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$\frac{1}{2}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{4}{3}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{\text{763}}{\mathrm{1,}\text{000}}$
$\frac{\frac{3}{4}}{\frac{5}{6}}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{\frac{1}{3}}{2}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{6}{\frac{9}{\text{10}}}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{4+\frac{3}{8}}{7-\frac{5}{6}}$
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 [link] .
Convert each of the following complex fractions to a simple fraction.
$\frac{\frac{3}{8}}{\frac{\text{15}}{\text{16}}}$
Convert this complex fraction to a simple fraction by performing the indicated division.
$\begin{array}{cccc}\hfill \frac{\frac{3}{8}}{\frac{\text{15}}{\text{16}}}& =& \frac{3}{8}\xf7\frac{\text{15}}{\text{16}}\hfill & \text{The divisor is}\frac{\text{15}}{\text{16}}.\text{Invert}\frac{\text{15}}{\text{16}}\text{and multiply.}\hfill \\ & =& \frac{\stackrel{1}{\overline{)3}}}{\underset{1}{\overline{)8}}}\cdot \frac{\stackrel{2}{\overline{)16}}}{\underset{5}{\overline{)15}}}=\frac{1\cdot 2}{1\cdot 5}=\frac{2}{5}\hfill & \end{array}$
$\begin{array}{cc}\frac{\frac{4}{9}}{6}\hfill & \text{Write 6 as}\frac{6}{1}\text{and divide.}\hfill \end{array}$
$\begin{array}{ccc}\hfill \frac{\frac{4}{9}}{\frac{6}{1}}& =& \frac{4}{9}\xf7\frac{6}{1}\hfill \\ & =& \frac{\stackrel{2}{\overline{)4}}}{9}\cdot \frac{1}{\underset{3}{\overline{)6}}}=\frac{2\cdot 1}{9\cdot 3}=\frac{2}{\text{27}}\hfill \end{array}$
$\begin{array}{cc}\frac{5+\frac{3}{4}}{\text{46}}\hfill & \text{Simplify the numerator.}\hfill \end{array}$
$\begin{array}{cc}\frac{\frac{4\cdot 5+3}{4}}{\text{46}}=\frac{\frac{\text{20}+3}{4}}{\text{46}}=\frac{\frac{\text{23}}{4}}{\text{46}}\hfill & \text{Write 46 as}\frac{\text{46}}{1}.\hfill \end{array}$
$\begin{array}{ccc}\hfill \frac{\frac{\text{23}}{4}}{\frac{\text{46}}{1}}& =& \frac{\text{23}}{4}\xf7\frac{\text{46}}{1}\hfill \\ & =& \frac{\stackrel{1}{\overline{)23}}}{4}\cdot \frac{1}{\underset{2}{\overline{)46}}}=\frac{1\cdot 1}{4\cdot 2}=\frac{1}{8}\hfill \end{array}$
$\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}+\frac{\text{13}}{\text{24}}}=\frac{\frac{2}{8}+\frac{3}{8}}{\frac{\text{12}}{\text{24}}+\frac{\text{13}}{\text{24}}}=\frac{\frac{2+3}{8}}{\frac{\text{12}+\text{13}}{\text{24}}}=\frac{\frac{5}{8}}{\frac{\text{25}}{\text{24}}}=\frac{5}{8}\xf7\frac{\text{25}}{\text{24}}$
$\frac{5}{8}\xf7\frac{\text{25}}{\text{24}}=\frac{\stackrel{1}{\overline{)5}}}{\underset{1}{\overline{)8}}}\cdot \frac{\stackrel{3}{\overline{)24}}}{\underset{5}{\overline{)25}}}=\frac{1\cdot 3}{1\cdot 5}=\frac{3}{5}$
$\begin{array}{ccc}\hfill \frac{4+\frac{5}{6}}{7-\frac{1}{3}}=\frac{\frac{4\cdot 6+5}{6}}{\frac{7\cdot 3-1}{3}}=\frac{\frac{\text{29}}{6}}{\frac{\text{20}}{3}}& =& \frac{\text{29}}{6}\xf7\frac{\text{20}}{3}\hfill \\ & =& \frac{\text{29}}{\underset{2}{\overline{)6}}}\cdot \frac{\stackrel{1}{\overline{)3}}}{\text{20}}=\frac{\text{29}}{\text{40}}\hfill \end{array}$
$\frac{\text{11}+\frac{3}{\text{10}}}{4\frac{4}{5}}=\frac{\frac{\text{11}\cdot \text{10}+3}{\text{10}}}{\frac{4\cdot 5+4}{5}}=\frac{\frac{\text{110}+3}{\text{10}}}{\frac{\text{20}+4}{5}}=\frac{\frac{\text{113}}{\text{10}}}{\frac{\text{24}}{5}}=\frac{\text{113}}{\text{10}}\xf7\frac{\text{24}}{5}$
$\frac{\text{113}}{\text{10}}\xf7\frac{\text{24}}{5}=\frac{\text{113}}{\underset{2}{\overline{)10}}}\cdot \frac{\stackrel{1}{\overline{)5}}}{\text{24}}=\frac{\text{113}\cdot 1}{2\cdot \text{24}}=\frac{\text{113}}{\text{48}}=2\frac{\text{17}}{\text{48}}$
Convert each of the following complex fractions to a simple fraction.
