A square with a side length of 4s has a smaller square inscribed. The vertices of the smaller square are at the midpoints of the sides of the larger square. What is the ratio of the larger square to the area of the smaller square? Express your answer in simplest form
Let the larger square have a side length of 4s. Then, the smaller square will have a side length of 2s (since the vertices of the smaller square are at the midpoints of the sides of the larger square).
The area of the larger square is (4s)^2 = 16s^2 and the area of the smaller square is (2s)^2 = 4s^2.
The ratio of the larger square to the area of the smaller square is:
16s^2 / 4s^2 = 16/4 = 4
Therefore, the ratio of the larger square to the area of the smaller square is 4:1.