No. of words using letters of the word 'INTEGER' $= \large\frac{7!}{2}$
No. of words in which I and N are together $=\large\frac{6!}{2}$
$\therefore$ No. of words in which I and N are never together $=\large\frac{7!}{2}$$-\large\frac{6!}{2}$
$\Rightarrow\:x=\large\frac{6.6!}{2}$
No. of words which begin with I and end with R=$\large\frac{5!}{2}$
$\Rightarrow\:y=\large\frac{5!}{2}$
$\Rightarrow\:\large\frac{x}{y}$$=36$