Question

If $\sqrt{0.04\times 0.4\times a}=0.4\times 0.04\times \sqrt{b}$ then find the ratio $\frac{a}{b}$.

Answer 1

$$\begin{gathered} \sqrt{0.04\times 0.4\times a}=0.4\times 0.04\times \sqrt{b} \\ \Rightarrow \sqrt{0.04\times 0.4}\times \sqrt{a}=0.4\times 0.04\times \sqrt{b} \\ \Rightarrow \frac{\sqrt{a}}{\sqrt{b}}=\frac{{\left(\sqrt{0.4\times 0.04}\right)}^2}{\sqrt{0.4\times 0.04}} \\ \Rightarrow \sqrt{\frac{a}{b}}=\sqrt{0.4\times 0.04} \end{gathered}$$
$$\begin{gathered} \Rightarrow \frac{a}{b}=0.4\times 0.04 \\ \Rightarrow \frac{a}{b}=\frac{4}{10}\times \frac{4}{100}=\frac{16}{1000} \\ \Rightarrow \frac{a}{b}=\frac{2}{125} \end{gathered}$$
∴ a : b = 2 : 125