1. **State the problem:** Solve the equation $$\frac{\frac{1}{2}X^{-1/2}}{2} = \frac{1}{4}$$ for $X$. 2. **Rewrite the equation:** The numerator is $\frac{1}{2}X^{-1/2}$ and the denominator is $2$, so the left side is $$\frac{\frac{1}{2}X^{-1/2}}{2} = \frac{1}{2}X^{-1/2} \times \frac{1}{2} = \frac{1}{4}X^{-1/2}.$$ Thus, the equation becomes $$\frac{1}{4}X^{-1/2} = \frac{1}{4}.$$ 3. **Multiply both sides by 4 to clear the fraction:** $$4 \times \frac{1}{4}X^{-1/2} = 4 \times \frac{1}{4}$$ $$\cancel{4} \times \frac{1}{\cancel{4}} X^{-1/2} = \cancel{4} \times \frac{1}{\cancel{4}}$$ $$X^{-1/2} = 1.$$ 4. **Rewrite the negative exponent:** $$X^{-1/2} = \frac{1}{X^{1/2}} = 1.$$ 5. **Solve for $X^{1/2}$:** $$\frac{1}{X^{1/2}} = 1 \implies X^{1/2} = 1.$$ 6. **Square both sides to solve for $X$:** $$\left(X^{1/2}\right)^2 = 1^2$$ $$X = 1.$$ **Final answer:** $$X = 1.$$