Problems On Pipes and Cistern

Solved Problems 

1.  Two pipes can fill a tank in 10 hrs and 12 hrs respectively. While a third pipe empties the full tank in 20 hrs. If all the three pipes operate together, in how much time the tank will be filled?

Answer: 7 hrs 30 mins
Explanation:
Net part filled in 1 hr.
= ( 1 /10 ) + ( 1 / 12 ) + ( 1 / 20 )
= 8 / 60
= 2 / 15
The tank will be full in 15 / 2 hrs.
= 7.5 hrs
= 7 hrs 30 mins.


2. 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?

Answer: 30 min
Explanation:
Part filled by ( A + B ) in 1 minute = ( 1 / 60 ) + ( 1 / 40 )
= 1 / 24
Suppose the tank is filled in x minutes.
⇒ x / 2 [ ( 1 / 24 ) + ( 1 / 40 ) ] = 1
⇒ ( x / 2 ) * ( 1 / 15 ) = 1
x = 30 min.


3. 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?

Answer: 6 hr
Explanation:
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 )
( x + 6 + x ) / [ x ( x + 6 ) ] = ( 1 / 4 )
⇒ 8x + 24 = x2 + 6x
⇒ x2 + 6x - 8x - 24 = 0
⇒ x2 - 2x - 24 = 0
⇒ ( x - 6 )( x + 4 ) = 0
∴ x = 6 ( neglecting the negative value of x )
x = 6 hr.


4. Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are opened simultaneousiy, how much time will be taken to fill the tank?

Answer: 20 hours
Explanation:
Part A filled ny 1 hour = 1 / 36
Part B filled by 2 hour = 1 / 45
Part ( A + B ) filled in 1 hour = ( 1 / 36 ) + ( 1 / 45 )
= ( 5 + 4 ) / 180
= 9 / 180
= 1 / 20
= 20 hours
Hence, both the pipes together will fill the tank in 20 hours.


5. 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 cistern?

Answer: 10 minutes
Explanation:
Work done by the waste pipe in x minute.
= ( x / 20 ) - [ ( x / 12 ) + ( x / 15 ) ]
= ( x / 20 ) - [ ( 5x + 4x) / 60 ]
= ( x / 20 ) - ( 9x / 60 )
= ( 3x - 9x ) / 60
= -5x / 60
= -x / 10
x = 10
Waste pipe will empty the fill cistern in 10 minutes.


6. Pipe A can fill a tank in 5 hours, pipe B in 10 hours and pipe C in 30 hours. If all the pipes are open, in how many hours will the tank be filled?

Answer: 3 hours
Explanation:
Part filled by ( A + B + C ) in 1 hour = [ ( 1 / 5 ) + ( 1 / 10 ) + ( 1 / 30 ) ]
= ( 6 + 3 + 1 ) / 30
= 10 / 30
= 1 / 3
All the three pipes together will fill the tank in 3 hours.


7. A pump can fill a tank with water in 2 hrs. Because of a leak, it took [ 2 * ( 1 / 3 ) ] hrs to fill the tank. Find the leak can drain all the water of the tank?

Answer: 14 hrs
Explanation:
Pump can fill a watertank = 2 hr
Water leak = [ 2 * ( 1 / 3 ) ] hrs
= ( 7 / 3 ) hrs
Work done by the leak in 1 hour = ( 1 / 2 ) - ( 3 / 7 )
= ( 7 - 6 ) / 14
= 1 / 14
Leak will empty the tank in 14 hrs.


8. 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?

Answer: 35 hrs
Explanation:
Let pipe A alone takes x hours to fill the tank.
Pipes B and C will take ( x / 2 ) and ( x / 4 ) hours respectively to fill the tank.
⇒ ( 1 / x ) + ( 2 / x ) + ( 4 / x ) = ( 1 / 5 )
⇒ ( 7 / x ) = ( 1 / 5 )
x = 5 * 7
x = 35 hrs.


9. 3 pumps, working 8 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?

Answer: 12
Explanation:
Let the required number of working hours per day be x.
More pumps, Less working hours per day ( ∴ Indirect Proportion )
Less days, More working hours per day ( ∴ Indirect Proportion )
Pumps = 4 : 3 :: 8 : x
Days = 1 * 2 :: 8 : x
⇒ ( 4 * 1 * x ) = ( 3 * 2 * 8 )
x = ( 3 * 2 * 8 ) / ( 4 * 1 * x )
= 3 * 4
= 12.

10.   Two pipes A and B can fill a tank in 2 and 6 minutes respectively. If both the pipes are used together, then how long will it take to fill the tank?

Answer: 1.5 minutes
Explanation:
Part filled by the first pipe in 1 minute = 1 / 2
Part filled by the second pipe in 1 minute = 1 / 6
Net part filled by pipe A and pipe B in 1 minute = ( 1 / 2 ) + ( 1 / 6 )
= 2 / 3
Pipe A and B together can fill the tank = ( 3 / 2 ) minutes
= 1.5 minutes.