Pipes and Cisterns

Important Facts: 

1.INLET:A pipe connected with a tank or cistern or a reservoir, 
        that fills it, it is known as Inlet.

 OUTLET:A pipe connected with a tank or a cistern or a reservoir, 
         emptying it, is known as Outlet.

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

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

   iii)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).

  iv)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 filled in 1 hour=(1/y -1/x).

  v) If two pipes A and B can fill a tank in x hours and y hours 
     respectively. If both the pipes are opened simultaneously, part
     filled by A+B in 1 hour= 1/x +1/y.
                                                         

Simple Problems

1)Two pipes A& 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?

Sol:        Part filled by A in 1 hour=1/36
            Part filled by B in 1 hour= 1/45;
            Part filled by (A+B)'s in 1 hour=1/36 +1/45= 9/180 =1/20
        Hence, both the pipes together will fill the tank in 20 hours.

2)Two pipes can fill a tank in 10 hours & 12 hours respectively. While 
3rd pipe empties the full tank n 20 hours. If all the three pipes
operate simultaneously,in how much  time will the tank be filled?

Sol:        Net part filled in 1 hour=1/10 +1/12 -1/20
                                             =8/60=2/15
            The tank be filled in 15/2hours= 7 hrs 30 min

3)A cistern can be filled by a tap in 4 hours while it can be emptied 
by another tap in 9 hours. If both the taps are opened simultaneously,
then after how much time will the cistern get filled?

Sol:      Net part filled in 1 hour= 1/4 -1/9= 5/36
      Therefore the cistern will be filled in 36/5 hours or 7.2 hours.

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

Sol:        Let the reservoir be filled by the 1st pipe in x hours.
            The second pipe will fill it in (x+10) hours
            1/x + (1/(x+10))= 1/12
        =>  (2x+10)/((x)*(x+10))= 1/12
        =>  x=20
       So, the second pipe will take 30 hours to fill the reservoir.

5)A cistern has two taps which fill it in 12 min and 15 min respectively.
There is also a waste pipe in the cistern. When all the three are opened,
the empty cistern is full in 20 min. How long will the waste pipe take to 
empty the full cistern?

Sol:  Work done by a waste pipe in 1 min
          =1/20 -(1/12+1/15)= -1/10 (-ve means emptying)

6)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?

Sol:   Time taken by one tap to fill the half of the tank =3 hours
       Part filled by the four taps in 1 hour=4/6=2/3
           Remaining part=1 -1/2=1/2
           Therefore, 2/3:1/2::1:x
           or x=(1/2)*1*(3/2)=3/4 hours.
           i.e 45 min
           So, total time taken= 3hrs 45min.

7)A water tank is two-fifth full. Pipe A can fill a tank in 10 min. And B 
can empty it in 6 min. If both pipes are open, how long will it take to 
empty or fill the tank completely ?

Sol:  Clearly, pipe B is faster than A and So, the tank will be emptied.
            Part to be emptied=2/5.
            Part emptied by (A+B) in 1 min= 1/6 -1/10=1/15
            Therefore, 1/15:2/5::1:x or x=((2/5)*1*15)=6 min.
             So, the tank be emptied in 6 min.

8)Bucket P has thrice the capacity as Bucket Q. It takes 60 turns for 
Bucket P to fill the empty drum. How many turns it will take for both the 
buckets P&Q, having each turn together to fill the empty drum?

Sol:    Let the capacity of P be x lit. 
        Then capacity of Q=x/3 lit
        Capacity of the drum=60x lit
        Required number of turns= 60x/(x+(x/3))= 60x*3/4x=45
                                                         

Complex Problems

1)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 32min more to fill the cistern. When the cistern is full,
in what time will the leak empty it? 

Sol:        Work done by the two pipes in 1 hour= 1/14+1/16=15/112
            Time taken by these two pipes to fill the tank=112/15 hrs.
            Due to leakage, time taken = 7 hrs 28 min+ 32 min= 8 hours
            Therefore, work done by (two pipes + leak) in 1 hr= 1/8
            work done by leak n 1 hour=15/112 -1/8=1/112
            Leak will empty full cistern n 112 hours.

2)Two pipes A&B can fill a tank in 30 min. First, A&B are opened. After
7 min, C also opened. In how much time, the tank s full.

Sol:    Part filled n 7 min = 7*(1/36+1/45)=7/20
        Remaining part= 1-7/20=13/20
 Net part filled in 1 min when A,B and C are opened=1/36 +1/45- 1/30=1/60
        Now, 1/60 part is filled in 1 min.
        13/20 part is filled n (60*13/20)=39 min
        Total time taken to fill the tank=39+7=46 min 

3)Two pipes A&B can fill a tank in 24 min and 32 min respectively. If 
both the pipes are opened simultaneously, after how much time B should
be closed so that the tank is full in 18 min. 

