The operation with fractions that performs the operation with both the numerators and the denominators is addition and subtraction.
When you add or subtract fractions, you generally need a common denominator. Once the fractions have a common denominator, you combine the numerators and keep the common denominator.
For example, to add the fractions \( \frac{a}{b} + \frac{c}{d} \), you would find a common denominator (often \( bd \)), and rewrite the fractions as:
\[ \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad + bc}{bd} \]
Here, you add the numerators \( (ad + bc) \) and keep the common denominator \( (bd) \).
Similarly, for subtraction:
\[ \frac{a}{b} - \frac{c}{d} = \frac{ad}{bd} - \frac{bc}{bd} = \frac{ad - bc}{bd} \]
In both cases, the operations are performed on the numerators while also maintaining the denominator.