The difference between a two-digit number and the number obtained by interchanging the digits is $36$. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is $1 : 2$?

### Concept & Strategy Break the problem into two sequential steps: first, find the actual digits using the reversal rule and the given ratio, then compute the required algebraic operation on those digits. $$ \text{Digit Difference} = \frac{\text{Number Difference}}{9} $$ ### Step-by-Step Solution **Given:** * The difference between the number and its reverse is $36$. * The ratio of the digits is $1 : 2$. **Calculation:** * First, find the difference between the digits: $36 / 9 = 4$. * Let the digits be $a$ and $2a$ (since their ratio is $1:2$). * The difference between these digits is $|2a - a| = a$. * Therefore, $a = 4$. * The digits are $4$ and $2(4) = 8$. (The number is either $48$ or $84$). * The question asks for the difference between the sum of the digits and the difference of the digits. * Sum of the digits = $8 + 4 = 12$. * Difference of the digits = $8 - 4 = 4$. * Difference between the sum and the difference = $12 - 4 = 8$. ### Exam Strategy & Shortcut **Algebraic Shortcut:** Let the digits be $x$ and $y$ (where $x > y$). The sum is $(x + y)$ and the difference is $(x - y)$. The difference between the sum and the difference is: $(x + y) - (x - y) = 2y$. This means the answer will always simply be twice the smaller digit. Since the difference between the digits is $4$ and they are in a $1:2$ ratio, the digits are $4$ and $8$. The smaller digit is $4$. Twice the smaller digit is $2 \times 4 = 8$. ### Common Pitfall Students often compute the digits ($4$ and $8$) and then lose track of the final nested question ("difference between the sum and the difference"). They might incorrectly mark $4$ (the difference of the digits) or $12$ (the sum) by failing to read the final sentence carefully. ### Final Answer **Therefore, the correct answer is 8.**