Question

Will division of a rational number by another rational number always give a rational number?

Answer 1

The addition operation is closed under the set of natural numbers. That means, on adding two natural numbers, we always get another natural number.
But, the subtraction operation is not closed under the set of natural numbers. Why? Because, 2−5 is not defined when we limit ourselves to natural numbers.
Your question can be rephrased as:
Is the division operation is closed under the set of rational numbers?
No.
p/q is a rational number as long as q≠0
So, a rational number divided by another rational number is not always a rational number.