You do not "solve" rational numbers. Rational numbers are not a puzzle nor a question nor an equation, so there is nothing to solve.
The question is like asking how do you solve a person's name.
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The greatest common factor is used to reduce fractions.
To reduce fractions
When you need to reduce fractions.
One common application of greatest common factors is to simplify fractions. Note that you don't necessarily need the GREATEST common factor; you can simplify by dividing both numbers by any common factor, and then continue looking for additional factors.
When doing fractions it is the greatest common factor (GCF) and the least common multiple (LCM). You want the GCF when you are reducing fractions to their simplest form. When changing the denominators to a common one, you want the LCM.