Let P and Q be two rational numbers
P = a/b
Q = c/d
where a,b,c,d are integers. Also, b and d cannot be zero.
When we multiply P and Q, we get
P*Q = (a/b)*(c/d)
P*Q = (a*c)/(b*d)
Showing that we multiply the numerators together, and the denominators are multiplied together.
The expression a*c is the result of multiplying integers a and c
The expression b*d is the result of multiplying integers b and d
Therefore, multiplying rational numbers involves multiplying integers (we just do two sets of multiplications).
After multiplying P and Q, it's often a good idea to reduce the fraction as much as possible.