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Q.

Pipe $A$ can fill a tank in '$a$' hours. On account of a leak at the bottom of the tank it takes thrice as long to fill the tank. How long will the leak at the bottom of the tank take to empty a full tank, when pipe $A$ is kept closed?

 A.

\(\dfrac{3a}{2}\) hours

 B.

\(\dfrac{2a}{3}\) hours

 C.

\(\dfrac{4a}{3}\) hours

 D.

\(\dfrac{3a}{4}\) hours

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Solution:
Option(A) is correct

Pipe $A$ fills the tank in '$a$' hours.

Therefore, \(\dfrac{1}{a}\) of the tank gets filled in an hour.

On account of the leak it takes $3a$ hours to fill the tank.

Therefore, \(\dfrac{1}{3a}\)of the tank gets filled in an hour. 
Let the leak at the bottom of the tank take '$x$' hours to empty the tank. 

Hence, \(\dfrac{1}{x}\)of the tank gets emptied every hour.

\(\dfrac{1}{a}-\dfrac{1}{x}=\dfrac{1}{3a}\)

\(\dfrac{1}{x}=\dfrac{1}{a}=\dfrac{1}{3a}=\dfrac{2}{3a}\)

Hence, $x = \dfrac{3a}{2}$


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