Rational equations — find a ratio (ACT). Game on.

Cross-multiply to clear the fraction: $2(4x - y) = 3(x + 2y)$, so $8x - 2y = 3x + 6y$. Collect like terms to get $5x = 8y$. Divide both sides by $y$, then by $5$, to obtain $\dfrac{x}{y} = \dfrac{8}{5}\textsf{.}$

Cross-multiply to clear the fraction: $5(x + 4y) = 3(2x + y)$, so $5x + 20y = 6x + 3y$. Collect like terms to get $x = 17y$. Divide both sides by $y$ to obtain $\dfrac{x}{y} = 17\textsf{.}$

Cross-multiply to clear the fraction: $4(x - 2y) = 1(3x + y)$, so $4x - 8y = 3x + y$. Collect like terms to get $x = 9y$. Divide both sides by $y$ to obtain $\dfrac{x}{y} = 9\textsf{.}$

Cross-multiply to clear the fraction: $5(4x - 3y) = 2(x + y)$, so $20x - 15y = 2x + 2y$. Collect like terms to get $18x = 17y$. Divide both sides by $y$, then by $18$, to obtain $\dfrac{x}{y} = \dfrac{17}{18}\textsf{.}$

Cross-multiply to clear the fraction: $3(2x + 5y) = 4(3x - y)$, so $6x + 15y = 12x - 4y$. Collect like terms to get $6x = 19y$. Divide both sides by $y$, then by $6$, to obtain $\dfrac{x}{y} = \dfrac{19}{6}\textsf{.}$

Cross-multiply to clear the fraction: $3(7x - y) = 5(x + 2y)$, so $21x - 3y = 5x + 10y$. Collect like terms to get $16x = 13y$. Divide both sides by $y$, then by $16$, to obtain $\dfrac{x}{y} = \dfrac{13}{16}\textsf{.}$

Cross-multiply to clear the fraction: $2(6x + y) = 5(2x - 3y)$, so $12x + 2y = 10x - 15y$. Collect like terms to get $2x = -17y$. Divide both sides by $y$, then by $2$, to obtain $\dfrac{x}{y} = -\dfrac{17}{2}\textsf{.}$

Cross-multiply to clear the fraction: $3(5x - 2y) = 2(x + 3y)$, so $15x - 6y = 2x + 6y$. Collect like terms to get $13x = 12y$. Divide both sides by $y$, then by $13$, to obtain $\dfrac{x}{y} = \dfrac{12}{13}\textsf{.}$