Time and Work
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Q.

Pipe A fills a tank of 700 litres capacity at the rate of 40 litres a minute. Another pipe B fills the same tank at the rate of 30 litres a minute. A pipe at the bottom of the tank drains the tank at the rate of 20 litres a minute. If pipe A is kept open for a minute and then closed and pipe B is kept open for a minute and then closed and then pipe C is kept open for a minute and then closed and the cycle repeated, how long will it take for the empty tank to overflow?

 A.

42 minutes

 B.

14 minutes

 C.

39 minutes

 D.

None of these

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Solution:
Option(D) is correct

Pipe $A$ fills the tank at the rate of 40 litres a minute. Pipe $B$ at the rate of 30 litres a minute and Pipe $C$ drains the tank at the rate of 20 litres a minute.

If each of them is kept open for a minute in the order $A-B-C$, the tank will have 50 litres of water at the end of 3 minutes.

After 13 such cycles, the tank will have $13×50=650$ litres of water.

It will take $13×3=39$ minutes for the 13 cycles to be over.

At the end of the $39^{th}$ minute, Pipe $C$ will be closed and Pipe $A$ will be opened. It will add 40 litres to the tank. 

Therefore, at the end of the $40^{th}$ minute, the tank will have $650+40=690$ litres of water.

At the end of the $40^{th}$ minute, Pipe $A$ will be closed and Pipe $B$ will be opened. It will add 30 litres of water in a minute. 

Therefore, at the end of the $41^{th}$ minute, the tank will have $690+30=720$ litres of water.

But then at 700 litres, the tank will overflow. Therefore, Pipe $B$ need not be kept open for a full minute at the end of 40 minutes.

Pipe $B$ needs to add 10 more litres of water at the end of 40 minutes. It will take \(\left(\dfrac{1}{3}\right)^{th}\) of a minute to fill 10 litres of water.

Therefore, the total time taken for the tank to overflow

$= 40$ minutes \(+\dfrac{1}{3}\) of a minute or 40 minutes 20 seconds


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