Answer: Option C
$$\eqalign{
& \sqrt {0.03 \times 0.3a} = {\text{0}}{\text{.3}} \times {\text{0}}{\text{.3}} \times \sqrt b \cr
& {\text{Squaring both sides}} \cr
& \Rightarrow {\text{0}}{\text{.03}} \times {\text{0}}{\text{.3a}} = {\left( {0.3} \right)^2} \times {\left( {0.3} \right)^2} \times b \cr
& \Rightarrow \frac{3}{{100}} \times \frac{3}{{10}}a = \frac{9}{{100}} \times \frac{9}{{100}} \times b \cr
& \Rightarrow 9a = \frac{{81}}{{10}}b \cr
& \Rightarrow 90a = 81b \cr
& \Rightarrow 10a = 9b \cr
& \Rightarrow \frac{a}{b} = \frac{9}{{10}} \cr
& \Rightarrow \frac{a}{b} = 0.9 \cr} $$