work =  1  +  1  =  9  =  3  .  
12  15  60  20 
(A + C)’s hour’s work =  1  +  1  =  8  =  2  .  
12  20  60  15 
Part filled in 2 hrs =  3  +  2  =  17  .  
20  15  60 
Part filled in 6 hrs =  3 x  17  =  17  .  
60  20 
Remaining part =  1 –  17  =  3  .  
20  20 
Now, it is the turn of A and B and  3  part is filled by A and B in 1 hour. 
20 
Total time taken to fill the tank = (6 + 1) hrs = 7 hrs.
Answer:15 Option C
Explanation:
Part filled in 2 hours =  2  =  1 
6  3 
Remaining part =  1 –  1  =  2  .  
3  3 
(A + B)’s 7 hour’s work =  2 
3 
(A + B)’s 1 hour’s work =  2 
21 
C’s 1 hour’s work = { (A + B + C)’s 1 hour’s work } – { (A + B)’s 1 hour’s work }
=  1  –  2  =  1  
6  21  14 
C alone can fill the tank in 14 hours.
Answer:16 Option E
Explanation:
 Time taken to fill the cistern without leak = 9 hours.
Part of cistern filled without leak in 1 hour =  1 
9 
 Time taken to fill the cistern in presence of leak = 10 hours.
Net filling in 1 hour =  1 
10 
Work done by leak in 1 hour =  1  –  1  =  1  
9  10  90 
Leak will empty the full cistern in 90 hours.
Clearly, both I and II are necessary to answer the question.
Correct answer is (E).
Answer:17 Option E
Explanation:
I. A’s 1 minute’s filling work =  1 
16 
II. B’s 1 minute’s filling work =  1 
8 
(A + B)’s 1 minute’s emptying work =  1  –  1  =  1  
8  16  16 
Tank will be emptied in 16 minutes.
Thus, both I and II are necessary to answer the question.
Correct answer is (E).
Answer:18 Option B
Explanation:
II. Part of the tank filled by A in 1 hour =  1 
4 
III. Part of the tank filled by B in 1 hour =  1 
6 
(A + B)’s 1 hour’s work =  1  +  1  =  5  
4  6  12 
A and B will fill the tank in  12  hrs = 2 hrs 24 min. 
5 
So, II and III are needed.
Correct answer is (B).