Question

Please answer my maths Question.

Two pipes A and B can fill a cistern in 20 and 30 minutes respectively. When the cistern was empty, two pipes are turned on at the same time. After some time pipe A was closed and the cistern was filled completely in 18 minutes. When was the pipe A closed?

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Answer 1

I suppose the answer is this,

LCM(20,30)=60

Suppose the capacity of the tank is 60 litre. Then,

quantity filled by pipe A in 1 min =60/20=3 Litre

quantity filled by pipe B in 1 min =60/30=2 Litre

Quantity filled by both the pipes together in x minutes=

x(3+2)=5x

Remaining quantity to be filled = 60-5x

Time required for pipe B alone to fill this remaining quantity is 18 minutes

Therefore, (60-5x)/2=18 minutes

x=4.8 minutes

Total time=4.8+18=22.8 minutes

Answer 2

Time required for pipe A alone to fill the cistern = 20 minutes

So, part of cistern filled in 1 minute by pipe A alone = 1/20.

Time required for pipe B alone to fill the cistern = 30 minutes

So, part of cistern filled in 1 minute by pipe B alone = 1/30.

So the part of cistern filled in 1 minute by both pipes A and B together turned on at the same time is,

1/20 + 1/30  =  3/60 + 2/60  =  5/60  =  1/12

∴  The cistern will be filled completely in 12 minutes by the two pipes A and B together turned on at the same time.

But given that pipe A was closed after some time. We have to find this time.

Let this time be x minutes.

When the cistern was empty, the two pipes A and B were turned on at the same time. Since 1/12 part of the cistern would be filled by both pipes in 1 minute, in first x minutes, x/12 part of the cistern is filled.

But after these x minutes, pipe A was closed, So the remaining  1 - (x/12) = (12 - x)/12  part of the cistern was filled by pipe B alone, and the cistern was completely filled in 18 minutes.

So we get that the remaining (12 - x)/12 part of the cistern was filled by pipe B alone in (18 - x) minutes.

Part of cistern filled in 1 minute by pipe B alone = 1/30.

Part of cistern filled in (18 - x) minutes by pipe B alone = (18 - x)/30, which is, (12 - x)/12.

So,

    (12 - x) / 12  =  (18 - x) / 30

⇒  30(12 - x)  =  12(18 - x)

⇒  360 - 30x = 216 - 12x

⇒  30x - 12x = 360 - 216

⇒  18x = 144

⇒  x = 8

Hence, the pipe A was closed after first 8 minutes.