Mean-to-variance ratio for a nested uniform draw

First $X$ is drawn uniformly from $(0,1)$. Given $X=x$, a second number $Y$ is drawn uniformly from $(x,1)$. Compute $\mathbb{E}[Y]/\operatorname{Var}(Y)$ as a fraction in lowest terms.

$\tfrac{108}{7}$. Since $\mathbb{E}[Y\mid X]=\tfrac{1+X}{2}$, we get $\mathbb{E}[Y]=\tfrac34$. By the law of total variance, $\operatorname{Var}(Y)=\operatorname{Var}\big(\tfrac{1+X}{2}\big)+\mathbb{E}\big[\tfrac{(1-X)^2}{12}\big]=\tfrac1{48}+\tfrac1{36}=\tfrac{7}{144}$, so the ratio is $\tfrac34\cdot\tfrac{144}{7}=\tfrac{108}{7}$.