Two-third of a positive number and $\frac{25}{216}$ of its reciprocal are equal. The number is

### Concept & Formula Equating fractional parts of a variable and its reciprocal allows us to isolate the square of the variable. Let the positive number be $x$. ### Step-by-Step Solution * **Given:** $\frac{2}{3}$ of $x$ equals $\frac{25}{216}$ of $\frac{1}{x}$. * **Calculation:** Write down the equation: $$\frac{2}{3}x = \frac{25}{216} \times \frac{1}{x}$$ Multiply both sides by $x$ to group the variables: $$\frac{2}{3}x^2 = \frac{25}{216}$$ Isolate $x^2$ by multiplying both sides by $\frac{3}{2}$: $$x^2 = \frac{25}{216} \times \frac{3}{2}$$ Simplify the fraction: $$x^2 = \frac{25 \times 1}{72 \times 2}$$ $$x^2 = \frac{25}{144}$$ Take the square root of both sides. Since $x$ is a positive number, we take the principal root: $$x = \sqrt{\frac{25}{144}} = \frac{5}{12}$$ ### Exam Strategy & Shortcut When equating $A \cdot x = B \cdot \frac{1}{x}$, always remember it immediately simplifies to $x^2 = \frac{B}{A}$. Jump straight to: $x^2 = \frac{25}{216} \times \frac{3}{2}$. Simplifying gives $\frac{25}{144}$. Taking the square root gives $\frac{5}{12}$ instantly without writing down intermediate steps. ### Common Pitfall A major pitfall is forgetting to take the square root at the very end. Notice that $\frac{25}{144}$ is explicitly offered as Option (c) to trap students who stop calculating too early. ### Final Answer **Therefore, the correct answer is 5/12.**