If $a\%$ of $x$ is equal to $b\%$ of $y$, then $c\%$ of $y$ is what $\%$ of $x$?

### Concept & Formula This is a substitution problem involving percentages. The key is to isolate one variable in terms of the other from the first statement, and then substitute it into the second statement. Remember that $a\%$ mathematically means $\frac{a}{100}$. ### Step-by-step Solution **Given:** $a\%$ of $x = b\%$ of $y$ **Step 1: Translate the given statement into an algebraic equation** $\frac{a}{100} \times x = \frac{b}{100} \times y$ **Step 2: Simplify and isolate $y$** Multiply both sides by $100$ to remove the denominators: $ax = by$ Now, isolate $y$: $y = \frac{a}{b}x$ **Step 3: Evaluate the target expression** We need to find "$c\%$ of $y$". Target = $\frac{c}{100} \times y$ **Step 4: Substitute the value of $y$** Replace $y$ with $\frac{a}{b}x$ in our target expression: Target = $\frac{c}{100} \times \left(\frac{a}{b}x\right)$ Target = $\left(\frac{ac}{100b}\right)x$ **Step 5: Convert back to a percentage format** To express this as a percentage of $x$, we factor out the $\frac{1}{100}$: Target = $\left(\frac{ac}{b}\right) \times \frac{1}{100} \times x$ Target = $\frac{ac}{b}\%$ of $x$ ### Exam Strategy & Shortcut Use smart number picking. Let $x = 100$, $a = 20$, and $b = 10$. $20\%$ of $100 = 10\%$ of $y \Rightarrow 20 = 0.1y \Rightarrow y = 200$. Now let $c = 30$. We need $30\%$ of $y$, which is $30\%$ of $200 = 60$. The question asks: $60$ is what $\%$ of $x$ (which is $100$)? It's $60\%$. Now plug $a=20, b=10, c=30$ into the options to see which yields $60$. (c) $\frac{ac}{b} = \frac{20 \times 30}{10} = \frac{600}{10} = 60$. This matches perfectly. ### Common Pitfall Students often get confused by the sequence of abstract variables and attempt to cross-multiply incorrectly, leading to option (b) $\frac{bc}{a}\%$. Systematically writing out the fractions ($\frac{a}{100}$) prevents algebraic missteps. ### Final Answer **Therefore, the correct answer is $\frac{ac}{b}\%$.**