The sum of the digits of a two-digit number is $1/5$ of the difference between the number and the number obtained by interchanging the positions of the digits. What is definitely the difference between the digits of that number?

### Concept & Formula This problem bridges the sum of the digits with the algebraic expression for the difference between a number and its reversed form. $$ \text{Sum of digits} = (x + y) $$ $$ \text{Difference of reversed numbers} = 9(x - y) $$ ### Step-by-Step Solution **Given:** * $(x + y) = \frac{1}{5} \times \text{Difference of the numbers}$ **Calculation:** * We know the difference between a two-digit number ($10x + y$) and its reverse ($10y + x$) is $9(x - y)$. * Substitute this into the given condition: $(x + y) = \frac{1}{5} \times 9(x - y)$. * Multiply both sides by $5$ to clear the fraction: $5(x + y) = 9(x - y)$. * Expand the terms: $5x + 5y = 9x - 9y$. * Rearrange to group variables: $5y + 9y = 9x - 5x$. * Simplify: $14y = 4x$, which reduces to $7y = 2x$. * This gives us a ratio: $x / y = 7 / 2$. * Since $x$ and $y$ must be single-digit integers (from $0$ to $9$) and $x$ cannot be $0$ for a two-digit number, the only valid integer solution is $x = 7$ and $y = 2$. * The original number is $72$ and its reverse is $27$. * The question asks for the difference between the digits: $x - y = 7 - 2 = 5$. ### Exam Strategy & Shortcut Translate the word problem directly into the simplified ratio: $5(x+y) = 9(x-y)$. Rearranging gives $7y = 2x$. The moment you see a ratio between single digits like $x:y = 7:2$, you know the digits must be exactly $7$ and $2$. The difference is instantly $5$. ### Common Pitfall Students might arrive at the equation $7y = 2x$ and incorrectly conclude that the data is inadequate because it's a single equation with two variables. However, the hidden constraint that $x$ and $y$ must be single-digit integers ($1$ through $9$) forces a unique solution. ### Final Answer **Therefore, the correct answer is 5.**