$25\%$ of $A$'s income is equal to $35\%$ of $B$'s income. The ratio of the incomes of $A$ and $B$ is

### Concept & Ratios from Percentages When a percentage of one quantity equals a percentage of another, you can form a direct equation to isolate the ratio of the two quantities. $$ x\% \text{ of } A = y\% \text{ of } B \implies \frac{A}{B} = \frac{y}{x} $$ ### Step-by-Step Solution * **Given:** $25\%$ of $A = 35\%$ of $B$. * **Formulate the equation:** Write it mathematically using fractions or decimals. $\frac{25}{100} \times A = \frac{35}{100} \times B$ * **Simplify the equation:** The denominators ($100$) cancel out on both sides. $25A = 35B$ * **Isolate the ratio $\frac{A}{B}$:** Rearrange the terms by dividing both sides by $25B$. $\frac{A}{B} = \frac{35}{25}$ * **Reduce the fraction:** Divide the numerator and denominator by their highest common factor, which is $5$. $\frac{A}{B} = \frac{7}{5}$ * **Format as a ratio:** $A : B = 7 : 5$ ### Exam Strategy & Shortcut Skip writing out the hundreds. If $25A = 35B$, then the ratio $A:B$ is simply the reverse of their coefficients: $35:25$. Mentally divide both by $5$ to instantly arrive at $7:5$. ### Common Pitfall A very common trap is writing the ratio in the exact order the numbers appear in the text ($25 : 35 \implies 5 : 7$), completely inverting the correct mathematical relationship. ### Final Answer Therefore, the correct answer is **$7 : 5$**.