If $x$% of 500 = $y$% of 300 and $x$% of $y$% of 200 = 60, then $x =$

### Concept & Formula This problem tests your ability to solve a system of non-linear equations derived from percentages. The core logic involves simplifying the percentage terms immediately to establish a linear ratio between $x$ and $y$, and then substituting that ratio into the second, more complex equation to solve for $x^2$. $$ a\% = \frac{a}{100} $$ ### Step-by-Step Solution * **Given:** * Equation 1: $x$% of $500 = y$% of $300$ * Equation 2: $x$% of $y$% of $200 = 60$ * **Calculation:** First, simplify Equation 1 to find the relationship between $x$ and $y$: $$ \frac{x}{100} \times 500 = \frac{y}{100} \times 300 $$ $$ 5x = 3y \implies y = \frac{5x}{3} $$ Next, expand Equation 2 using the percentage definition: $$ \left(\frac{x}{100}\right) \times \left(\frac{y}{100}\right) \times 200 = 60 $$ Simplify the constants: $$ \frac{x \cdot y \cdot 200}{10000} = 60 $$ $$ \frac{2xy}{100} = 60 \implies 2xy = 6000 \implies xy = 3000 $$ Now, substitute the value of $y = \frac{5x}{3}$ from our first step into this simplified second equation: $$ x \left( \frac{5x}{3} \right) = 3000 $$ $$ \frac{5x^2}{3} = 3000 $$ Solve for $x^2$: $$ 5x^2 = 9000 \implies x^2 = 1800 $$ Find $x$ by taking the square root. Extract perfect squares to simplify: $$ x = \sqrt{1800} = \sqrt{900 \times 2} = 30\sqrt{2} $$ ### Exam Strategy & Shortcut Simplify fractions mentally to save time. When you see $x$% of $500 = y$% of $300$, instantly drop the `%` and the zeros. Think "$5x = 3y$". For the second equation, "$x$% of $y$% of $200 = 60$", recognize that the double percentage means dividing by $10,000$. The zeros cancel fast: $2xy / 100 = 60$, so $xy = 3000$. Substituting a linear ratio into a product immediately gives a square term. Master this algebraic flow to solve in under 45 seconds. ### Common Pitfall Students often get confused by the nested percentages in the phrase "$x$% of $y$% of 200". A common mistake is to only divide by 100 once instead of twice (yielding $10,000$ in the denominator). Always remember that each "%" symbol demands its own division by 100. ### Final Answer Therefore, the correct answer is **$30\sqrt{2}$**.