I found ths on 'Mind Your Decisions' who got it from 'CueMath' and I don't know where 'CueMath' got it. I hadn't seen it before but as is often the case I'd just been missing out, because a search shows it's in lots of places on the web. Anyway, here is my solution, I haven't looked at other answers because I wanted to solve it the way Euclid might have. It's a really nice problem.

We are given a square in which is inscribed a quarter-circle and inside that a semi-crcle tangent to the quarter-circle, as shown. Length PQ = 4. What is the ratio of the portion shaded light purple (the semi-circle) to the area shaded green (the area of the quarter-circle not also in the semi-circle)?

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The Line of Centres Theorem If two circles are tangent to one another, the line passing through their centres passes the the point at which they are tangent. ^[1]

So we can draw in line PQR passing through these points.

Thales's theorem  If the hypotenuse of a right-angles triangle is the diameter of a circle, 

Since ST is a diameter of the semi-circle and P a right angle, P must lie on the circle through R, S and T. Therefore PQ is a radius of that circle and equal in length to QR which is also a radius. This makes PR, a radius of the quarter-circle, equal to 2QR, a radius of the semi-circle.

So radius of quarter-circle : radius of semi-circle = 2:1

Area of of quarter-circle : area of semi-circle = : = 1 : = 2 : 1.

So the quarter-circle as a whole is exactly twice the semi-circle, making the light purple and green areas the same. The length 4 is irrelevant, a pure red herring!

References

[1] Euclid, Elements, Book III, Proposition 11.

[2] Euclid, Elements, Book III, Proposition 31.