A small circle makes exactly four rotations around the inside of a larger circle before returning to its starting position. How many times larger is the area of the large circle?

Imagine that the small blue circle paints a blue line as it rolls around the big green circle. From the information in the question we see that it paints this blue line four times: so the line around the green circle -- its circumference -- is four times the circumference of the blue circle. The circumference of a circle is proportional to its radius, the distance from its centre to its edge. So the radius of the green circle is 4 times that of the blue circle. The area of a circle is proportional to the square of the radius (the radius multiplied by itself). So the area of the green circle must be 16 times the area of the blue circle (because 16 = 4 x 4 ).