# 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.

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## Properties of Fractions

Fractions are real numbers and they therefore have the same properties of commutatitivity, associativity and distributivity as the 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 that when using the above distributivity from right to left, we say that we are factoring out (writing as a product) the fraction $\dfrac{a}{b}$
$\left(\color{red}{\dfrac{a}{b}} \times \dfrac{c}{d} \right) + \left( \color{red}{\dfrac{a}{b}} \times \dfrac{e}{f} \right ) = \color{red}{\dfrac{a}{b}} \left(\dfrac{c}{d} + \dfrac{e}{f} \right)$

The above properties and other rules of fractions are be used to simplify and factor expressions including fractions.