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Question 12 : Rohit purchased a cistern which had a leakage. The cistern can be filled by two inlet pipes, which can individually fill the cistern in 12 min and 15 min respectively. Despite leakage, the two inlet pipes together can fill the cistern in 20 min. How long will it take to empty the completely full the cistern due to leakage?

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The cistern has 2 inlet pipes P_{1} and P_{2} and a Leakage L. The total flow out of these three pipes gives the net flow.

It is given that pipes P_{1} and P_{2} fill the tank in 12 and 15 minutes respectively.

So, in 1 minute P_{1} will fill \\frac{1}{12}) ^{th} of the tank.

And in 1 minute P_{2} will fill \\frac{1}{15}) ^{th} of the tank.

The total flow in 1 minute will be \\frac{1}{20}) ^{th} of the tank.

Lets say the L is \\frac{1}{L}). Because it is emptying the cistern, we take it as negative.

Then,

\\frac{1}{12}) + \\frac{1}{15}) - \\frac{1}{L}) = \\frac{1}{20})

\\frac{5}{60}) + \\frac{4}{60}) - \\frac{3}{60}) = \\frac{1}{L})

\\frac{1}{10}) = \\frac{1}{L})
Which means that the Leaking pipe empties 1/10^{th} each minute and so, the completely filled cistern is emptied in 10 minutes.

The question is **"How long will it take to empty the completely full the cistern due to leakage?" **

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