If 25% of a number is subtracted from a second number, the second number reduces to its five-sixths. What is the ratio of the first number to the second number?

### Concept & Formula When subtracting a part of one variable from another causes a proportional reduction in the second, you can equate the subtracted amount directly to the amount lost. $$ \text{Amount Subtracted} = \text{Initial Value} - \text{Final Value} $$ ### Step-by-Step Solution * Let the first number be $A$ and the second number be $B$. * **Given:** Subtracting $25\%$ of $A$ from $B$ results in $\frac{5}{6}$ of $B$. * Translate this into an algebraic equation: $$ B - 25\% \text{ of } A = \frac{5}{6}B $$ * Convert the percentage to a fraction ($25\% = \frac{1}{4}$): $$ B - \frac{1}{4}A = \frac{5}{6}B $$ * Group the $B$ terms on one side and the $A$ terms on the other: $$ B - \frac{5}{6}B = \frac{1}{4}A $$ * Factor out $B$ on the left side: $$ B \left(1 - \frac{5}{6}\right) = \frac{1}{4}A $$ $$ \frac{1}{6}B = \frac{1}{4}A $$ * We need the ratio of the first number to the second number, which is $\frac{A}{B}$. * Rearrange the equation to isolate $\frac{A}{B}$: $$ \frac{A}{B} = \frac{\frac{1}{6}}{\frac{1}{4}} $$ $$ \frac{A}{B} = \frac{1}{6} \times 4 = \frac{4}{6} = \frac{2}{3} $$ ### Exam Strategy & Shortcut Read carefully: "reduces **to** its five-sixths". This implies it lost $\frac{1}{6}$ of its value ($1 - \frac{5}{6} = \frac{1}{6}$). What caused this loss of $\frac{1}{6}$ of the second number? The subtraction of $25\%$ ($\frac{1}{4}$) of the first number. Therefore, directly equate them: $\frac{1}{4} \text{ of First} = \frac{1}{6} \text{ of Second}$. $\frac{\text{First}}{\text{Second}} = \frac{4}{6} = 2:3$. You can solve this mentally in seconds without writing the full equation. ### Common Pitfall A very common mistake is confusing "reduces **to** its five-sixths" with "reduces **by** its five-sixths", which would lead to a completely different (and incorrect) equation: $B - \frac{1}{4}A = B - \frac{5}{6}B$. Always pay close attention to prepositions in word problems. ### Final Answer **Therefore, the correct answer is 2: 3.**