Answer should be C.

Let side of Square = \(a\)

If AEC is 1/4 of circumference it means that Angle ADC is 90 as the circle is cut into 4 quarters. (also evident from the fact that ADC is the angle of a square)
Since AD is the radius as well, we can find the area of the arc AEC. Similarly we can find the area of the Arc AFC. Now if you add the two areas up, you notice that the shaded region is counted twice. To remove this double counting and excess area you subtract the area of the square. This gives us the area of the shaded region.

Area of Shaded portion = \(2*\frac{\pi*a^2}{4}-a^2\)\(=\) \(\frac{\pi*a^2}{2}-a^2 =\frac{\pi*a^2-2*a^2}{2}\)

Area of Unshaded portion = Area of Square - Area of Shaded portion

= \(a^2 - (2*\frac{\pi*a^2}{4}-a^2)\)\(=\) \(2*a^2 - \frac{\pi*a^2}{2}=\frac{4*a^2 - \pi*a^2}{2}\)

\(\frac{Area of Unshaded portion}{Area of Shaded portion}=\frac{4*a^2 - \pi*a^2}{\pi*a^2-2*a^2}=\frac{4 - \pi}{\pi-2}\)

Putting \(\pi = \frac{22}{7}\) in above ratio , we get \(\frac{Area of Unshaded portion}{Area of Shaded portion}= \frac{(28-22)}{(22-14)} =\frac{6}{8}=\frac{3}{4}\)

Answer C

Now if you add the two areas up, you notice that the shaded region is counted twice. To remove this double counting and excess area you subtract the area of the square

Area of Shaded portion = \(2*\frac{\pi*a^2}{4}-a^2\)\(=\) \(\frac{\pi*a^2}{2}-a^2 =\frac{\pi*a^2-2*a^2}{2}\)


Please can you simplify this,it's quite confusing.

Join AC, it divides the square into 2 equal parts..
Let us take one part ABFC, ABC is isosceles right angled triangle with sides, say 2, so area = (1/2)*2*2=2
Area of ABFC.. ABFC is 1/4th of a circle with side 2, so area = \(\frac{\pi*2^2}{4}=\pi\)
so shaded portion = area of ABFC - area of ABC = \(\pi-2\)
But there are two shade portion, one each side of AC, so area = 2(\(\pi-2\))=\(2\pi-4\)

therefore, the area of unshaded portion = area of square - area of shaded portion = \(2*2-2(\pi-2)=4-2\pi+4\)=\(8-2\pi\)

Ratio unshaded to shaded = \(8-2\pi\):\(2\pi-4\)=\(4-\pi\):\(\pi-2\)=\(\frac{6}{7}:\frac{8}{7}=6:8=3:4\)
C

Many thanks! @chetan2u

Hello from the GRE Prep Club BumpBot!

Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.