Given, the ratio of dispersive powers of crown glass and flint glass is $\frac{\omega_1}{\omega_2} = \frac{1}{2}$. For an achromatic combination of two thin lenses in contact, the condition is: $$ \begin{aligned} \frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} &= 0 \\ \Rightarrow \frac{\omega_1}{\omega_2} &= -\frac{f_1}{f_2} \\ \Rightarrow \frac{1}{2} &= -\frac{f_1}{f_2} \\ \Rightarrow f_1 &= -\frac{1}{2}f_2 \end{aligned} $$ The equivalent focal length $f$ of the combination is $30\text{ cm}$. The formula for the combination of two thin lenses is: $$ \begin{aligned} \frac{1}{f_1} + \frac{1}{f_2} &= \frac{1}{f} \\ \Rightarrow \frac{1}{-\frac{1}{2}f_2} + \frac{1}{f_2} &= \frac{1}{30} \\ \Rightarrow -\frac{2}{f_2} + \frac{1}{f_2} &= \frac{1}{30} \\ \Rightarrow -\frac{1}{f_2} &= \frac{1}{30} \\ \Rightarrow f_2 &= -30\text{ cm} \end{aligned} $$ Now, substituting the value of $f_2$ to find $f_1$: $$ \begin{aligned} f_1 &= -\frac{1}{2}(-30\text{ cm}) \\ &= 15\text{ cm} \end{aligned} $$ Thus, the focal lengths of the lenses are $15\text{ cm}$ and $-30\text{ cm}$.