The diameters of two circular coins are in the ratio of 1:3. The smaller coin is made to roll around the bigger coin till it returns to the position from where the process of rolling started. How many times the smaller coin rolled around the bigger coin?

The correct answer is C. The number of rotations is determined by the ratio of the distances traveled. Let the radius of the smaller coin be r and the radius of the larger coin be 3r, based on the 1 to 3 diameter ratio. When the smaller coin rolls around the larger coin, its center moves in a circle. The radius of this path is the sum of the radii of both coins, which is r plus 3r, equaling 4r. The total distance the center of the smaller coin travels is the circumference of this path, which is 2 times pi times 4r. The distance covered in one full rotation of the smaller coin is its own circumference, which is 2 times pi times r. To find the number of rotations, divide the total distance traveled by the distance of one rotation. Dividing 2 times pi times 4r by 2 times pi times r equals 4. However, one of these rotations is due to the circular path itself, similar to how the moon rotates once while orbiting the earth. The question asks how many times the coin rolled around the bigger coin relative to the surface. Since the ratio of the circumferences is 3 to 1, the smaller coin completes 3 full rotations relative to the surface of the larger coin to cover the distance.