Sol:  Let B be closed after x min, then part filled by (A+B) in x min+ 
                              part filled by A in (18-x) min=1
         x(1/24+1/32) +(18-x)1/24 =1
      => x=8
         Hence B must be closed after 8 min.

4)Two pipes A& 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?

Sol:       Let the cistern be filled by pipe A alone in x hours.
           Pipe B will fill it in x+6 hours
           1/x + 1/x+6=1/4
           Solving this we get x=6.
           Hence, A takes 6 hours to fill the cistern separately.

5)A tank is filled by 3 pipes with uniform flow. The first two pipes 
operating simultaneously fill the tan in the same time during which
the tank is filled by the third pipe alone. The 2nd pipe fills the tank
5 hours faster than first pipe and 4 hours slower than third pipe. The
time required by first pipe is :

Sol:      Suppose, first pipe take x hours to fill the tank then 
          B & C will take (x-5) and (x-9) hours respectively.
          Therefore, 1/x +1/(x-5) =1/(x-9)
          On solving, x=15 
          Hence, time required by first pipe is 15 hours.
                                                         
6)A large tanker can be filled by two pipes A& B in 60min and 40 min 
respectively. How many minutes will it take to fill the tanker from 
empty state if B is used for half the time & A and B fill it together for 
the other half?

 Sol:        Part filled by (A+B) n 1 min=(1/60 +1/40)=1/24
             Suppose the tank is filled in x minutes
             Then, x/2(1/24+1/40)=1
              =>  (x/2)*(1/15)=1
              =>      x=30 min.

7)Two pipes A and B can fill a tank in 6 hours and 4 hours respectively.
If they are opened on alternate hours and if pipe A s opened first, in 
how many hours, the tank shall be full.

Sol:       (A+B)'s 2 hours work when opened alternatively =1/6+1/4 =5/12
           (A+B)'s 4 hours work when opened alternatively=10/12=5/6
           Remaining part=1 -5/6=1/6.
           Now, it is A's turn  and 1/6 part is filled by A in 1 hour.
           So, total time taken to fill the tank=(4+1)= 5 hours.

8)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 
alternatively, the tank will be full in.

Sol:        (A+B)'s 1 hour's work=1/12+1/15=9/60=3/20
            (A+C)'s 1 hour's work=1/20+1/12=8/60=2/15
            Part filled in 2 hours=3/20+2/15=17/60
            Part filled in 2 hours=3/20+2/15= 17/60
            Part filled in 6 hours=3*17/60 =17/20
            Remaining part=1 -17/20=3/20
  Now, it is the turn of A & B and 3/20 part is filled by A& B in 1 hour.
            Therefore, total time taken to fill the tank=6+1=7 hours

9)A Booster pump can be used for filling as well as for emptying a tank. 
The capacity of the tank is 2400 m3. The emptying capacity of the tank is
10 m3 per minute higher than its filling capacity and the pump needs 8
minutes lesser to empty the tank than it needs to fill it. What is the 
filling capacity of the pump?

Sol:          Let, the filling capacity of the pump be x m3/min
              Then, emptying capacity of the pump=(x+10) m3/min.
              So,2400/x – 2400/(x+10) = 8         
                    on solving x=50.

10)A leak in the bottom of a tank can empty the full tan in 8 hr. An inlet
pipe fills water at the rate of 6 lits a minute. When the tank is full,
the inlet is opened and due to the leak, the tank is empty in 12 hrs.
How many liters does the cistern hold?

Sol:        Work done by the inlet in 1 hr= 1/8 -1/12=1/24
            Work done by the inlet in i min= (1/24)*(1/60)=1/1440
            Therefore, Volume of 1/1440 part=6 lit
            Volume of whole=(1440*6) lit=8640 lit.

11)Two pipes A and B can fill a cistern in 37 ½ min and 45 minutes 
respectively. Both the pipes are opened. The cistern will be filled in 
just half an hour, if the pipe B is turned off  after:

sol:         Let B be turned off after x min. Then,
     Part filled by (A+B) in x min+ part filled by A in (30-x)min=1
             Therefore,   x(2/75+1/45)+(30-x)(2/75)=1
                    11x/225 + (60-2x)/75=1
                    11x+ 180-6x=225
                    x=9.
             So, B must be turned off after 9 minutes.