When a smaller circle rolls around a larger circle without slipping, the number of rotations the smaller circle makes is determined by the ratio of their circumferences and the distance it travels. If the diameter of the smaller coin is 'd' and the larger coin is '3d', their radii are 'r' and '3r' respectively. The center of the smaller coin travels a circular path with a radius equal to the sum of the radii of both coins (r + 3r = 4r). The circumference of this path is 2 * pi * (4r) = 8 * pi * r. The circumference of the smaller coin is 2 * pi * r. The number of rotations is the distance traveled by the center divided by the circumference of the smaller coin, plus one rotation due to the revolution around the larger circle itself. However, a simpler way to consider this specific problem, where the smaller coin rolls around the *outside* of the larger coin, is that the number of rotations is the ratio of the circumference of the larger coin to the circumference of the smaller coin. Since the diameters are in the ratio 1:3, the circumferences are also in the ratio 1:3. Thus, the smaller coin completes 3 revolutions around the larger coin.