Can the ratio of a parent's age to his or her child's age ever be exactly 10 : 9?

Yes, if the parents had the child at a very young age. ex. Parents: 80 years old Child: 72 years old

Mathematically, yes.

Realistically, no.

The difference in their ages is the parent's age when the child was born.

Let's say that the youngest possible is 12.

Then the ratio would be 10:9 when the parent is 120 and the child is 108.

Not likely.