business mathematics

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.

PIPES AND CISTERNS practice problems:

For PIPES AND CISTERNS concept click: http://business-maths.blogspot.com/2008/08/pipes-and-cisterns.html

For solved problems click: http://business-maths.blogspot.com/2008/11/pipes-and-cisterns-problems.html

Practice problems:

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?
a)10 hrs b)15 hrs c)20 hrs d)25 hrs

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?
a)7 hrs 30 min b)5 hrs c)7 hrs d)none of these

3.If two pipes function simultaneously,the reserviour will be filled in 12 hours.One
pipe fills the reservoir 10 hours faster than the other.How many hours does it take
the second pipe to fill the reservior?
a)10 b)20 c)30 d)40

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 are opened,the empty cistern
is full in 20 minutes.How long will the waste pipe take to empty the full?
a)5 min b)10 min c)15 min d)20 min

5.An electric pump can fill a tank in 3 hours.Because of a leak in the tank it took 3 1/2 hours to fill the tank.If the tank is full,how much time will the leak take to empty
it?
a)17 b)18 c)20 d)21

6.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?
a)110 hrs b)111 hrs c)112 hrs d)none of these

7.Two pipes A and B can fill a tank in 36 min and 45 min resp.A water pipe C can empty the tank in 30 min.First A and B are are opened.After 7 min,C is also opened.In how much time,the tank is full?
a)19 min b)29 min c)39 min d)49 min

8.Two pipes A and B can fill a tank in 24 min and 32 min. resp.If both the pipes are
opened simultaneously,after how much time B should be closed so that the tank is full
in 18 min?
a)5 min b)8 min c)16 min d)29 min

9.Two pipes A and B can fill a tank in 20 and 30 min resp.If both the pipes are used
together , then how long will it take to fill the tank?
a)12 min b)15 min c)25 min d)50 min

10.Three pipes A,B and C can fill a tank in 6 hours.After working at it together for 2 hrs, 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
a)10 b)12 c)14 d)16

11.A cistern can be filled by a tap in 4 hours while it can be emptied by another tap in 9 hrs.If both the taps are opened simulataneously,then after how much time will the cistern get filled?
a)4.5 hrs b)5 hrs c)6.5 hrs d)7.2 hrs

12.Two pipes A and B can fill a cistern in 37 1/2 minutes and 45 minutes resp.Both pipes are opened.The cistern will be filled in just half an hour,if the pipe B is turned off after?
a)5 min b)9 min c)10 min d)15 min

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?
a)3 hrs 15 min b)3 hrs 45 min c)4 hrs d)4 hrs 15 min

14.Two pipes can fill a tank in 20 and 24 min resp. and a waste pipe can empty 3 gallons per minute.All the three pipes working together can fill the tank in 15 min.The capacity of the tank is
a)60 gallons b)100 gallons c)120 gallons d)180 gallons

15.A water tank is two fifth full.Pipe A can fill a tank in 10 min and pipe B can empty it in 6 min.If both the pipes are open,how long will it take to empty or fill the tank completely?
a)6 min to empty b)6 min to fill c)9 min to empty d)9 min to fill

16.A leak in the bottom of a tank can empty the full tank in 8 hours.An inlet pipe fills water at the rate of 6 litres a min.When the tank is full,the inlet is opened and due to the leak,the tank is empty in 12 hrs.How many litres does the cistern hold?
a)7580 b)7960 c)8290 d)8640

17.Pipe A can fill a tank in 5 hrs,pipe B in 10 hrs and Pipe C in 30 hrs.If all the pipes are open,in how many hours will the tank be filled?
a)2 b)2.5 c)3 d)3.5

18.A booster pump can be used for filling as well as for emptying a tank.The capacity of the tank is 2400 cubic meter.The emptying capacity of the tank is 10 m^3 per min higher than its filling capacity and the pump needs 8 min lesser to empty the tank than it needs to fill it.What is the filling capacity of the pump?
a)50 b)60 c)72 d)none of these

19.Pipes A and B can fill a tank in 5 and 6 hours respectively.Pipe C can empty it in 12 hrs. If all the three pipes are opened together,then the tank will be filled in
a)1 13/17 hrs b)2 8/11 hrs c)3 9/17 hrs d)4 1/2 hrs

20.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
a)6 hrs b)6 2/3 hrs c)5 hrs d)7 1/2 hrs

Pipes and Cisterns problems

Pipes and Cisterns problems

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.