If $x$% of $a$ is the same as $y$% of $b$, then $z$% of $b$ is

### Concept & Logic This algebraic aptitude question requires isolating a variable from a proportional equivalence and substituting it into a new expression. Maintaining the percentage denominators structure clear until the final step is key to avoiding algebraic tangle. $$ \text{Translation: } p\% \text{ of } q \iff \frac{p \times q}{100} $$ ### Step-by-Step Solution * **Given:** $x$% of $a = y$% of $b$ * **Calculation:** First, write the given statement algebraically: $$ \frac{x}{100} \times a = \frac{y}{100} \times b $$ Multiply both sides by 100 to eliminate the denominators: $$ xa = yb $$ The question asks us to evaluate an expression involving $b$, and the options are expressed in terms of $a$. This implies we must solve our initial equation for $b$. Isolate $b$: $$ b = \frac{xa}{y} $$ Now, evaluate the target expression: "$z$% of $b$": $$ z\% \text{ of } b = \frac{z}{100} \times b $$ Substitute our isolated expression for $b$ into this target: $$ z\% \text{ of } b = \frac{z}{100} \times \left( \frac{xa}{y} \right) $$ Rearrange the terms to match the format of the options (which are formatted as "something% of a"): $$ = \frac{z \times x}{y} \times \frac{a}{100} $$ $$ = \frac{xz}{y} \times \frac{a}{100} $$ $$ = \frac{xz}{y}\% \text{ of } a $$ ### Exam Strategy & Shortcut Treat the percentage signs as non-existent during the initial equivalence. $xa = yb \implies b = \frac{xa}{y}$. The question wants $z$ applied to $b$. Just multiply both sides by $z$: $zb = z \left(\frac{xa}{y}\right) = \frac{xza}{y}$. Group the constants with the percentage: $\frac{xz}{y}$ applied to $a$. The percent sign drops naturally into place based on the options, giving $\frac{xz}{y}$% of $a$. ### Common Pitfall A frequent algebra mistake is solving for $a$ instead of $b$, or accidentally moving a variable to the wrong side of the fraction during substitution (e.g., getting $\frac{yz}{x}$). Always pause to confirm which variable needs to be substituted out. The target expression is "z% of **b**", meaning **b** is the variable you want to replace. ### Final Answer Therefore, the correct answer is **$\frac{xz}{y}$% of $a$**.