Properties of Fractions
The properties of commutativity, associatitivy and distributivity of fractions are presented. These properties along with other rules of fractions may be used to simplify and factor expressions including fractions.
Do NOT use the calculator to answer the questions.
Properties of Fractions
Fractions are real numbers and they therefore have the same properties of commutatitivity, associativity and distributivity as the real numbers.

Commutativity of Addition of Fractions
\( \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{c}{d} + \dfrac{a}{b}\)

Commutativity of Multiplication of Fractions
\( \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{c}{d} \times \dfrac{a}{b}\)

Associativity of Addition of Fractions
\( \dfrac{a}{b} + \dfrac{c}{d} + \dfrac{e}{f} = \left(\dfrac{a}{b} + \dfrac{c}{d} \right) + \dfrac{e}{f} = \dfrac{a}{b} + \left(\dfrac{c}{d} + \dfrac{e}{f}\right) \)

Associativity of Multiplication of Fractions
\( \dfrac{a}{b} \times \dfrac{c}{d} \times \dfrac{e}{f} = \left(\dfrac{a}{b} \times \dfrac{c}{d} \right) \times \dfrac{e}{f} = \dfrac{a}{b} \times \left(\dfrac{c}{d} \times \dfrac{e}{f}\right) \)

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 that when using the above distributivity from right to left, we say that we are factoring out (writing as a product) the fraction \( \dfrac{a}{b} \)
\( \left(\color{red}{\dfrac{a}{b}} \times \dfrac{c}{d} \right) + \left( \color{red}{\dfrac{a}{b}} \times \dfrac{e}{f} \right ) = \color{red}{\dfrac{a}{b}} \left(\dfrac{c}{d} + \dfrac{e}{f} \right) \)
More References and Links
fractionssimplify fractions
factor fractions
rules of fractions