Question

What is the rule in dividing positive and negative rational numbers?

Answer 1

If the numbers have the same sign (both positive or both negative), then the answer is positive.
If the numbers have opposite signs (one is positive and the other is negative), then the answer is negative.

One way of explaining this:

The rule for dividing is the as same rule for multiplying positive and negative numbers.
The rule is the same because division is multiplying by the reciprocal.

The reciprocal of a positive number is positive and the reciprocal of a negative number is negative.

The reciprocal of `p/q` is `1/(p/q)` which is the same as `q/p`.

The reciprocal of a number is the number you have to multiply by to get `1`.

Not every number has a reciprocal. `0` does not have a reciprocal (because `0` times any number is `0`).