In this section, you will learn the shortcuts which can be useful to solve problems on pipes and cisterns.

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 emty 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 ?

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.

So the right choice would be (A) or (B)

Total capacity of the tank = 60 units. (L.C.M of 10,6)

The tank is already two-fifth full.

That is,quantity of water is in the tank =(2/5)X60 = 24 units

If both the pipes are opened together, this 24 units will be emptied.

work done by pipe A = 60/10 = 6 units/min

work done by pipe B = 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)= 24/4=6 min

Time taken to empty the tank is 6 min. Option (A) is correct.

That is, pipe B is faster than pipe A.

When two pipes are opened together, the tank will emptied.

So the right choice would be (A) or (B)

Total capacity of the tank = 60 units. (L.C.M of 10,6)

The tank is already two-fifth full.

That is,quantity of water is in the tank =(2/5)X60 = 24 units

If both the pipes are opened together, this 24 units will be emptied.

work done by pipe A = 60/10 = 6 units/min

work done by pipe B = 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)= 24/4=6 min

Time taken to empty the tank is 6 min. Option (A) is correct.

**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 ?

Total work = 60 units. (L.C.M 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)

1st hour: (A+B) = 10 units/hr

2nd hour: (A+C) = 9 units/hr

3rd hour: (A+B) = 10 units/hr

4th hour: (A+C) = 9 units/hr

5th hour: (A+B) = 10 units/hr

6th 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)

Hence, time taken to fill the tank = 6.5 hours.

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)

1st hour: (A+B) = 10 units/hr

2nd hour: (A+C) = 9 units/hr

3rd hour: (A+B) = 10 units/hr

4th hour: (A+C) = 9 units/hr

5th hour: (A+B) = 10 units/hr

6th 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)

Hence, time taken to fill the tank = 6.5 hours.

**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 ?

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 = 108 units (L.C.M 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 = 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 = 108/3 = 36 turns

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 = 108 units (L.C.M 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 = 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 = 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 ?

Total capacity of the tank = 80x12.5 = 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.

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 ?

Total work = 60 units (L.C.M 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 = 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 = 9x4 = 36 units.

Remaining work to be done = 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 min

Hence, further time taken by B to fill the tank is 6 min.

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 = 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 = 9x4 = 36 units.

Remaining work to be done = 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 min

Hence, further time taken by B to fill the tank is 6 min.

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