compare the 2 circumferences ...
(circumference A) / (circumference B)
= (2 x pi x radius A) / (2 x pi x radius B)
= (radius A) / (radius B)
= 1 / 3
this means that ...
circumference B = 3 x circumference A

at first glance, it looks like the answer to this question is 3 times. the correct answer is actually 4 times, because as circle A travels around a circular path, it is equivalent to rotating an additional round.

you can visualize this better with 2 coins of the same size. hold 1st coin steady, and roll the 2nd coin around the first coin, without slipping, just like in this question. you may think that since the 2 coins are the same size, and the circumferences are the same, then the 2nd coin should rotate only 1 round. by observing carefully, you should see that the 2nd coin rotates 2 full rounds.

in general, for problems involving one object rolling around another object, you need to consider the ratio of their perimeter/ circumference, then add 1, to get the total rounds of rotation of the moving object.