$\frac{3}{4} : \frac{1}{2} : : 27y : $x$

### Concept & Proportion with Fractions The rule of proportion applies exactly the same way when fractions and variables are involved. Product of extremes = product of means. $$ a : b : : c : d \implies ad = bc $$ ### Step-by-Step Solution * **Given:** $\frac{3}{4} : \frac{1}{2} : : 27y : x$ * **Apply the rule:** $(\frac{3}{4}) \times x = (\frac{1}{2}) \times 27y$ * **Simplify the right side:** $\frac{3x}{4} = \frac{27y}{2}$ * **Solve for $x$:** Multiply both sides by $\frac{4}{3}$. $x = (\frac{27y}{2}) \times (\frac{4}{3})$ * **Calculate:** $x = \frac{108y}{6} = 18y$ ### Exam Strategy & Shortcut Simplify the ratio on the left first. $\frac{3/4}{1/2} = \frac{3}{4} \times 2 = \frac{3}{2}$. So the ratio is $3 : 2$. Now apply this to the right side: $27y : x = 3 : 2$. $3$ parts $= 27y$ (so $1$ part $= 9y$). $2$ parts $= 2 \times 9y = 18y$. ### Common Pitfall Mismanaging fractions during cross-multiplication, particularly forgetting to take the reciprocal when moving a fraction to the other side of the equation. ### Final Answer Therefore, the correct answer is **$18y$**.