Even though it can be easier to solve it by plugging in numbers as suggest on

Manhattan Prep, I think I find a way to solve it generally. Here is my try.

Quantity A is \(\frac{2r_1^2}{2r_2^2}\), where \(r_1\) is the radius of the larger circle and \(r_2\) the radius of the smaller one.

Quantity B is \(2*\frac{\pi*r_2^2}{\pi*r_1^2}\), keeping the same notation.

Now, let's try to get some information from the figure to simplify those ratios.

It is easy to notice that the diagonal of the smaller square is the diameter of the larger circle. That's could be a nice relationship to use.

In formula, this means that \(2r_1 = 2r_2\sqrt(2)\).

Substituting in the quantities, they become

Q1: \(\frac{(2r_2\sqrt(2))^2}{2r_2^2}\)

Q2: \(2*\frac{\pi*r_2^2}{\pi(r_2\sqrt(2))^2}\)

Simplifying we get

Q1: \(\frac{(8r_2^2}{4r_2^2}\)

Q2: \(2*\frac{r_2^2}{2r_2^2}\)

Simplifyng once more,

Q1: \(2\)

Q2: \(1\)

Thus, Q1 is larger than Q2. Answer A!