To solve the expression \(\left(\frac{x^{3}}{y^{3}}+\frac{1}{2}\right) \div \left(\frac{1}{2} - \frac{x^{2}}{y^{2}}\right)\) given that \(x = \frac{y}{2}\), we will follow a step-by-step approach. ### Step 1: Substitute \(x\) in terms of \(y\) Given: \[ x = \frac{y}{2} \] We can express \(x^2\) and \(x^3\) in terms of \(y\): \[ x^2 = \left(\frac{y}{2}\right)^2 = \frac{y^2}{4} \] \[ x^3 = \left(\frac{y}{2}\right)^3 = \frac{y^3}{8} \] ### Step 2: Substitute into the expression Now, we substitute \(x^2\) and \(x^3\) into the original expression: 1. **Numerator**: \[ \frac{x^3}{y^3} = \frac{\frac{y^3}{8}}{y^3} = \frac{1}{8} \] Therefore, the numerator becomes: \[ \frac{x^3}{y^3} + \frac{1}{2} = \frac{1}{8} + \frac{1}{2} \] To add these fractions, we need a common denominator. The common denominator of 8 and 2 is 8: \[ \frac{1}{2} = \frac{4}{8} \] Thus: \[ \frac{1}{8} + \frac{4}{8} = \frac{5}{8} \] 2. **Denominator**: \[ \frac{x^2}{y^2} = \frac{\frac{y^2}{4}}{y^2} = \frac{1}{4} \] Therefore, the denominator becomes: \[ \frac{1}{2} - \frac{x^2}{y^2} = \frac{1}{2} - \frac{1}{4} \] Again, we need a common denominator, which is 4: \[ \frac{1}{2} = \frac{2}{4} \] Thus: \[ \frac{2}{4} - \frac{1}{4} = \frac{1}{4} \] ### Step 3: Combine the results Now we can substitute the results of the numerator and denominator back into the expression: \[ \left(\frac{5}{8}\right) \div \left(\frac{1}{4}\right) \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{5}{8} \times \frac{4}{1} = \frac{5 \times 4}{8 \times 1} = \frac{20}{8} = \frac{5}{2} \] ### Conclusion The final answer is: \[ \frac{5}{2} \] ### Explanation of Other Options - **Option A: \(\frac{5}{8}\)** - This is the result of the numerator alone before division. It does not account for the division by the denominator. - **Option C: \(\frac{5}{32}\)** - This value is incorrect as it does not reflect any part of the calculations we performed. - **Option D: \(\frac{5}{16}\)** - This is also incorrect and does not match any of the calculations we derived. ### Revision Summary - Substitute \(x\) in terms of \(y\) to simplify the expression. - Calculate \(x^2\) and \(x^3\) in terms of \(y\) to find the necessary fractions. - Use common denominators to add and subtract fractions correctly. - Remember that dividing by a fraction is equivalent to multiplying by its reciprocal. The correct answer is **B: \(\frac{5}{2}\)**.