A circle is inscribed in a half circle with a diameter of

This question needs a revision, because there are infinitely many circles possible to inscribe in a semicircle. Consider just two cases below:Now, intended meaning of the question is the case with the circle which inscribed right in the middle but it should be specified.

Anyway, for that case we don't really need any formula to get the ratio:
Since the radius of the big circle (\(\frac{\pi}{2}\)) is twice the radius of the inscribed circle (\(\frac{\pi}{4}\)) then its area is 4 times greater then the area of the inscribed circle (because in the area formula the radius is squared. For example the area of a circle with radius of 2 is \(4\pi\), which is 4 times greater than the radius of a circle with the radius of 1, which is \(\pi\)).

Thus the are of the semicircle is 4/2=2 times greater than the area of the inscribed circle: the area of the semicircle 2 units, the area of the inscribed circle 1 unit, and the area of the semicircle not covered by the inscribed circle is also 1. Ratio: 2/1.

Answer: E.

Hope it's clear.