1. **Stating the problem:** Given the ratios $x : y = 4 : 6$ and $2 : x = 1 : 2$, find the value of $y$. 2. **Understanding the ratios:** The ratio $x : y = 4 : 6$ means $\frac{x}{y} = \frac{4}{6} = \frac{2}{3}$. 3. The ratio $2 : x = 1 : 2$ means $\frac{2}{x} = \frac{1}{2}$. 4. **Find $x$ from the second ratio:** $$\frac{2}{x} = \frac{1}{2} \implies 2 \times 2 = x \implies x = 4$$ 5. **Use $x$ to find $y$ from the first ratio:** $$\frac{x}{y} = \frac{2}{3} \implies \frac{4}{y} = \frac{2}{3} \implies y = \frac{4 \times 3}{2} = 6$$ 6. **Final answer:** $$y = 6$$