Answer
$\dfrac{1}{10}$ and $1: 10$
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:
$\dfrac{a^2}{b^2}=\dfrac{2\pi}{200 \pi} \\ \dfrac{a}{b}=\sqrt {\dfrac{2\pi}{200 \pi}}=\sqrt {\dfrac{1}{100}}$
This implies that the square factor is $ \dfrac{a}{b}=\dfrac{1}{10}$
So, the ratio of the perimeters is equal to the scale factor, which is $1:10$.
