1. If a pipe can fill a tank in ‘m’ hours, it can fill (1/m) part of the tank in 1 hour.
2. If (1/m) part of the tank is filled by a pipe in 1 hour, time taken by the pipe to fill the entire tank is "m" hours.
3. If a pipe can empty a tank in ‘x’ hours, it can empty (1/x) part of the tank in 1 hour.
4. If (1/m) part of the tank is emptied by a pipe in 1 hour, time taken by the pipe to empty the entire tank is "x" hours
5. If a pipe can fill a tank in "x" hours and another pipe can empty the full tank in "y" hours (x > y), then on opening both the pipes, the net part emptied in 1 hour is
= 1/y - 1/x
Problem 1 :
A water tank is two-fifth full. Pipe A can fill a tank in 10 minutes and pipe B can empty in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely ?
Solution :
From the question, we have to consider an important thing.
That is, pipe B is faster than pipe A.
When two pipes are opened together, the tank will emptied.
Total capacity of the tank is 60 units. (LCM of 10, 6)
The tank is already two-fifth full.
That is, quantity of water in the tank is
= (2/5) x 60
= 24 units
If both the pipes are opened together, this 24 units will be emptied.
Work done by pipe A is
= 60 / 10
= 6 units/min
Work done by pipe B is
= 60 / 6
= -12 units/min (emptying the tank)
Adding the above two equations, we get
A + B = -4 units/min
That is, 4 units will be emptied per minute when both the pipes are opened together
Time taken to empty 24 units (2/5 of the tank) is
= 24 / 4
= 6 minutes
Time taken to empty the tank is 6 minutes.
Problem 2 :
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, how long will it take for the tank to be filled ?
Solution :
Total work = 60 units (LCM of 10, 15, 20)
Work done by the pipe A = 60/10 = 6 units/hr
Work done by the pipe B = 60/15 = 4 units/hr
Work done by the pipe C = 60/20 = 3 units/hr
(Given : A is open all the time, B and C are alternately)
1^{st} hour : (A + B) = 10 units/hr
2^{nd} hour: (A + C) = 9 units/hr
3^{rd} hour: (A + B) = 10 units/hr
4^{th} hour: (A + C) = 9 units/hr
5^{th} hour: (A + B) = 10 units/hr
6^{th} hour: (A + C) = 9 units/hr
When we add the above units, we get the total 57 units.
Apart from the 6 hours of operation, to get the total work 60 units, A has to work for half an hour.
Because in one of hour work of A, we will get 6 units.
So, time taken to fill the tank is
6.5 hours
or
6 hours 30 minutes
Problem 3 :
Bucket A has twice the capacity as bucket B. It takes 54 turns for bucket A to fill the empty cistern. How many turns will it take for both the buckets A and B, having each turn together to fill the empty cistern ?
Solution :
It takes 54 turns for bucket A to fill the empty cistern.
Bucket A has twice the capacity as bucket B.
So, it will take 108 turns for bucket B to fill the empty cistern.
Total work is 108 units (LCM of 54, 108)
Work done by bucket A in 1 turn = 108/54 = 2 units
Work done by bucket B in 1 turn = 108/108 = 1 unit
If both the buckets are used simultaneously, word done in 1 turn is
= 3 units (2+1 = 3)
No.of turns taken for both the buckets A and B, having each turn together to fill the empty cistern is
= 108/3
= 36 turns
Problem 4 :
80 buckets of water fill a tank when the capacity of each bucket is 12. 5 liters. How many buckets will be needed to fill the same tank if the capacity of each bucket is 10 liters ?
Solution :
Total capacity of the tank is
= 80 x 12.5
= 1000 liters
If the capacity of the bucket is 10 liters, no. of buckets will be needed to fill the tank
= 1000/10
= 100 buckets
Problem 5 :
Taps A and B can fill a cistern in 12 minutes and 15 minutes respectively. If both are opened and A is closed after 4 minutes, how much further time would it take for B to fill the cistern ?
Solution :
Total work is 60 units (LCM of 12, 15).
Word done by tap A = 60/12 = 5 units/min
Word done by tap B = 60/15 = 4 units/min
Word done by A and B together is
= 9 units/min (5+4 = 9)
Both are opened together and A is closed after 4 minutes.
Word done by both A and B in 4 minutes is
= 9 x 4
= 36 units
Remaining work to be done is
= 60 - 36
= 24 units
This 24 units of work is completed by B alone.
In 1 min, B can do 4 units of work.
B can do 24 units in
= 24/4
= 6 minutes
So, further time taken by B to fill the tank is 6 min.
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