\(\dfrac{x-y}{x+2y}=\dfrac{3}{4}\Leftrightarrow4\left(x-y\right)=3\left(x+2y\right)\\ \Leftrightarrow4x-4y=3x+6y\\ \Leftrightarrow x=10y\Leftrightarrow\dfrac{x}{y}=10\)
\(\dfrac{x-y}{x+2y}=\dfrac{3}{4}\)
\(\Rightarrow4\left(x-y\right)=3\left(x+2y\right)\)
\(\Rightarrow4x-4y=3x+6y\)
\(\Rightarrow x=10y\)
\(\dfrac{x}{y}=\dfrac{10y}{y}=10\)