Answer
$\frac{I_A}{I_B} = \frac{1}{9}$
Work Step by Step
Since the belt does not slip, the linear velocity $v$ on the rim of each wheel must be equal.
Let the angular speed of wheel A be $\omega_A$
$\omega_A = \frac{v}{R_A}$
We can find an expression for the angular speed of wheel B:
$\omega_B = \frac{v}{R_B}$
$\omega_B = \frac{v}{3R_A}$
$\omega_B = \frac{\omega_A}{3}$
We can find the ratio of $\frac{I_A}{I_B}$ if the two wheels have the same rotational kinetic energy:
$K_A=K_B$
$\frac{1}{2}~I_A~\omega_A^2 = \frac{1}{2}~I_B~\omega_B^2$
$\frac{I_A}{I_B} = \frac{\omega_B^2}{\omega_A^2}$
$\frac{I_A}{I_B} = \frac{(\frac{\omega_A}{3})^2}{\omega_A^2}$
$\frac{I_A}{I_B} = \frac{1}{9}$
