For what value of $x$ the statement $\left( \frac{x}{15} \right) \left( \frac{x}{135} \right) = 1$ is true?

Concept & Logic When multiplying fractions involving the same unknown variable, combine the numerators and denominators to form a quadratic equation, then isolate the variable. Step-by-Step Solution * **Given:** $\left( \frac{x}{15} \right) \left( \frac{x}{135} \right) = 1$ * **Calculation:** Multiply the fractions on the left side: $\frac{x^2}{15 \times 135} = 1$ * Cross-multiply to isolate $x^2$: $x^2 = 15 \times 135$ * To avoid large multiplication, factor $135$ into smaller, manageable pieces to easily find the square root: $x^2 = 15 \times (15 \times 9)$ $x^2 = 15^2 \times 3^2$ * Take the square root of both sides: $x = 15 \times 3$ * $x = 45$ Exam Strategy & Shortcut Instead of multiplying $15 \times 135$ to get $2025$ and then struggling to find its square root, break $135$ down immediately into $15 \times 9$. Taking the square root of $(15 \times 15 \times 9)$ visually reduces straight to $15 \times 3 = 45$. Common Pitfall Multiplying the denominators completely ($15 \times 135 = 2025$) and then spending excessive time trying to extract the square root of a large number without a calculator. Final Answer **Therefore, the correct answer is 45.**