## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$\dfrac{2cy}{b}$
$\bf{\text{Solution Outline:}}$ To simplify the complex fraction, $\dfrac{\dfrac{45xyz}{24ab}}{\dfrac{30xz}{32ac}} ,$ multiply the numerator by the reciprocal of the denominator. Then cancel any common factors between the numerator and the denominator. $\bf{\text{Solution Details:}}$ Using $\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}=\dfrac{a}{b}\div\dfrac{c}{d},$ the given expression is equivalent to \begin{array}{l}\require{cancel} \dfrac{45xyz}{24ab}\div\dfrac{30xz}{32ac} .\end{array} To divide the fractions above, get the reciprocal of the divisor and change the operator to multiplication. Hence, the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{45xyz}{24ab}\cdot\dfrac{32ac}{30xz} \\\\= \dfrac{45\cancel{x}y\cancel{z}}{24\cancel{a}b}\cdot\dfrac{32\cancel{a}c}{30\cancel{x}\cancel{z}} \\\\= \dfrac{45y}{24b}\cdot\dfrac{32c}{30} \\\\= \dfrac{\cancel{15}(3)y}{24b}\cdot\dfrac{32c}{\cancel{15}(2)} \\\\= \dfrac{3y}{24b}\cdot\dfrac{32c}{2} \\\\= \dfrac{3y}{\cancel{8}(3)b}\cdot\dfrac{\cancel{8}(4)c}{2} \\\\= \dfrac{3y}{3b}\cdot\dfrac{4c}{2} \\\\= \dfrac{\cancel{3}y}{\cancel{3}b}\cdot\dfrac{\cancel{2}(2)c}{\cancel{2}} \\\\= \dfrac{2cy}{b} .\end{array}