In a two-digit positive number, the digit in the unit's place is equal to the square of the digit in ten's place, and the difference between the number and the number obtained by interchanging the digits is $54$. What is $40\%$ of the original number?

### Concept & Logic This problem combines non-linear digit relationships (squares) with standard linear reversal properties. $$ \text{Units digit } y = x^2 $$ ### Step-by-Step Solution **Given:** * Unit's digit ($y$) is the square of the ten's digit ($x$): $y = x^2$. * The absolute difference between the number and its reverse is $54$. **Calculation:** * From the reversal rule, the difference between the digits is $54 / 9 = 6$. So, $|x - y| = 6$. * Substitute $y = x^2$ into the equation. We have two cases for the absolute difference: **Case 1:** $x - x^2 = 6 \Rightarrow x^2 - x + 6 = 0$. This quadratic has no real roots. **Case 2:** $x^2 - x = 6 \Rightarrow x^2 - x - 6 = 0$. * Factor the quadratic equation: $(x - 3)(x + 2) = 0$. * Since $x$ must be a positive single digit, $x = 3$. * Find the unit's digit: $y = 3^2 = 9$. * The original number is $39$. * The question asks for $40\%$ of the original number. * $40\% \text{ of } 39 = 0.4 \times 39 = 15.6$. ### Exam Strategy & Shortcut **Logical Deduction:** The unit's digit is a perfect square of the ten's digit. The possible digits ($x, y$) where $y = x^2$ are $(1, 1)$, $(2, 4)$, and $(3, 9)$. Check the differences between these digits: * For $(1, 1)$, difference is $0$. * For $(2, 4)$, difference is $2$. * For $(3, 9)$, difference is $6$. We need a difference of digits equal to $54 / 9 = 6$. Therefore, the digits must be $3$ and $9$. The number is $39$. Calculate $40\%$ of $39$ mentally: $10\%$ is $3.9$, so $40\%$ is $3.9 \times 4 = 15.6$. ### Common Pitfall A major pitfall is ignoring the final part of the question. After correctly identifying the original number as $39$, students often immediately look for $39$ in the options and select option (d). The question specifically asks for $40\%$ of that original number. ### Final Answer **Therefore, the correct answer is 15.6.**