A two-digit number becomes five-sixth of itself when its digits are reversed. The two digits differ by one. The number is

### Concept & Strategy By setting up the algebraic ratio between a two-digit number and its reversed form, we can simplify the equation to find the exact ratio between the two digits. $$ 10y + x = \frac{5}{6}(10x + y) $$ ### Step-by-Step Solution **Given:** * The reversed number is $\frac{5}{6}$ of the original number. * The digits differ by 1. **Calculation / Deduction:** * Let the original number be $10x + y$. The reversed number is $10y + x$. * Set up the equation: $10y + x = \frac{5}{6}(10x + y)$. * Multiply both sides by 6 to clear the fraction: $60y + 6x = 50x + 5y$. * Rearrange to group the $x$ and $y$ terms: $60y - 5y = 50x - 6x$. * Simplify: $55y = 44x$. * Divide by 11: $5y = 4x$, which gives the ratio $x/y = 5/4$. * We are given that the digits differ by 1 ($x - y = 1$). Since the digits are in a 5:4 ratio, the only single-digit values that fit are $x = 5$ and $y = 4$. * The original number is 54. ### Exam Strategy & Shortcut **Option Elimination via Fractions:** The reversed number becomes $\frac{5}{6}$ of itself, meaning the reversed number must be smaller. (a) 45 reverses to 54 (Bigger, eliminate). (b) 54 reverses to 45 (Smaller, check it: is 45 exactly $\frac{5}{6}$ of 54? Yes, $54 \times \frac{5}{6} = 9 \times 5 = 45$). (c) 56 reverses to 65 (Bigger, eliminate). (d) 65 reverses to 56 (Smaller, check it: $65 \times \frac{5}{6}$ is a fraction, not 56). ### Common Pitfall Confusing the order of the fraction multiplication, which leads to setting up $\frac{5}{6}(10y + x) = 10x + y$. This will yield negative or non-integer digit values, causing confusion. ### Final Answer **Therefore, the correct answer is 54.**