If $x : y = 3 : 1$, then $x^3 - y^3 : x^3 + y^3 = $x$

### Concept & Cubes in Ratios When dealing with a ratio of sums and differences of cubes, check for homogeneity. Since all terms are of degree $3$, direct substitution of the ratio values is valid and efficient. ### Step-by-Step Solution - **Given:** $x : y = 3 : 1$. - **Substitute:** Let $x = 3$ and $y = 1$. - **Expression:** $(x^3 - y^3) : (x^3 + y^3)$ - **Calculate terms:** - $x^3 = 3^3 = 27$ - $y^3 = 1^3 = 1$ - **Numerator (Difference):** $27 - 1 = 26$ - **Denominator (Sum):** $27 + 1 = 28$ - **Simplify the ratio:** $26 : 28 = 13 : 14$. ### Exam Strategy & Shortcut Calculate the cubes mentally ($27$ and $1$). The required ratio is simply their difference divided by their sum: $(27 - 1) / (27 + 1) = 26 / 28 = 13 / 14$. ### Common Pitfall A frequent error is placing the sum in the numerator and the difference in the denominator by not reading the question order carefully, which would lead to the trap option $14 : 13$. ### Final Answer Therefore, the correct answer is **$13 : 14$**.