Important results of Ratio

i). Duplicate ratio of (a:b) is

$$\begin{aligned} (a^2:b^2) \end{aligned}$$

ii). Sub-duplicate ratio of (a : b) is
$$\begin{aligned} (\sqrt{a}:\sqrt{b}) \end{aligned}$$

iii). Triplicate ratio of (a : b) is
$$\begin{aligned} ({a}^3:{b}^3) \end{aligned}$$

iv). Sub Triplicate ratio of (a : b) is
$$\begin{aligned} ({a}^\frac{1}{3}:{b}^\frac{1}{3}) \end{aligned}$$

v).
$$\begin{aligned} if& \frac{a}{b} = \frac{c}{d}, \\ then \frac{a+b}{a-b} = \frac{c+d}{c-d} \end{aligned}$$