Reverse regression slope from the forward slope

Two variables have $\operatorname{Var}(X)=10$ and $\operatorname{Var}(Y)=20$. An ordinary least squares fit of $Y$ on $X$ (with intercept), $y\approx\alpha_x+\beta_x x$, returns slope $\beta_x=1$. If you instead regress $X$ on $Y$, $x\approx\alpha_y+\beta_y y$, what slope $\beta_y$ do you get?

$1/2$. OLS slopes are $\beta_x=\operatorname{Cov}(X,Y)/\operatorname{Var}(X)$ and $\beta_y=\operatorname{Cov}(X,Y)/\operatorname{Var}(Y)$. The forward fit gives $\operatorname{Cov}=10$, so $\beta_y=10/20$. Note $\beta_x\beta_y=r^2$, so the reverse slope is not $1/\beta_x$ unless $|r|=1$.