Choice A is correct. It's given that $w$ represents the total wall area, in square feet. Since the walls of the room will be painted twice, the amount of paint, in gallons, needs to cover $2w$ square feet. It’s also given that one gallon of paint will cover $220$ square feet. Dividing the total area, in square feet, of the surface to be painted by the number of square feet covered by one gallon of paint gives the number of gallons of paint that will be needed. Dividing $2w$ by $220$ yields  $\frac{2w}{220}$, or $\frac{w}{110}$. Therefore, the equation that represents the total amount of paint $P$, in gallons, needed to paint the walls of the room twice is $P=\frac{w}{110}$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from finding the amount of paint needed to paint the walls once rather than twice.

Choice D is incorrect and may result from conceptual or calculation errors.