Robust Approximate Variance of a Ratio via the Delta Method

Let $X$ and $Y$ be independent random variables with $E[X] = 10$, $\operatorname{Var}(X) = 4$, $E[Y] = 5$, and $\operatorname{Var}(Y) = 1$. Using the delta method (first-order Taylor expansion), derive an approximation for $\operatorname{Var}(X/Y)$ and evaluate it numerically.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.