Question : Two inlet pipes can fill a cistern in 10 and 12 hours respectively and an outlet pipe can empty 80 gallons of water per hour. All three pipes working together can fill the empty cistern in 20 hours. What is the capacity (in gallons) of the tank?
Option 1: 360
Option 2: 300
Option 3: 600
Option 4: 900
Correct Answer: 600
Solution :
Let the time taken by the outlet pipe to empty the tank be $x$ hour.
One hour's work of pipe A = $\frac{1}{10}$
One hour's work of pipe B = $\frac{1}{12}$
One hour's work of pipe C = $-\frac{1}{x}$
When all pipes work together, the tank will be emptied in 20 hours.
So, $\frac{1}{10}+\frac{1}{12}-\frac{1}{x} =\frac{1}{20}$
⇒$\frac{1}{10}+\frac{1}{12}-\frac{1}{20} =\frac{1}{x}$
⇒$\frac{1}{x}=\frac{6+5-3}{60}$
⇒$\frac{1}{x}=\frac{8}{60}$
$\therefore x = \frac{15}{2}=7.5$
Therefore, the outlet pipe can empty the tank in 7.5 hours.
In one hour, it empties 80 gallons.
In 7.5 hours, it empties 80 × 7.5 = 600 gallons
So, the capacity of the tank is 600 gallons.
Hence, the correct answer is 600.
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