The properties of commutativity, associativity, and distributivity of fractions are presented. These properties along with other rules of fractions may be used to simplify and factor expressions including fractions.
Note: Do NOT use a calculator to answer the questions.
Properties of Fractions
Fractions are real numbers and therefore have the same properties of commutativity, associativity, and distributivity as real numbers.
1. Commutativity of Addition of Fractions
\[ \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{c}{d} + \dfrac{a}{b} \]2. Commutativity of Multiplication of Fractions
\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{c}{d} \times \dfrac{a}{b} \]3. Associativity of Addition of Fractions
\[ \dfrac{a}{b} + \dfrac{c}{d} + \dfrac{e}{f} = \left(\dfrac{a}{b} + \dfrac{c}{d} \right) + \dfrac{e}{f} = \dfrac{a}{b} + \left(\dfrac{c}{d} + \dfrac{e}{f}\right) \]4. Associativity of Multiplication of Fractions
\[ \dfrac{a}{b} \times \dfrac{c}{d} \times \dfrac{e}{f} = \left(\dfrac{a}{b} \times \dfrac{c}{d} \right) \times \dfrac{e}{f} = \dfrac{a}{b} \times \left(\dfrac{c}{d} \times \dfrac{e}{f}\right) \]5. Distributivity of Multiplication Over Addition of Fractions
\[ \color{red}{\dfrac{a}{b}} \left(\dfrac{c}{d} + \dfrac{e}{f} \right) = \left(\color{red}{\dfrac{a}{b}} \times \dfrac{c}{d} \right) + \left( \color{red}{\dfrac{a}{b}} \times \dfrac{e}{f} \right ) \]NOTE: When using the above distributivity from right to left, we say that we are factoring out (writing as a product) the fraction \( \dfrac{a}{b} \):
\[ \left(\color{red}{\dfrac{a}{b}} \times \dfrac{c}{d} \right) + \left( \color{red}{\dfrac{a}{b}} \times \dfrac{e}{f} \right ) = \color{red}{\dfrac{a}{b}} \left(\dfrac{c}{d} + \dfrac{e}{f} \right) \]The above properties and other rules of fractions are used to simplify and factor expressions including fractions.