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Two pipes $P$ and $Q$ can fill a tank in $6$ hours and $9$ hours respectively, while a third pipe $R$ can empty the tank in $12$ hours.  Initially, $P$ and $R$ are open for $4$ hours, Then $P$ is closed and $Q$ is opened. After $6$ more hours $R$ is closed. The total time taken to fill the tank (in hours) is  ____

1. $13.50$
2. $14.50$
3. $15.50$
4. $16.50$

edited | 426 views
Migrated from GO Mechanical 8 months ago by Arjun

P can fill the tank in $6$ hours

$\Rightarrow$ In $1$ hr P can fill $\frac{1}{6}$ of the tank.

Q can fill the tank in $9$ hours

$\Rightarrow$ In $1$ hr Q can fill $\frac{1}{9}$ of the tank.

R can empty the tank in $12$ hours

$\Rightarrow$ In $1$ hr R can empty $\frac{1}{12}$ of the tank.

P and R are opened for $4$ hours

$\Rightarrow$ They fill $4*\left ( \frac{1}{6}-\frac{1}{12} \right ) = 4*\frac{1}{12}=\frac{1}{3}$ of the tank.

$\Rightarrow$ $1-\frac{1}{3}=\frac{2}{3}$ of the tank is still empty.

Then P is closed and Q is opened. After $6$ more hours R is closed.

$\Rightarrow$ Q and R are opened together for $6$ hours.

$\Rightarrow$ They fill $6*\left ( \frac{1}{9}-\frac{1}{12} \right ) = 6*\frac{1}{36}=\frac{1}{6}$ of the tank.

$\Rightarrow$ $\frac{2}{3}-\frac{1}{6}=\frac{4-1}{6}=\frac{1}{2}$ of the tank is still empty.

Now only Q is opened

$\because$ Q can fill a tank in $9$ hr

$\Rightarrow$ Q can fill $\frac{1}{2}$ of the tank in $9*\frac{1}{2}$ hours $=4.5$ hours.

$\therefore$ Total Time to fill the tank $=4$ hours $+6$ hours $+ 4.5$ hours $=14.5$ hours

So, Option B. $14.50$  is the correct answer.

by Boss (25.1k points)
edited by
+2
There is a mistake in the last step. Q can fill tank in 9 hours. So, answer would be 4+6+4.5=14.5 hours. Hence, B is the answer.
+1
Corrected now :)

by Active (5k points)

$Let\ Capacity=36L$

$P\rightarrow 6H-\ 6L/H$

$Q\rightarrow 9H-\ 4L/H$

$R\rightarrow 12H-3L/H$

$|\underbrace{4H:P+R=3L/H}||\underbrace{6H:Q+R=1L/H}||\underbrace{?H:Q=4L/H}|$

$12L$                              $6L$                        $18L$

$1H\leftarrow4L$

$?\leftarrow18L$

$?=4.5H$

$ans:4+6+4.5=14.5H$

ago by Loyal (5.5k points)