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}