5.5: Complex Fractions

Learning Objectives

• be able to distinguish between simple and complex fractions
• be able to convert a complex fraction to a simple fraction

Simple Fractions and Complex Fractions

Simple Fraction
A simple fraction is any fraction in which the numerator is any whole number and the denominator is any nonzero whole number. Some examples are the following:

\(\dfrac{1}{2}, \dfrac{4}{3}, \dfrac{763}{1,000}\)

Complex Fraction
A complex fraction is any fraction in which the numerator and/or the denomina­tor is a fraction; it is a fraction of fractions. Some examples of complex fractions are the following:

\(\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}}, \dfrac{\dfrac{1}{3}}{2}, \dfrac{6}{\dfrac{9}{10}}, \dfrac{4 + \dfrac{3}{8}}{7 - \dfrac{5}{6}}\)

Converting Complex Fractions to Simple Fractions

The goal here is to convert a complex fraction to a simple fraction. We can do so by employing the methods of adding, subtracting, multiplying, and dividing fractions. Recall from [link] that a fraction bar serves as a grouping symbol separating the fractional quantity into two individual groups. We proceed in simplifying a complex fraction to a simple fraction by simplifying the numerator and the denom­inator of the complex fraction separately. We will simplify the numerator and denominator completely before removing the fraction bar by dividing. This tech­nique is illustrated in problems 3, 4, 5, and 6 of Sample Set A.

Sample Set A

Convert each of the following complex fractions to a simple fraction.

\(\dfrac{\dfrac{3}{8}}{\dfrac{15}{16}}\)

Convert this complex fraction to a simple fraction by performing the indicated division.

Solution

\(\begin{array} {rcll} {\dfrac{\dfrac{3}{8}}{\dfrac{15}{16}}} & = & {\dfrac{3}{8} \div \dfrac{15}{16}} & {\text{The divisor is } \dfrac{15}{16}. \text{Invert } \dfrac{15}{16} \text{ and multiply.}} \\ {} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^2} \\ {\cancel{16}} \end{array}}{\begin{array} {c} {\cancel{15}} \\ {^5} \end{array}} = \dfrac{1 \cdot 2}{1 \cdot 5} = \dfrac{2}{5}} & {} \end{array}\)

Sample Set A

\(\dfrac{\dfrac{4}{9}}{6}\) Write 6 as \(\dfrac{6}{1}\) and divide.

Solution

\(\begin{array} {rcl} {\dfrac{\dfrac{4}{9}}{\dfrac{6}{1}}} & = & {\dfrac{4}{9} \div \dfrac{6}{1}} \\ {} & = & {\dfrac{\begin{array} {c} {^2} \\ {\cancel{4}} \end{array}}{9} \cdot \dfrac{1}{\begin{array} {c} {\cancel{6}} \\ {^3} \end{array}} = \dfrac{2 \cdot 1}{9 \cdot 3} = \dfrac{2}{27}} \end{array}\)

Sample Set A

\(\dfrac{5 + \dfrac{3}{4}}{46}\) Simplify the numerator.

Solution

\(\dfrac{\dfrac{4 \cdot 5 + 3}{4}}{46} = \dfrac{\dfrac{20 + 3}{4}}{46} = \dfrac{\dfrac{23}{4}}{46}\) Write 46 as \(\dfrac{46}{1}\).

\(\begin{array} {rcl} {\dfrac{\dfrac{23}{4}}{\dfrac{46}{1}}} & = & {\dfrac{23}{4} \div \dfrac{46}{1}} \\ {} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{23}} \end{array}}{4} \cdot \dfrac{1}{\begin{array} {c} {\cancel{46}} \\ {^2} \end{array}} = \dfrac{1 \cdot 1}{4 \cdot 2} = \dfrac{1}{8}} \end{array}\)

Sample Set A

\(\dfrac{\dfrac{1}{4} + \dfrac{3}{8}}{\dfrac{1}{2} + \dfrac{13}{24}} = \dfrac{\dfrac{2}{8} + \dfrac{3}{8}}{\dfrac{12}{24} + \dfrac{13}{24}} = \dfrac{\dfrac{2 + 3}{8}}{\dfrac{12 + 13}{24}} = \dfrac{\dfrac{5}{8}}{\dfrac{25}{24}} = \dfrac{5}{8} \div \dfrac{25}{24}\)

