If P and Q are circular regions, what is the sum of their areas?

to know the sum of the areas of 2 circles we need to know their respective radii otherwise there's an infinite number of posibilities.

from:
(1) A sector of the smaller circle enclosing an angle of 120 degrees has the same area of a sector of the larger circle enclosing an angle of 30 degrees.
this tells us that the total area of the smaller circle is 1/4th of the total area of the larger circle
(3x120 = 360 --> 3x30 = 90 = 1/4 of 360)
meaning PI*R1^2 = 4*PI*R2^2
however i have 2 variables and 1 equation, so i can find an infinite number of solutions since there is no limitations (other than that R has to be positive)
Not Suff

(2) The sum of the squares of the radii is 100.
this basically tells us that R1^2 + R2^2 = 100
a quick listing gives us only 2 possible combinations

R1 = 6 and R2 = 8
or
R1= 8 and R2 = 6

since the question is asking us about the total surface area, it doesn't matter which circle is largest since both cases result in the same total area.

don't get fooled into thinking you need (1) to limit the number of cases to 1


kudos if this was useful