Circle a has 1/3 the radius of circle b, and circle a rolls one trip around circle
b. how many times will circle a revolve in total?

Let the radius of circle a be R, then the radius of circle b is 3R.

The circumference of circle b is 2π*(radius b)=2π*3R= 6πR.

The circumference of circle a is 2π*(radius a)=2π*R=2πR.

This means that one complete revolution of circle a covers 2πR distance. Since there is a total 6πR distance in circle b:

Circle a revolves in total 6πR/2πR=3 many times.

3

Let r = radius of circle a.
Then the radius of circle b is 3r.

One revolution around circle b is 2π*(3r) = 6πr
One revolution of circle a is a distance of 2πr.

Therefore when circle a rolls one trip around circle b, it will make
(6πr)/(2πr) = 3 revolutions.

 Circle a will revolve 3 times

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