Answer: D. For a mass–spring system, $T=2\pi\sqrt{m/k}$ and $\omega=2\pi/T$

$$\frac{T_{\text{new}}}{T} =\sqrt{\frac{(m/4)/(9k)}{m/k}} =\frac{1}{6}, \qquad \frac{\omega_{\text{new}}}{\omega} =\frac{T}{T_{\text{new}}} =6.$$

A: gives the period ratio but does not take its reciprocal for angular frequency.
B: applies the mass change alone, giving a factor of $2$ and omits the spring change.
C: applies the spring change alone, giving a factor of $3$ and omits the mass change.
Remember: combine both changes when finding the period, then use the inverse relation between angular frequency and period.