1. The problem is to find the ratio $x:y$ given the equation $2x = 3y$. 2. Start by expressing $x$ in terms of $y$: from $2x = 3y$, divide both sides by $2$ to get $x = \frac{3y}{2}$. 3. The ratio $x:y$ can be written as $\frac{x}{y}$. 4. Substitute $x = \frac{3y}{2}$ into the ratio: $\frac{x}{y} = \frac{\frac{3y}{2}}{y}$. 5. Simplify the fraction by canceling $y$: $\frac{\frac{3y}{2}}{y} = \frac{3}{2}$. 6. Therefore, the ratio $x:y = \frac{3}{2}$.