Lesson 8 · from §67 of the 1910 text
Dividing Fractions
To divide by a fraction, multiply by it upside down. In arithmetic, \frac34 \div \frac57 = \frac34\cdot\frac75 = \frac{21}{20}, and \frac34 \div 1\frac15 = \frac34 \div \frac65 = \frac34\cdot\frac56 = \frac58.
In algebra: \frac{a}{b} \div \frac{c}{d} = \frac{a}{b}\cdot\frac{d}{c} = \frac{ad}{bc}, \qquad \frac{a}{b} \div n = \frac{a}{bn}, \qquad \frac{a}{b} \div \left(c + \frac{n}{d}\right) = \frac{a}{b}\cdot\frac{d}{cd + n} = \frac{ad}{bcd + bn}.
When several divisions and multiplications appear in a row, work from left to right: invert each divisor, then multiply everything.
Rule. Change whole and mixed expressions to fractions. Then invert the divisor (or divisors) and proceed as in multiplication.
Quick check
\displaystyle \frac{a^2-4}{3a}\div\frac{a+2}{6a^2}
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.
Perform the indicated operations. Work from left to right: invert each divisor, then multiply.
\displaystyle \frac{6a}{3b}\div\frac{4a}{12b^2}
\displaystyle \frac{9a^2}{6b^2}\div\frac{14ab}{8x}
\displaystyle \left(\frac{2ax}{3c}\right)^2\div\frac{6ax^2}{9c^4}
\displaystyle \frac{a}{b}\div\frac{c}{d}\div\frac{ad^2}{bc^2}
\displaystyle 2a\cdot\left(\frac{2a^2}{3x}\right)^2\div\frac{(4a^2)^2}{18x^2}
\displaystyle \frac{5x}{5y}\div\frac{10x^2}{12y^3}\div\frac{4ax}{3by^2}
\displaystyle \frac{6}{(2x)^2}\div\frac{12}{(4x^2)^3}\cdot\left(\frac{2}{x}\right)^2
\displaystyle \frac{3m^2}{5n}\div\frac{21m^3}{10mn^2}\div {14m^4n}
\displaystyle \left(\frac{4x^2}{3a}\right)^2\div\left(\frac{2x}{3a}\right)^3\cdot\left(\frac{a}{x}\right)^2
\displaystyle \frac{6a-3}{5x}\div\frac{2a-1}{15bx^3}
Polynomials: factor everything first.
\displaystyle \frac{4-x^2}{3x^2+x^3}\div\frac{4-4x+x^2}{x^3-x^2-12x}
\displaystyle \frac{a^2-7a+12}{a-1}\div\frac{a^2-16}{1-a^2}
\displaystyle (-14-5c+c^2)\div\frac{ac^2-49a}{bc^2+9bc+14b}
\displaystyle \frac{4a^2-4ab-3b^2}{8a^3x}\div\left(a-\frac{9b^2}{4a}\right)
\displaystyle \left(\frac{2a}{x}\right)^4\div\left(\frac{-6a^2c}{5x}\right)^3\cdot\left(\frac{3c^3}{10ax}\right)^2
\displaystyle \left(\frac{-a^2}{x}\right)\div\left(\frac{-a}{x^2}\right)^5\cdot\left(\frac{a}{x}\right)^3\cdot\left(\frac{-1}{x^2}\right)^3
\displaystyle \left(\frac{m^2}{n^2}-\frac{n^2}{m^2}\right)\div\left(\frac{m^4+2m^2n^2+n^4}{m^2n-4mn^2}\right)
\displaystyle \frac{9x^2+6xy-8y^2}{2x+y}\div(3x-2y)^2\div\left(2-\frac{x-2y}{2x+y}\right)
\displaystyle \left(\frac{a}{c}+\frac{c}{a}\right)\div\left(\frac{a^6+c^6}{a^3c^3}\right)\cdot\left(a^2-c^2+\frac{c^4}{a^2}\right)\cdot\frac{1}{ac^2}
\displaystyle \frac{2a-5}{2a^2+2}\div\frac{4a^4-25a^2}{4-4a^4}\cdot\left(3a^2-\frac{a^3-8a^2}{a-1}\right)
\displaystyle \frac{8m^3-125n^3}{m^2+mn}\div\left(\frac{2m}{5n}-\frac{5n}{2m}\right)\div\left[3mn\left(2m+5n+\frac{25n^2}{2m}\right)\right]
\displaystyle \left(\frac{4y}{x}-\frac{15y^2}{x^2}+4\right)\div\left(4-\frac{16y}{x}+\frac{15y^2}{x^2}\right)\cdot\left(3-\frac{4x+20y}{2x+5y}\right)
\displaystyle \left(\frac{9c^2-4d^2}{6c^2}\right)\div\left(d+4c+\frac{15c^2}{4d}\right)\cdot\frac{15c^2d+6cd^2}{9c^2d+24cd^2-20d^3}
\displaystyle \left(6x-11-\frac{7}{x}\right)\div\left(2+\frac{11}{x}+\frac{5}{x^2}\right)\div\left(\frac{1}{3x^3-75x}\right)
\displaystyle \left(x^2-y^2+\frac{4xy(y+x)}{x-y}\right)\div\left(\frac{x^2+y(y+2x)}{2x^2-3xy+y^2}\right)