Welcome to the forum.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
You haven't proven the closure of the rational numbers under addition nor multiplication: you've only shown some examples. You'd need to show that the sum and product of two rational numbers a/b and c/d is also rational.