Section 6.7
Complex Fractions
A complex fraction is a fraction in which either the numerator or the denominator (or both) contains a fraction itself. Although complex fractions look daunting, there is, in fact, a straightforward way to simplify them: Multiply top and bottom by the least common denominator of all the interior fractions.
Examples:
a. $\dfrac{1\ +\ \frac{1}{x}}{3}$ is a complex fraction because the numerator contains the fraction $\dfrac{1}{x}$.
Multiply both the numerator and the denominator of this fraction by $x$:
$\dfrac{\left( 1\ +\ \frac{1}{x} \right)\cdot x}{3\cdot x}=\dfrac{x+1}{3x}$
b. $\dfrac{\tfrac{1}{2}-\frac{1}{3}}{\frac{1}{4}-\frac{1}{5}}=\dfrac{\left( \frac{1}{2}-\frac{1}{3} \right)\cdot 60}{\left( \frac{1}{4}-\frac{1}{5} \right)\cdot 60} =\dfrac{30-20}{15-12} = \dfrac{10}{3}$
c. ${{\left( {{x}^{-1}}+{{3}^{-1}} \right)}^{-1}}\ \ =\ \ \dfrac{1}{\ \tfrac{1}{3}+\tfrac{1}{x}\ }\ \ =\dfrac{(1)\cdot 3x}{\left( \ \tfrac{1}{3}+\tfrac{1}{x}\ \right)\cdot 3x}\ \ \ =\ \ \dfrac{3x}{x+3}$
d. $\dfrac{1-\tfrac{a}{b}}{\tfrac{b}{a}-1}=\dfrac{\left( 1-\tfrac{a}{b} \right)\cdot ab}{\left( \tfrac{b}{a}-1 \right)\cdot ab}\ =\ \ \dfrac{ab-{{a}^{2}}}{{{b}^{2}}-ab}\ \ =\ \ \dfrac{a(b-a)}{b(b-a)}\ \ =\ \ \dfrac{a}{b}$
Exercises
Simplify each of the following.\begin{array}{l l l}
1. \;\;\frac{1}{\tfrac{1}{x}-2}\;\;& 2. \;\;\frac{1-\tfrac{4}{{{x}^{2}}}}{\tfrac{1}{x}+\tfrac{2}{{{x}^{2}}}}\;\;& 3. \;\;\frac{1}{\tfrac{1}{a}-\tfrac{2}{b}}\;\; \\
4. \;\;\frac{1-\tfrac{1}{{{x}^{2}}}}{x-\tfrac{1}{x}}\;\;& 5. \;\;\frac{2-\tfrac{1}{x-1}}{\tfrac{2}{x-1}+3}\;\;& 6. \;\;\frac{2x+\tfrac{x}{x-2}}{2x-\tfrac{x}{x-2}}\;\; \\
7. \;\;{{\left( {{2}^{-1}}+{{3}^{-1}} \right)}^{-1}}\;\;& 8. \;\;\frac{1+\tfrac{2}{m}}{1-\tfrac{4}{{{m}^{2}}}}\;\;& 9. \;\;\frac{\tfrac{3}{m}-1}{1-\tfrac{9}{{{m}^{2}}}}\;\; \\
10. \;\;\frac{a-\tfrac{a}{{{x}^{2}}}}{\tfrac{1}{x}-1}\;\;& 11. \;\;\frac{\tfrac{x}{y}-\tfrac{y}{x}}{\tfrac{1}{x}-\tfrac{1}{y}}\;\;& 12. \;\;\frac{{{2}^{-1}}-{{3}^{-1}}}{{{2}^{-1}}+{{3}^{-1}}}\;\; \\
13. \;\;\frac{\tfrac{1}{x-1}-\tfrac{1}{x+1}}{\tfrac{2}{x-1}+\tfrac{2}{x+1}}\;\;& 14. \;\;\frac{{{a}^{-1}}-{{b}^{-1}}}{{{a}^{-1}}+{{b}^{-1}}}\;\;& 15. \;\;\frac{1-\tfrac{1}{x-1}}{1+\tfrac{1}{x-1}}\;\; \\
16. \;\;\frac{1-\tfrac{3}{x}+\tfrac{2}{{{x}^{2}}}}{1-\tfrac{4}{{{x}^{2}}}}\;\;& 17. \;\;\frac{\tfrac{4x-y}{x-y}-\tfrac{y}{x+y}}{\tfrac{x}{x+y}+\tfrac{x}{x-y}}\;\;& 18. \;\;1+\frac{1}{1+\tfrac{1}{1+\tfrac{1}{x}}}\;\; \\
19. \;\;\frac{\tfrac{1}{x-1}-1}{2+\tfrac{1}{x-1}}\;\;& 20. \;\;1+\frac{x}{1+x+\tfrac{2{{x}^{2}}}{1-x}}\;\;& 21. \;\;1-\frac{1}{1-\tfrac{1}{1-x}}\;\;
\end{array}