Comparing Areas

To solve this problem, let's say the largest circle, which is circle $3$ has a radius $2x$ . Then the area of this circle would be $4x^2\pi$ . If Circle 2 was the same as Circle 1, then each circle would have a radius equal to $x$ . But Circle $2$ is larger than Circle $1$ , so we are going to consider the radius of Circle $2$ to be $(x+y)$ and the radius of Circle $1$ to be $(x-y)\ for \ y>0$ . Then the sum of the areas of the two circles is :

$(x+y)^2\pi+(x-y)^2\pi=$

$(x^2+2xy+y^2)\pi+(x^2-2xy+y^2)\pi=$

$(2x^2+2y^2)\pi$

which is more than $2x^2\pi$ . We can say that the shaded region is more than half the total region which is $4x^2\pi$ . Thus, the shaded region is larger than the unshaded region which represents Quantity $A$ .