\(\dfrac{5}{8} \div \dfrac{25}{24} = \dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^3} \\ {\cancel{24}} \end{array}}{\begin{array} {c} {\cancel{25}} \\ {^5} \end{array}} = \dfrac{1 \cdot 3}{1 \cdot 5} = \dfrac{3}{5}\)

Sample Set A

\(\begin{array} {rcl} {\dfrac{4 + \dfrac{5}{6}}{7 - \dfrac{1}{3}} = \dfrac{\dfrac{4 \cdot 6 + 5}{6}}{\dfrac{7 \cdot 3 - 1}{3}} = \dfrac{\dfrac{29}{6}}{\dfrac{20}{3}}} & = & {\dfrac{29}{6} \div \dfrac{20}{3}} \\ {} & = & {\dfrac{29}{\begin{array} {c} {\cancel{6}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{20} = \dfrac{29}{40}} \end{array}\)

Sample Set A

\(\dfrac{11 + \dfrac{3}{10}}{4 \dfrac{4}{5}} = \dfrac{\dfrac{11 \cdot 10 + 3}{10}}{\dfrac{4 \cdot 5 + 4}{5}} = \dfrac{\dfrac{110 + 3}{10}}{\dfrac{20 + 4}{5}} = \dfrac{\dfrac{113}{10}}{\dfrac{24}{5}} = \dfrac{113}{10} \div \dfrac{24}{5}\)

\(\dfrac{113}{10} \div \dfrac{24}{5} = \dfrac{113}{\begin{array} {c} {\cancel{10}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{24} = \dfrac{113 \cdot 1}{2 \cdot 24} = \dfrac{113}{48} = 2 \dfrac{17}{48}\)

Practice Set A

Convert each of the following complex fractions to a simple fraction.

\(\dfrac{\dfrac{4}{9}}{\dfrac{8}{15}}\)

Answer

\(\dfrac{5}{6}\)

Practice Set A

\(\dfrac{\dfrac{7}{10}}{28}\)

Answer

\(\dfrac{1}{40}\)

Practice Set A

\(\dfrac{5 + \dfrac{2}{5}}{3 + \dfrac{3}{5}}\)

Answer

\(\dfrac{3}{2}\)

Practice Set A

\(\dfrac{\dfrac{1}{8} + \dfrac{7}{8}}{6 - \dfrac{3}{10}}\)

Answer

\(\dfrac{10}{57}\)

Practice Set A

\(\dfrac{\dfrac{1}{6} + \dfrac{5}{8}}{\dfrac{5}{9} - \dfrac{1}{4}}\)

Answer

\(2 \dfrac{13}{22}\)

Practice Set A

\(\dfrac{16 - 10 \dfrac{2}{3}}{11 \dfrac{5}{6} - 7 \dfrac{7}{6}}\)

Answer

\(1 \dfrac{5}{11}\)

Exercises

Simplify each fraction.

Exercise \(\PageIndex{1}\)

\(\dfrac{\dfrac{3}{5}}{\dfrac{9}{15}}\)

Answer

1

Exercise \(\PageIndex{2}\)

\(\dfrac{\dfrac{1}{3}}{\dfrac{1}{9}}\)

Exercise \(\PageIndex{3}\)

\(\dfrac{\dfrac{1}{4}}{\dfrac{5}{12}}\)

Answer

\(\dfrac{3}{5}\)

Exercise \(\PageIndex{4}\)

\(\dfrac{\dfrac{8}{9}}{\dfrac{4}{15}}\)

Exercise \(\PageIndex{5}\)

\(\dfrac{6 + \dfrac{1}{4}}{11 + \dfrac{1}{4}}\)

Answer

\(\dfrac{5}{9}\)

Exercise \(\PageIndex{6}\)

\(\dfrac{2 + \dfrac{1}{2}}{7 + \dfrac{1}{2}}\)

Exercise \(\PageIndex{7}\)

\(\dfrac{5 + \dfrac{1}{3}}{2 + \dfrac{2}{15}}\)

Answer

\(\dfrac{5}{2}\)

Exercise \(\PageIndex{8}\)

\(\dfrac{9 + \dfrac{1}{2}}{1 + \dfrac{8}{11}}\)

