One-fourth of sixty percent of a number is equal to two-fifths of twenty percent of another number. What is the respective ratio of the first number to the second number?

### Concept & Formula To find the ratio between two unknown variables described by a mathematical statement, we can translate the verbal percentages and fractions directly into an algebraic equation. The basic translation rules are: "of" means multiplication, and percentages can be written as fractions over $100$. $$ \text{Percentage} = \frac{\text{Value}}{100} $$ ### Step-by-Step Solution 1. Let the first number be $x$ and the second number be $y$. 2. Translate the first part of the sentence: "One-fourth of sixty percent of a number" becomes $\frac{1}{4} \times \frac{60}{100} \times x$. 3. Translate the second part of the sentence: "two-fifths of twenty percent of another number" becomes $\frac{2}{5} \times \frac{20}{100} \times y$. 4. Equate the two expressions as stated in the problem: $$ \frac{1}{4} \times \frac{60}{100} \times x = \frac{2}{5} \times \frac{20}{100} \times y $$ 5. Simplify the fractions on both sides: $$ \frac{15}{100} x = \frac{8}{100} y $$ 6. Multiply both sides by $100$ to eliminate the denominators: $$ 15x = 8y $$ 7. Rearrange to find the ratio of $x$ to $y$ ($\frac{x}{y}$): $$ \frac{x}{y} = \frac{8}{15} $$ 8. The ratio of the first number to the second number is $8 : 15$. 9. Since $8 : 15$ is not present in options (a), (b), or (c), the correct choice is "None of these". ### Exam Strategy & Shortcut Instead of writing out the full $100$ in the denominator for percentages, note that percentages on both sides of an equation will cancel out. You can calculate directly: $\frac{1}{4} \times 60 \times x = \frac{2}{5} \times 20 \times y$ $15x = 8y \Rightarrow \frac{x}{y} = \frac{8}{15}$. This saves time writing and simplifying. ### Common Pitfall A very common mistake is to misread the final equation $15x = 8y$ as a ratio of $15 : 8$. Remember that to find the ratio $\frac{x}{y}$, you must divide both sides by $y$ and by $15$, yielding $\frac{8}{15}$. ### Final Answer Therefore, the correct answer is **None of these**.