With reference to the origin $O$, the points $A, B, C$ and $D$ are such that $\overrightarrow{O A}=\mathbf{a}, \overrightarrow{O B}=\mathbf{b}, \overrightarrow{O C}=2 \mathbf{a}+4 \mathbf{b}$ and $\overrightarrow{O D}=\mathbf{b}+5 \mathbf{a}$. The lines $B D$ and $A C$ cross at $X$ (see diagram).

(i) Express $\overrightarrow{O X}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.


The point $Y$ lies on $C D$ and is such that the points $O, X$ and $Y$ are collinear.

(ii) Express $\overrightarrow{O Y}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ and find the ratio $O X: O Y$.