Exercise \(\PageIndex{9}\)

\(\dfrac{4 + \dfrac{10}{13}}{\dfrac{12}{39}}\)

Answer

\(\dfrac{31}{2}\)

Exercise \(\PageIndex{10}\)

\(\dfrac{\dfrac{1}{3} + \dfrac{2}{7}}{\dfrac{26}{21}}\)

Exercise \(\PageIndex{11}\)

\(\dfrac{\dfrac{5}{6} - \dfrac{1}{4}}{\dfrac{1}{12}}\)

Answer

7

Exercise \(\PageIndex{12}\)

\(\dfrac{\dfrac{3}{10} + \dfrac{4}{12}}{\dfrac{19}{90}}\)

Exercise \(\PageIndex{13}\)

\(\dfrac{\dfrac{9}{16} + \dfrac{7}{3}}{\dfrac{139}{48}}\)

Answer

1

Exercise \(\PageIndex{14}\)

\(\dfrac{\dfrac{1}{288}}{\dfrac{8}{9} - \dfrac{3}{16}}\)

Exercise \(\PageIndex{15}\)

\(\dfrac{\dfrac{27}{429}}{\dfrac{5}{11} - \dfrac{1}{13}}\)

Answer

\(\dfrac{1}{6}\)

Exercise \(\PageIndex{16}\)

\(\dfrac{\dfrac{1}{3} + \dfrac{2}{5}}{\dfrac{3}{5} + \dfrac{17}{45}}\)

Exercise \(\PageIndex{17}\)

\(\dfrac{\dfrac{9}{70} + \dfrac{5}{42}}{\dfrac{13}{30} - \dfrac{1}{21}}\)

Answer

\(\dfrac{52}{81}\)

Exercise \(\PageIndex{18}\)

\(\dfrac{\dfrac{1}{16} + \dfrac{1}{14}}{\dfrac{2}{3} - \dfrac{13}{60}}\)

Exercise \(\PageIndex{19}\)

\(\dfrac{\dfrac{3}{20} + \dfrac{11}{12}}{\dfrac{19}{7} - 1 \dfrac{11}{35}}\)

Answer

\(\dfrac{16}{21}\)

Exercise \(\PageIndex{20}\)

\(\dfrac{2 \dfrac{2}{3} - 1 \dfrac{1}{2}}{\dfrac{1}{4} + 1 \dfrac{1}{16}}\)

Exercise \(\PageIndex{21}\)

\(\dfrac{3 \dfrac{1}{5} + 3 \dfrac{1}{3}}{\dfrac{6}{5} - \dfrac{15}{63}}\)

Answer

\(\dfrac{686}{101}\)

Exercise \(\PageIndex{22}\)

\(\dfrac{\dfrac{1 \dfrac{1}{2} + 15}{5 \dfrac{1}{4} - 3 \dfrac{5}{12}}}{\dfrac{8 \dfrac{1}{3} - 4 \dfrac{1}{2}}{11 \dfrac{2}{3} - 5 \dfrac{11}{12}}}\)

Exercise \(\PageIndex{23}\)

\(\dfrac{\dfrac{5 \dfrac{3}{4} + 3 \dfrac{1}{5}}{2 \dfrac{1}{5} + 15 \dfrac{7}{10}}}{\dfrac{9 \dfrac{1}{2} - 4 \dfrac{1}{6}}{\dfrac{1}{8} + 2 \dfrac{1}{120}}}\)

Answer

\(\dfrac{1}{3}\)

Exercises for Review

Exercise \(\PageIndex{24}\)

Find the prime factorization of 882.

Exercise \(\PageIndex{25}\)

Convert \(\dfrac{62}{7}\) to a mixed number.

Answer

\(8 \dfrac{6}{7}\)

Exercise \(\PageIndex{26}\)

Reduce \(\dfrac{114}{342}\) to lowest terms.

Exercise \(\PageIndex{27}\)

Find the value of \(6 \dfrac{3}{8} - 4 \dfrac{5}{6}\)

Answer

\(1 \dfrac{13}{24}\) or \(\dfrac{37{24}\)

Exercise \(\PageIndex{28}\)

Arrange from smallest to largest: \(\dfrac{1}{2}\), \(\dfrac{3}{5}\), \(\dfrac{4}{7}\).