Two pipes can fill a cistern in 55 min and 66 min respectively and a waste pipe can drain off 1320 gallons per minute. If all of them can fill the cistern in 2 hours, then what is the capacity of the cistern(in gallons) ? 

52800

Let the capacity of the cistern be $x$ gallons.

Step 1: Filling rates of the two pipes

• First pipe fills in 55 min $⇒\text{Rate} = \frac{x}{55}\text{ gallons/min}$.
• Second pipe fills in 66 min $⇒\text{Rate} = \frac{x}{66}\text{ gallons/min}$.

Waste pipe drains 1320 gallons/min.

Net filling time = 2 hours = 120 min.

So,

$\frac{x}{55} + \frac{x}{66} - 1320 = \frac{x}{120}$

Step 2: Simplify

LCM of 55 and 66 is 330.

$\frac{6x + 5x}{330} - 1320 = \frac{x}{120}$

$\frac{11x}{330} - 1320 = \frac{x}{120}$

$\frac{x}{30} - 1320 = \frac{x}{120}$

Multiply both sides by 120:

$4x - 158400 = x$

$3x = 158400$

$x = 52800$