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 numbers, translate the given percentage and fraction conditions into an algebraic equation and simplify to find the ratio $\frac{A}{B}$. $$ \text{Fraction}_1 \times \text{Percentage}_1 \times A = \text{Fraction}_2 \times \text{Percentage}_2 \times B $$ ### Step-by-Step Solution * Let the first number be $A$ and the second number be $B$. * Translate the first condition: "One-fourth of sixty percent of a number" $$ \frac{1}{4} \times 60\% \times A $$ * Translate the second condition: "two-fifths of twenty percent of another number" $$ \frac{2}{5} \times 20\% \times B $$ * Equate both sides: $$ \frac{1}{4} \times \left(\frac{60}{100}\right) \times A = \frac{2}{5} \times \left(\frac{20}{100}\right) \times B $$ * Simplify the percentages: $$ \frac{1}{4} \times \left(\frac{3}{5}\right) \times A = \frac{2}{5} \times \left(\frac{1}{5}\right) \times B $$ $$ \frac{3}{20} \times A = \frac{2}{25} \times B $$ * Rearrange to find the ratio $\frac{A}{B}$: $$ \frac{A}{B} = \frac{\frac{2}{25}}{\frac{3}{20}} $$ $$ \frac{A}{B} = \frac{2}{25} \times \frac{20}{3} = \frac{40}{75} $$ * Simplify the fraction by dividing the numerator and denominator by 5: $$ \frac{A}{B} = \frac{8}{15} $$ * The required ratio is $8:15$. Since this is not among the options (a), (b), or (c), the answer is "None of these". ### Exam Strategy & Shortcut Ignore the percentage signs on both sides since they cancel out (effectively dividing both sides by 100). The equation immediately becomes: $\frac{1}{4} \times 60 \times A = \frac{2}{5} \times 20 \times B$ $15A = 8B$ $\frac{A}{B} = \frac{8}{15}$ This directly yields the ratio $8:15$ in seconds. ### Common Pitfall Students often incorrectly match the ratio values, calculating $\frac{B}{A}$ instead of $\frac{A}{B}$, or they get lost simplifying fractions on opposite sides of the equation. Also, since $8:15$ is not in the options, a common mistake is second-guessing and choosing a close option instead of confidently marking "None of these". ### Final Answer **Therefore, the correct answer is None of these.**