In a quarter circle of radius R, a circle of radius r is inscribed. What is the ratio of R to r?

Let centre of quarter circle be O, and centre of inscribed circle C at distance d from O along the angle bisector. Inscribed circle is tangent to both radii and to the arc internally. Distance from C to each radius = r, so d sin 45° = r, i.e., d = r√2. Also distance from O to C plus r = R (tangency with arc), so d + r = R. Thus R = r√2 + r = r(√2 + 1). So R : r = (√2 + 1) : 1.