5.5: Complex Fractions
- 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.
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
\(\begin{array} {rcll} {\dfrac{\dfrac{3}{8}}{\dfrac{15}{16}}} & = & {\dfrac{3}{8} \div \dfrac{15}{16}} & {\text{The divisor is } \dfrac{15}{16}. \text{Invert } \dfrac{15}{16} \text{ and multiply.}} \\ {} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^2} \\ {\cancel{16}} \end{array}}{\begin{array} {c} {\cancel{15}} \\ {^5} \end{array}} = \dfrac{1 \cdot 2}{1 \cdot 5} = \dfrac{2}{5}} & {} \end{array}\)
\(\dfrac{\dfrac{4}{9}}{6}\) Write 6 as \(\dfrac{6}{1}\) and divide.
Solution
\(\begin{array} {rcl} {\dfrac{\dfrac{4}{9}}{\dfrac{6}{1}}} & = & {\dfrac{4}{9} \div \dfrac{6}{1}} \\ {} & = & {\dfrac{\begin{array} {c} {^2} \\ {\cancel{4}} \end{array}}{9} \cdot \dfrac{1}{\begin{array} {c} {\cancel{6}} \\ {^3} \end{array}} = \dfrac{2 \cdot 1}{9 \cdot 3} = \dfrac{2}{27}} \end{array}\)
\(\dfrac{5 + \dfrac{3}{4}}{46}\) 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 \(\dfrac{46}{1}\).
\(\begin{array} {rcl} {\dfrac{\dfrac{23}{4}}{\dfrac{46}{1}}} & = & {\dfrac{23}{4} \div \dfrac{46}{1}} \\ {} & = & {\dfrac{\begin{array} {c} {^1} \\ {\cancel{23}} \end{array}}{4} \cdot \dfrac{1}{\begin{array} {c} {\cancel{46}} \\ {^2} \end{array}} = \dfrac{1 \cdot 1}{4 \cdot 2} = \dfrac{1}{8}} \end{array}\)
\(\dfrac{\dfrac{1}{4} + \dfrac{3}{8}}{\dfrac{1}{2} + \dfrac{13}{24}} = \dfrac{\dfrac{2}{8} + \dfrac{3}{8}}{\dfrac{12}{24} + \dfrac{13}{24}} = \dfrac{\dfrac{2 + 3}{8}}{\dfrac{12 + 13}{24}} = \dfrac{\dfrac{5}{8}}{\dfrac{25}{24}} = \dfrac{5}{8} \div \dfrac{25}{24}\)
\(\dfrac{5}{8} \div \dfrac{25}{24} = \dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^3} \\ {\cancel{24}} \end{array}}{\begin{array} {c} {\cancel{25}} \\ {^5} \end{array}} = \dfrac{1 \cdot 3}{1 \cdot 5} = \dfrac{3}{5}\)
\(\begin{array} {rcl} {\dfrac{4 + \dfrac{5}{6}}{7 - \dfrac{1}{3}} = \dfrac{\dfrac{4 \cdot 6 + 5}{6}}{\dfrac{7 \cdot 3 - 1}{3}} = \dfrac{\dfrac{29}{6}}{\dfrac{20}{3}}} & = & {\dfrac{29}{6} \div \dfrac{20}{3}} \\ {} & = & {\dfrac{29}{\begin{array} {c} {\cancel{6}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{20} = \dfrac{29}{40}} \end{array}\)
\(\dfrac{11 + \dfrac{3}{10}}{4 \dfrac{4}{5}} = \dfrac{\dfrac{11 \cdot 10 + 3}{10}}{\dfrac{4 \cdot 5 + 4}{5}} = \dfrac{\dfrac{110 + 3}{10}}{\dfrac{20 + 4}{5}} = \dfrac{\dfrac{113}{10}}{\dfrac{24}{5}} = \dfrac{113}{10} \div \dfrac{24}{5}\)
\(\dfrac{113}{10} \div \dfrac{24}{5} = \dfrac{113}{\begin{array} {c} {\cancel{10}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{24} = \dfrac{113 \cdot 1}{2 \cdot 24} = \dfrac{113}{48} = 2 \dfrac{17}{48}\)
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: \(\dfrac{1}{2}\), \(\dfrac{3}{5}\), \(\dfrac{4}{7}\).