Which number can replace both the question marks in the equation $\frac{4 \frac{1}{2}}{x} = \frac{x}{32}$.

Concept & Formula The problem requires solving a simple algebraic proportion involving fractions. Cross-multiplication is the key principle: if $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$. Step-by-Step Solution * Given equation with the unknown replaced by $x$: $\frac{4 \frac{1}{2}}{x} = \frac{x}{32}$ * Convert the mixed fraction $4 \frac{1}{2}$ to an improper fraction: $\frac{9}{2}$. * The equation becomes: $\frac{\frac{9}{2}}{x} = \frac{x}{32}$ * Cross-multiply to solve for $x$: $x \times x = \frac{9}{2} \times 32$ * Simplify the right side: $x^2 = 9 \times 16$ * $x^2 = 144$ * Take the square root of both sides: $x = 12$ Exam Strategy & Shortcut Instead of fully multiplying $9 \times 16 = 144$, leave it as factors: $x^2 = 9 \times 16$. Apply the square root directly to the perfect squares: $x = \sqrt{9} \times \sqrt{16} = 3 \times 4 = 12$. This saves calculation time and avoids arithmetic errors on larger numbers. Common Pitfall Students often miscalculate the mixed fraction conversion, writing $4 \frac{1}{2}$ as $\frac{5}{2}$ by adding instead of multiplying. Always remember: $(\text{Whole} \times \text{Denominator}) + \text{Numerator}$. Final Answer Therefore, the correct answer is None of these.