Question : Pipes A, B and C can fill a tank in 10, 15 and 30 hours, respectively. D is an emptying pipe which alone can empty the full tank in $x$ hours. A, B and C are opened together for 3 hours and then closed. Now D is opened which alone empties the tank in 30 hours. What is the value of $x$?
Option 1: 50
Option 2: 40
Option 3: 60
Option 4: 45
Correct Answer: 50
Solution :
Let the capacity of the tank as $V$.
The rates at which pipes A, B, and C fill the tank are $\frac{V}{10}$, $\frac{V}{15}$, and $\frac{V}{30}$ units per hour, respectively.
The rate at which pipe D empties the tank is $\frac{V}{x}$ units per hour.
In the first 3 hours, pipes A, B, and C together fill the tank with $3 \left(\frac{V}{10} + \frac{V}{15} + \frac{V}{30}\right)$ units of water.
After that, pipe D alone empties the tank in 30 hours, so it empties $30 \left(\frac{V}{x}\right)$ units of water.
Since the amount of water filled by pipes A, B, and C is equal to the amount of water emptied by pipe D.
$⇒3 \left(\frac{V}{10} + \frac{V}{15} + \frac{V}{30}\right) = 30 \left(\frac{V}{x}\right)$
$⇒ \left(\frac{1}{10} + \frac{1}{15} + \frac{1}{30}\right) = 10 \left(\frac{1}{x}\right)$
$⇒ \frac{1}{5} = 10 \left(\frac{1}{x}\right)$
$⇒x=50$
Hence, the correct answer is 50.
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