$\frac{\frac{7}{\text{10}}}{\text{28}}$
$\frac{1}{\text{40}}$
$\frac{\frac{1}{8}+\frac{7}{8}}{6-\frac{3}{\text{10}}}$
$\frac{\text{10}}{\text{57}}$
$\frac{\frac{1}{6}+\frac{5}{8}}{\frac{5}{9}-\frac{1}{4}}$
$2\frac{\text{13}}{\text{22}}$
$\frac{\text{16}-\text{10}\frac{2}{3}}{\text{11}\frac{5}{6}-7\frac{7}{6}}$
$1\frac{5}{\text{11}}$
Simplify each fraction.
$\frac{\frac{1}{3}}{\frac{1}{9}}$
$\frac{\frac{8}{9}}{\frac{4}{\text{15}}}$
$\frac{2+\frac{1}{2}}{7+\frac{1}{2}}$
$\frac{9+\frac{1}{2}}{1+\frac{8}{\text{11}}}$
$\frac{4+\frac{\text{10}}{\text{13}}}{\frac{\text{12}}{\text{39}}}$
$\frac{\text{31}}{2}$
$\frac{\frac{1}{3}+\frac{2}{7}}{\frac{\text{26}}{\text{21}}}$
$\frac{\frac{3}{\text{10}}+\frac{4}{\text{12}}}{\frac{\text{19}}{\text{90}}}$
$\frac{\frac{9}{\text{16}}+\frac{7}{3}}{\frac{\text{139}}{\text{48}}}$
1
$\frac{\frac{1}{\text{288}}}{\frac{8}{9}-\frac{3}{\text{16}}}$
$\frac{\frac{27}{\text{429}}}{\frac{5}{11}-\frac{1}{\text{13}}}$
$\frac{1}{6}$
$\frac{\frac{1}{3}+\frac{2}{5}}{\frac{3}{5}+\frac{\text{17}}{\text{45}}}$
$\frac{\frac{9}{\text{70}}+\frac{5}{\text{42}}}{\frac{\text{13}}{\text{30}}-\frac{1}{\text{21}}}$
$\frac{\text{52}}{\text{81}}$
$\frac{\frac{1}{\text{16}}+\frac{1}{\text{14}}}{\frac{2}{3}-\frac{\text{13}}{\text{60}}}$
$\frac{\frac{3}{\text{20}}+\frac{\text{11}}{\text{12}}}{\frac{\text{19}}{7}-1\frac{\text{11}}{\text{35}}}$
$\frac{\text{16}}{\text{21}}$
$\frac{2\frac{2}{3}-1\frac{1}{2}}{\frac{1}{4}+1\frac{1}{\text{16}}}$
$\frac{3\frac{1}{5}+3\frac{1}{3}}{\frac{6}{5}-\frac{\text{15}}{\text{63}}}$
$\frac{\text{686}}{\text{101}}$
$\frac{\frac{1\frac{1}{2}+\text{15}}{5\frac{1}{4}-3\frac{5}{\text{12}}}}{\frac{8\frac{1}{3}-4\frac{1}{2}}{\text{11}\frac{2}{3}-5\frac{\text{11}}{\text{12}}}}$
$\frac{\frac{5\frac{3}{4}+3\frac{1}{5}}{2\frac{1}{5}+\text{15}\frac{7}{\text{10}}}}{\frac{9\frac{1}{2}-4\frac{1}{6}}{\frac{1}{8}+2\frac{1}{\text{120}}}}$
$\frac{1}{3}$
( [link] ) Find the prime factorization of 882.
( [link] ) Convert $\frac{\text{62}}{7}$ to a mixed number.
$8\frac{6}{7}$
( [link] ) Reduce $\frac{\text{114}}{\text{342}}$ to lowest terms.
( [link] ) Find the value of $6\frac{3}{8}-4\frac{5}{6}$ .
$1\frac{\text{13}}{\text{24}}$ or $\frac{\text{37}}{\text{24}}$
( [link] ) Arrange from smallest to largest: $\frac{1}{2}$ , $\frac{3}{5}$ , $\frac{4}{7}$ .
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