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 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.

Answer: Circle a will revolve 3 times

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.

Answer: 3