From GMATPrep Quant:

A circular medallion is made of a circular piece of colored glass surrounded by a circular metal frame. If the radius of the medallion is r centimeters and the width of the metal frame is S centimeters, then in terms of S and r, what is the area of the metal frame in square centimeters?

A) Pi(r - s)^2
B) Pi(r^2 - s^2)
C) 2Pi(r - s)
D) rPi(2r - s)
E) sPi(2r - s)

I have re-done this problem several times and always end up with sPi(2r + s), which is close to the answer E, but not quite there. Your help is greatly appreciated.

Area of metal frame = Area of medallion - Area of the inner circular area of the medallion (except frame)

= Pi(r^2) - Pi(r-s)^2
= Pi(r^2) - Pi(r^2 + s^2 - 2rs)
= Pi(r^2) - Pi*r^2 - Pi*s^2 + Pi*2rs
= Pi*2rs - Pi*s^2
= Pi*s(2r-s)
Answer is (E)

Hope it helps

GMAT 2007

You variable guys live in a different world from us lowly numbers folk. As my intellect is slightly less evolved than that of Forrest Gump, I'm going with the plugging in numbers approach on this VIC (Variables In the answer Choices). Also, don't forget to draw the problem. If you don't draw, Stupid is as stupid does, just like Mama says.

If r = 6, the area of the entire medallion is 36pi
If s = 2, the area of the colored class is 16 pi (since the radius of glass = 4)
Thus, the area of the metal frame is 36pi - 16pi = 20pi

Now, just plug r = 6 and s = 2 into the answer choices, and choose the one that equals 20pi.

A) Pi(r - s)^2 = 16pi
B) Pi(r^2 - s^2) = 32 pi
C) 2Pi(r - s) = 8pi
D) rPi(2r - s) = 60pi
E) sPi(2r - s) = 20pi

The correct answer is E, and that's all I have to say about that.

-dan