Pipe and cistern
Pipe and cistern handwritten notes
Complete math notes
Question 1 : Two pipes A and B can fill a tank separately in 12 and 16 hours respectively. If both of them are opened together when the tank is initially empty, how much time will it take to completely fill the tank?
Solution : Part of tank filled by pipe A in one hour working alone = 1 / 12
Part of tank filled by pipe B in one hour working alone = 1 / 16
=> Part of tank filled by pipe A and pipe B in one hour working together = (1 / 12) + (1 / 16) = 7 / 48
Therefore, time taken to completely fill the tank if both A and B work together = 48 / 7 hours
Another Method
Let the capacity of tank be LCM (12, 16) = 48 units
=> Efficiency of pipe A = 48 / 12 = 4 units / hour
=> Efficiency of pipe B = 48 / 16 = 3 units / hour
=> Combined efficiency of pipes A and B = 7 units / hour
Therefore, time taken to completely fill the tank = 48 / 7 hours
Question 2 : Three pipes A, B and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours, B fills the tank in 12 hours and C empties the tank in 30 hours. If all three are opened simultaneously, how much time does it take to fill / empty the tank ?
Solution : Part of tank filled by pipe A in one hour working alone = 1 / 10
Part of tank filled by pipe B in one hour working alone = 1 / 12
Part of tank emptied by pipe C in one hour working alone = 1 / 30
=> Part of tank filled by pipes A,B and C in one hour working together = (1 / 10) + (1 / 12) – (1 / 30) = 3 / 20
Therefore, time taken to completely fill the tank if both A and B work together = 20 / 3 hours = 6 hours 40 minutes
Another Method
Let the capacity of tank be LCM (10, 12, 30) = 60 units
=> Efficiency of pipe A = 60 / 10 = 6 units / hour
=> Efficiency of pipe B = 60 / 12 = 5 units / hour
=> Efficiency of pipe C = – 60 / 30 = – 2 units / hour (Here, ‘-‘ represents outlet pipe)
=> Combined efficiency of pipes A, B and C = 6 + 5 – 2 = 9 units / hour
Therefore, time taken to completely fill the tank = 60 / 9 = 6 hours 40 minutes
Question 3 : Three pipes A, B and C are connected to a tank. Out of the three, A is the inlet pipe and B and C are the outlet pipes. If opened separately, A fills the tank in 10 hours, B empties the tank in 12 hours and C empties the tank in 30 hours. If all three are opened simultaneously, how much time does it take to fill / empty the tank ?
Solution : Part of tank filled by pipe A in one hour working alone = 1 / 10
Part of tank emptied by pipe B in one hour working alone = 1 / 12
Part of tank emptied by pipe C in one hour working alone = 1 / 30
=> Part of tank filled by pipes A, B and C in one hour working together = (1 / 10) – (1 / 12) – (1 / 30) = -1 / 60
Therefore, time taken to completely empty the tank if all pipes are opened simultaneously = 1 / 60 hours = 60 hours
Another Method
Let the capacity of tank be LCM (10, 12, 30) = 60 units
=> Efficiency of pipe A = 60 / 10 = 6 units / hour
=> Efficiency of pipe B = – 60 / 12 = – 5 units / hour (Here, ‘-‘ represents outlet pipe)
=> Efficiency of pipe C = – 60 / 30 = – 2 units / hour (Here, ‘-‘ represents outlet pipe)
=> Combined efficiency of pipes A, B and C = 6 – 5 – 2 = – 1 units / hour (Here, ‘-‘ represents outlet pipe)
Therefore, time taken to completely empty the tank = 60 / (1) = 60 hours
Question 4 : A cistern has two pipes. Both working together can fill the cistern in 12 minutes. First pipe is 10 minutes faster than the second pipe. How much time would it take to fill the cistern if only second pipe is used ?
Solution : Let the time taken by first pipe working alone be ‘t’ minutes.
=> Time taken by second pipe working alone = t + 10 minutes.
Part of tank filled by pipe A in one hour working alone = 1 / t
Part of tank filled by pipe B in one hour working alone = 1 / (t + 10)
=> Part of tank filled by pipe A and B in one hour working together = (1 / t) + (1 / t+10) = (2t + 10) / [t x (t + 10)]
But we are given that it takes 12 minutes to completely fill the cistern if both pipes are working together.
=> (2t + 10) / [t x (t + 10)] = 1 / 12
=> t x (t + 10) / (2t + 10) = 12
=> t2 + 10t = 24t + 120
=> t2 – 14t – 120 = 0
=> (t – 20) (t + 6) = 0
=> t = 20 minutes (Time cannot be negative)
Therefore, time taken by second pipe working alone = 20 + 10 = 30 minutes
Another Method
Let the time taken by first pipe working alone be ‘t’ minutes.
=> Time taken by second pipe working alone = t + 10 minutes.
Let the capacity of cistern be t x (t + 10) units.
=> Efficiency of first pipe = t x (t + 10) / t = (t + 10) units / minute
=> Efficiency of second pipe = t x (t + 10) / (t + 10) = t units / minute
=> Combined efficiency of pipes = (2t + 10) units / minute
=> Time taken to fill the cistern completely = t x (t + 10) / (2t + 10)
But we are given that it takes 12 minutes to completely fill the cistern if both pipes are working together.
=> t x (t + 10) / (2t + 10) = 12
=> t2 + 10t = 24t + 120
=> t2 – 14t – 120 = 0
=> (t – 20) (t + 6) = 0
=> t = 20 minutes (Time cannot be negative)
Therefore, time taken by second pipe working alone = 20 + 10 = 30 minutes
Question 5 : Three pipes A, B and C are connected to a tank. Out of the three, A and B are the inlet pipes and C is the outlet pipe. If opened separately, A fills the tank in 10 hours and B fills the tank in 30 hours. If all three are opened simultaneously, it takes 30 minutes extra than if only A and B are opened. How much time does it take to empty the tank if only C is opened?
Solution : Let the capacity of tank be LCM (10, 30) = 30 units
=> Efficiency of pipe A = 30 / 10 = 3 units / hour
=> Efficiency of pipe B = 30 / 30 = 1 units / hour
=> Combined efficiency of pipes A and B = 4 units/hour
Therefore, time taken to completely fill the tank if only A and B are opened = 30 / 4 = 7 hours 30 minutes
=> Time taken to completely fill the tank if all pipes are opened = 7 hours 30 minutes + 30 minutes = 8 hours
=> Combined efficiency of all pipes = 30 / 8 = 3.75 units / hour
Now, efficiency of pipe C = Combined efficiency of all three pipes – Combined efficiency of pipes A and B
Therefore, efficiency of pipe C = 4 – 3.75 = 0.25 units / hour
Thus, time taken to empty the tank if only C is opened = 30 / 0.25 = 120 hours
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