50 (or 2).
The larger rectangle, of course, is a square, since it circumscribes a circle. It sides are equal in length to the diameter of the inscribed circle.
Let's call the sides of the small rectangle x and 2x.
First, draw the radius from the center of the circle to the corner of the small rectangle. Call this length r (naturally). Then, extend one of the legs of the triangle, say the length (2x), into the circle. Then draw the right triangle that has that and the radius for sides.
The legs of this right triangle run parallel to the sides of the square (actually, to both rectangles). The shorter leg (the extension of the rectangle) has a length of r-2x, while the longer leg has a length of r-x. The hypotenuse of the triangle is the radius of the circle, or r.
The only right triangle in which the sides differ in length by a value of x is a 3x, 4x, 5x right triangle. So, the hypotenuse of the triangle (and the radius of the circle is 5x.
Since the circle is inscribed in the square, the side of the square is the diameter of the circle, which we now know must be 10x.
Therefore, 50 of the smaller rectangles will fit into the square.