Summary: Multiplying and Dividing Fractions

Key Concepts

  • Equivalent Fractions Property
    • If [latex]a,b,c[/latex] are numbers where [latex]b\ne 0[/latex] , [latex]c\ne 0[/latex] , then [latex]\Large\frac{a}{b}\normalsize=\Large\frac{a\cdot c}{b\cdot c}[/latex] and [latex]\Large\frac{a\cdot c}{b\cdot c}\normalsize=\Large\frac{a}{b}[/latex] .
  • Simplify a fraction.
    1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
    2. Simplify, using the equivalent fractions property, by removing common factors.
    3. Multiply any remaining factors.
  • Fraction Multiplication
    • If [latex]a,b,c[/latex], and [latex]d[/latex] are numbers where [latex]b\ne 0[/latex] and [latex]d\ne 0[/latex] , then [latex]\Large\frac{a}{b}\cdot \frac{c}{d}\normalsize=\Large\frac{ac}{bd}[/latex] .
  • Reciprocal
    • A number and its reciprocal have a product of [latex]1[/latex] . [latex]\Large\frac{a}{b}\cdot \frac{b}{a}\normalsize=1[/latex]
      Opposite Absolute Value Reciprocal
      has opposite sign is never negative has same sign, fraction inverts
  • Fraction Division
    • If [latex]a,b,c[/latex], and [latex]d[/latex] are numbers where [latex]b\ne 0[/latex] , [latex]c\ne 0[/latex] and [latex]d\ne 0[/latex] , then[latex]\Large\frac{a}{b}\normalsize+\Large\frac{c}{d}\normalsize=\Large\frac{a}{b}\cdot\Large\frac{d}{c}[/latex]
    • To divide fractions, multiply the first fraction by the reciprocal of the second.


The reciprocal of the fraction [latex]\Large\frac{a}{b}[/latex] is [latex]\Large\frac{b}{a}[/latex] where [latex]a\ne 0[/latex] and [latex]b\ne 0[/latex] .
simplified fraction
A fraction is considered simplified if there are no common factors in the numerator and denominator.