Lesson 9 · from §68 of the 1910 text
Complex Fractions
A complex fraction has a fraction in its numerator, its denominator, or both. It's just a division problem written vertically.
Example. Simplify \dfrac{x - 3 - \frac{10}{x}}{1 - \frac{2}{x} - \frac{15}{x^2}}. Make the top and bottom simple fractions: \frac{\frac{x^2 - 3x - 10}{x}}{\frac{x^2 - 2x - 15}{x^2}} = \frac{(x - 5)(x + 2)}{x}\cdot\frac{x^2}{(x - 5)(x + 3)} = \frac{x^2 + 2x}{x + 3}. Check with x = 1: \frac{1 - 3 - 10}{1 - 2 - 15} = \frac{-12}{-16} = \frac34, and \frac{1 + 2}{1 + 3} = \frac34. ✓
Rule. Reduce both the numerator and the denominator to simple fractions, then perform the indicated division.
A shortcut that often helps: multiply the top and bottom by the L.C.D. of all the little fractions. In the example, multiplying by x^2 gives \dfrac{x^3 - 3x^2 - 10x}{x^2 - 2x - 15} in one step.
Quick check
\displaystyle \frac{1+\frac{1}{x}}{1-\frac{1}{x^2}}
From the history of algebra
The slanted fraction bar, as in a/b, is called a solidus, after a Roman gold coin. Printers liked it because it fits on one line, which is exactly why you type fractions that way here. The idea of stacking fractions inside fractions goes back at least to the continued fractions studied in the 1500s and 1600s.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Simplify. These first ones are plain numbers: give a fraction like 3/2.
\displaystyle \frac{4-\frac{1}{4}}{2+\frac{1}{2}}
\displaystyle \frac{\frac{4}{9}+1}{\frac{7}{2}-2}
\displaystyle \frac{9-\left(\frac{1}{5}\right)^2}{4-\frac{4}{5}}
\displaystyle \frac{\frac{3}{5}+\frac{4}{7}}{2-\frac{3}{5}\cdot\frac{4}{7}}
\displaystyle \frac{\left(\frac{8}{5}\right)^2-2}{3\cdot\frac{8}{5}}
\displaystyle \frac{4-\frac{3}{2}+\frac{2}{3}}{3-\frac{2}{3}+\frac{3}{2}}
\displaystyle \frac{\frac{2^3}{27}+1}{\frac{9}{2}-1}
\displaystyle \frac{2\tfrac{1}{2}-3\tfrac{1}{3}+4\tfrac{3}{4}}{2\tfrac{1}{2}\cdot3\tfrac{1}{3}-4\tfrac{3}{4}}
\displaystyle \frac{2-\frac{9}{2}}{\left(\frac{1}{2}\right)^2-\left(\frac{1}{2}\right)^3-12\left(\frac{1}{2}\right)^4}
Now with letters. Give a single fraction in lowest terms.
\displaystyle \frac{a/b}{c/d}
\displaystyle \frac{\frac{1+x}{x}}{1-\frac{1}{x^2}}
\displaystyle \frac{c+\frac{c}{d}}{1+\frac{1}{d}}
\displaystyle \frac{9-\frac{b^2}{4a^2}}{1-\frac{b}{6a}}
\displaystyle \frac{\frac{a^2}{b^2}-\frac{b^2}{a^2}}{a+\frac{b^2}{a}}
\displaystyle \frac{a-\frac{9}{a}}{\frac{1}{a^2}-\frac{1}{a^3}-\frac{12}{a^4}}
\displaystyle \frac{\frac{m^3}{n^3}+1}{m+\frac{n^2}{m}-n}
\displaystyle \frac{2+\frac{1}{a-2}+a}{a+\frac{1}{a+2}-2}
\displaystyle \frac{\frac{a}{b}}{c}-\frac{a}{\frac{b}{c}}
\displaystyle \frac{1+\frac{b}{a}-\frac{20a}{b}}{\frac{b}{a}-\frac{8a}{b}-2}
\displaystyle \frac{\frac{16}{x}-x}{\frac{24}{x^4}+\frac{10}{x^3}+\frac{1}{x^2}}
\displaystyle \frac{x+\frac{x^2+y^2}{y}}{\frac{x}{y}-1}
\displaystyle \frac{2+\frac{a}{b}}{\frac{(a-2b)^2}{4ab}+2}
\displaystyle \frac{\left(\frac{2x^2}{x-y}\right)^2}{\frac{8x^3}{(x-y)^3}}
\displaystyle \frac{\frac{a-b}{a}-\frac{a+b}{b}}{\frac{a-b}{b}+\frac{a+b}{a}}
\displaystyle 1-\frac{\left(1-\frac{c^4}{9}\right)-\left(1-\frac{c^4}{16}\right)}{1-\frac{7c^4}{144}}
\displaystyle \frac{\frac{a}{1+a}+\frac{1-a}{a}}{\frac{a}{1+a}-\frac{1-a}{a}}
\displaystyle \frac{\frac{9}{8}x-\frac{9y}{8}-\frac{7y^2}{2x}}{\frac{3x}{4}+\frac{9}{4}y+\frac{5y^2}{3x}}
\displaystyle \frac{\frac{x}{2ax+3bx+6ab+x^2}}{\frac{1}{x+2a}-\frac{1}{x+3b}}
\displaystyle \frac{\left(\frac{3a+4b}{3a}\right)^2-\frac{6b}{a}}{8a-\frac{(3a+2b)^2}{3b}}
\displaystyle \frac{1-\frac{x^2-1}{5x^2-6x+1}}{1+\frac{3x+2}{5x-1}}