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 ?

For a circle of radius r rolling without slipping around a fixed circle of radius R, the rolling circle makes (R/r) rotations due to covering the larger circumference plus one extra rotation from the circular path — total = 1 + (R/r) [1]. Here the diameter ratio is 1:3, so R/r = 3 and the smaller coin should rotate 1 + 3 = 4 times [1]. None of the provided options equals 4. Option (3) = 3 is the common (but incomplete) answer one gets by counting only the distance-based rotations and neglecting the extra rotation from the circular motion; therefore the supplied choices are inconsistent with the correct result and require review.

Sources

1. [1] https://en.wikipedia.org/wiki/Coin_rotation_paradox