The difference between a two-digit number and the number obtained by interchanging the two digits is $63$. Which is the smaller of the two numbers?

### Concept & Logic The difference between a two-digit number and its reversed counterpart is directly proportional to the difference between its individual digits. $$ \text{Difference} = 9(x - y) $$ ### Step-by-Step Solution **Given:** * The difference between the two-digit number and its reversed form is $63$. **Calculation:** * Let the original number be $10x + y$ and the reversed number be $10y + x$. * The difference is $9(x - y) = 63$. * Solving for the difference of the digits: $(x - y) = 63 / 9 = 7$. * We need to find pairs of digits $(x, y)$ that have a difference of $7$. * The possible pairs of digits are $(8, 1)$ and $(9, 2)$. * This means the original number and its reverse could be $81$ and $18$, or $92$ and $29$. * The smaller number could be $18$ or $29$. Because there are multiple valid possibilities, a unique smaller number cannot be isolated. ### Exam Strategy & Shortcut Divide the difference by $9$ to get the difference between the digits ($63 / 9 = 7$). Immediately recognize that multiple pairs of digits satisfy a difference of $7$ ($9$ and $2$, $8$ and $1$). Since the exact digits are not fixed, the exact numbers aren't either. Mark "Cannot be determined" and move on. ### Common Pitfall A common mistake is finding one pair that works (like $9$ and $2$), generating the numbers $92$ and $29$, and prematurely selecting $29$ from the options without checking if other pairs (like $81$ and $18$) also satisfy the condition. ### Final Answer **Therefore, the correct answer is Cannot be determined.**