A certain number of two digits is three times the sum of its digits and if 45 be added to it, the digits are reversed. The number is

### Concept & Logic Translate the numerical value of a number into an equation proportional to its digit sum to quickly isolate a single definitive digit ratio. $$ 10x + y = 3(x + y) $$ ### Step-by-Step Solution **Given:** * The number is exactly 3 times the sum of its own digits. * Adding 45 to the number reverses its digits. **Calculation / Deduction:** * Let the original two-digit number be $10x + y$. * Set up the first condition: $10x + y = 3(x + y)$. * Expand the terms: $10x + y = 3x + 3y$. * Rearrange to find the ratio: $7x = 2y \Rightarrow x/y = 2/7$. * Because $x$ and $y$ must be single, positive integers (from 1 to 9), the only numbers that can satisfy a ratio of 2:7 are $x = 2$ and $y = 7$. * The original number is 27. * Verify using the second condition: $27 + 45 = 72$. 72 is indeed the reversed form of 27. ### Exam Strategy & Shortcut **Direct Ratio Mapping:** The moment you simplify $10x + y = 3(x + y)$ into $7x = 2y$, you can completely stop calculating. There are no other single-digit combinations that equal each other when multiplied by 7 and 2 respectively. The digits are locked to 2 and 7 instantly. ### Common Pitfall Getting bogged down trying to create a complex system of equations by utilizing the $+45$ condition ($9(y-x) = 45 \Rightarrow y-x = 5$). While correct, solving the simultaneous equations $7x = 2y$ and $y - x = 5$ wastes valuable time when the initial ratio already provides the exclusive answer. ### Final Answer **Therefore, the correct answer is 27.**