If $x$% of $y$ is equal to $z$, what percent of $z$ is $x$?

### Concept & Formula This problem involves algebraic manipulation of percentage formulas. You first express $z$ in terms of $x$ and $y$, and then substitute this relationship into the standard percentage formula to find what percentage $x$ is of $z$. $$ \text{Percentage} = \left( \frac{\text{Part}}{\text{Base}} \right) \times 100 $$ ### Step-by-Step Solution * **Given:** $x$% of $y = z$ * **Calculation:** Convert the given statement into an equation: $$ \frac{x}{100} \times y = z \implies z = \frac{xy}{100} $$ The question asks: "what percent of $z$ is $x$?". Here, $z$ is the base. Required Percentage $= \left(\frac{x}{z}\right) \times 100$ Substitute the expression for $z$ into this formula: $$ \text{Required Percentage} = \frac{x}{\left(\frac{xy}{100}\right)} \times 100 $$ Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: $$ \text{Required Percentage} = \left( x \times \frac{100}{xy} \right) \times 100 $$ The $x$ terms cancel out cleanly: $$ \text{Required Percentage} = \frac{100}{y} \times 100 = \frac{100^2}{y} $$ ### Exam Strategy & Shortcut Instead of getting lost in variables, pick smart numbers. Let $x = 20$ and $y = 50$. Then 20% of $50 = 10$, which means $z = 10$. The question then asks: what percent of $z$ (10) is $x$ (20)? $\frac{20}{10} \times 100 = 200\%$. Now plug your assumed value $y = 50$ into the options to see which one equals 200: (a) $50^2 / 100 = 25$ (b) $50 / 100^2 = 0.005$ (c) $100 / 50 = 2$ (d) $100^2 / 50 = 10000 / 50 = 200$. Option (d) matches perfectly. ### Common Pitfall A common error is confusing the base of the percentage. Students often incorrectly calculate $(z/x) \times 100$ instead of $(x/z) \times 100$, leading to incorrect algebraic manipulations. Always identify the word immediately following "of" as the denominator. ### Final Answer Therefore, the correct answer is **$\frac{100^2}{y}$**.