When $30\%$ of one number is subtracted from another number, the second number reduces to its four-fifths. What is the ratio of the first to the second number? (Bank P.O., 2010)

### Concept & Algebraic Translation Translate the word problem directly into a linear equation. Let the "one number" (the first number) be $x$ and "another number" (the second number) be $y$. ### Step-by-Step Solution 1. Express "$30\%$ of one number": $0.3x$ or $\frac{3}{10}x$. 2. "subtracted from another number": $y - \frac{3}{10}x$. 3. "the second number reduces to its four-fifths": The result equals $\frac{4}{5}y$. 4. Set up the equation: $$ y - \frac{3}{10}x = \frac{4}{5}y $$ 5. Group the $y$ terms on one side: $$ y - \frac{4}{5}y = \frac{3}{10}x $$ $$ \frac{1}{5}y = \frac{3}{10}x $$ 6. Cross-multiply to eliminate the denominators: $$ 10y = 15x $$ 7. Divide both sides by $5$: $$ 2y = 3x $$ 8. Find the ratio of the first number ($x$) to the second number ($y$): $$ \frac{x}{y} = \frac{2}{3} $$ The ratio is $2 : 3$. 9. Check the given options. The ratio $2 : 3$ is not among (a) $2:5$, (b) $3:2$, or (c) $4:7$. ### Exam Strategy & Shortcut Instead of fractions, work purely with percentages. The equation is $100\% \text{ of } y - 30\% \text{ of } x = 80\% \text{ of } y$. Subtracting $80\% \text{ of } y$ from both sides leaves $20\% \text{ of } y = 30\% \text{ of } x$. Drop the percentages and zeroes: $2y = 3x \implies x:y = 2:3$. ### Common Pitfall Reversing the final ratio. Many test-takers reach $2y = 3x$ and hastily choose option (b) $3 : 2$. However, the question specifically asks for "the first ($x$) to the second ($y$)", which is $x/y = 2/3$. ### Final Answer Therefore, the correct answer is **None of these**.