If 8% of $x = 4$% of $y$, then 20% of $x$ is

### Concept & Strategy This is an equivalence proportion problem. Because percentages are inherently linear scalar multipliers, if you know the ratio between a specific percentage of $x$ and a specific percentage of $y$, you can scale that relationship up or down proportionally without needing to find the absolute values of $x$ or $y$. $$ a\% \text{ of } x = b\% \text{ of } y \implies x = \frac{b}{a}y $$ ### Step-by-Step Solution * **Given:** 8% of $x = 4$% of $y$ * **Calculation:** First, establish the fundamental algebraic relationship between $x$ and $y$. Drop the % signs (as it divides both sides by 100): $$ 8x = 4y $$ Divide by 4 to simplify the ratio: $$ 2x = y \implies x = \frac{y}{2} $$ The question asks for the value of "20% of $x$". Substitute our finding for $x$ into this expression: $$ 20\% \text{ of } x = 20\% \text{ of } \left(\frac{y}{2}\right) $$ $$ = \frac{20}{100} \times \frac{y}{2} $$ $$ = \frac{10}{100} \times y $$ $$ = 10\% \text{ of } y $$ ### Exam Strategy & Shortcut Use the scalar multiplier trick. You are given the value of 8% of $x$ and need to find 20% of $x$. Look at the given equivalence: $8\%(x) = 4\%(y)$. Notice that the percentage on the right (4%) is exactly half of the percentage on the left (8%). This means any percentage of $x$ will equal half that percentage of $y$. Therefore, 20% of $x$ must simply be half of 20 applied to $y$. Half of 20 is 10. The answer is instantly 10% of $y$. ### Common Pitfall A common mistake is applying operations improperly when substituting, such as multiplying the 20 by 2 instead of dividing, leading to an incorrect choice of 40% (if it were an option) or 80%. Sticking strictly to proportional logic prevents this. ### Final Answer Therefore, the correct answer is **10% of $y$**.