Answer
$\dfrac{4}{1}$ and $4: 1$
Work Step by Step
We can see that the ratio of the perimeters of two smaller figures is $\dfrac{a}{b}$, and the ratio of their areas is the square of the ratio of their perimeters:
$\dfrac{a^2}{b^2}$.
Set up a proportion:
$\dfrac{a^2}{b^2}=\dfrac{16 \sqrt 3}{\sqrt 3} \\ \dfrac{a}{b}=\sqrt {\dfrac{16 \sqrt 3}{\sqrt 3}}$
This implies that the square factor is $ \dfrac{a}{b}=\dfrac{4}{1}$
So, the ratio of the perimeters is equal to the scale factor, which is $4:1$.