If $x$ is 80% of $y$, then what percent of $2x$ is $y$?

### Concept & Logic This problem tests ratio and proportion within the context of percentages. By establishing the ratio between $x$ and $y$, you can easily find the relative percentage of any multiples of these variables. $$ \text{Percentage} = \left( \frac{\text{Value}}{\text{Reference Base}} \right) \times 100 $$ ### Step-by-Step Solution * **Given:** $x = 80$% of $y$ * **Calculation:** Convert the percentage to a fraction to find the fundamental ratio between $x$ and $y$: $$ x = \frac{80}{100} y = \frac{4}{5} y $$ $$ \frac{y}{x} = \frac{5}{4} $$ The question asks: "what percent of $2x$ is $y$?". This translates algebraically to: $$ \text{Required Percentage} = \left( \frac{y}{2x} \right) \times 100 $$ We can rewrite this expression to isolate our known ratio: $$ \text{Required Percentage} = \frac{1}{2} \times \left( \frac{y}{x} \right) \times 100 $$ Substitute the established ratio $\frac{y}{x} = \frac{5}{4}$ into the expression: $$ \text{Required Percentage} = \frac{1}{2} \times \left( \frac{5}{4} \right) \times 100 $$ $$ \text{Required Percentage} = \frac{5}{8} \times 100 = \frac{500}{8} = 62.5\% $$ Convert the decimal to a mixed fraction format to match the given options: $62.5\% = 62\frac{1}{2}\%$ ### Exam Strategy & Shortcut Use the value assumption method for rapid solving. Let $y = 100$. Then, $x = 80$% of $100 = 80$. The question asks for $y$ as a percent of $2x$. $2x = 2 \times 80 = 160$. The required percentage is $\frac{100}{160} \times 100$, which simplifies to $\frac{5}{8} \times 100$. Knowing standard fraction-to-percentage conversions is vital for speed: $\frac{1}{8} = 12.5\%$, so $\frac{5}{8} = 5 \times 12.5\% = 62.5\%$. This avoids complex algebra completely. ### Common Pitfall Many students mistakenly calculate what percent $2x$ is of $y$, or they forget to multiply $x$ by 2 in the denominator, simply finding $y$ as a percentage of $x$ (which would incorrectly yield 125%). Always double-check the base. ### Final Answer Therefore, the correct answer is **$62\frac{1}{2}$%**.