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CAPITULO XIV

Operaciones con Fracciones

Ejercicio 128

Simplificar:

$\begin{matrix} \frac{x - 3}{4} - \frac{x + 2}{8} & = \frac{1}{4} [x - 3 - \frac{x + 2}{2}] \\ = \frac{1}{4} [\frac{2 (x - 3) - (x + 2)}{2}] \\ = \frac{1}{4} [\frac{2 x - 6 - x - 2}{2}] \\ = \frac{1}{4} [\frac{x - 8}{2}] \\ = \frac{x - 8}{8} \end{matrix}$
$\begin{matrix} \frac{a + 5 b}{a^{2}} - \frac{b - 3}{a b} & = \frac{1}{a} [\frac{a + 5 b}{a} - \frac{b - 3}{b}] \\ = \frac{1}{a} [\frac{b (a + 5 b) - a (b - 3)}{a b}] \\ = \frac{1}{a} [\frac{a b + 5 b^{2} - a b + 3 a}{a b}] \\ = \frac{1}{a} [\frac{3 a + 5 b^{2}}{a b}] \\ = \frac{3 a + 5 b^{2}}{a^{2} b} \end{matrix}$
$\begin{matrix} \frac{2}{3 m n^{2}} - \frac{1}{2 m^{2} n} & = \frac{1}{m n} [\frac{2}{3 n} - \frac{1}{2 m}] \\ = \frac{1}{m n} [\frac{2 (2 m) - 3 n}{6 m n}] \\ = \frac{1}{m n} [\frac{4 m - 3 n}{6 m n}] \\ = \frac{4 m - 3 n}{6 m^{2} n^{2}} \end{matrix}$
$\begin{matrix} \frac{a - 3}{5 a b} - \frac{4 - 3 a b^{2}}{3 a^{2} b^{3}} & = \frac{1}{a b} [\frac{a - 3}{5} - \frac{4 - 3 a b^{2}}{3 a b^{2}}] \\ = \frac{1}{a b} [\frac{3 a b^{2} (a - 3) - 5 (4 - 3 a b^{2})}{15 a b^{2}}] \\ = \frac{1}{a b} [\frac{3 a^{2} b^{2} - 9 a b^{2} - 20 + 15 a b^{2}}{15 a b^{2}}] \\ = \frac{1}{a b} [\frac{3 a^{2} b^{2} + 6 a b^{2} - 20}{15 a b^{2}}] \\ = \frac{3 a^{2} b^{2} + 6 a b^{2} - 20}{15 a^{2} b^{3}} \end{matrix}$
$\begin{matrix} \frac{2 a + 3}{4 a} - \frac{a - 2}{8 a} & = \frac{1}{4 a} [2 a + 3 - \frac{a - 2}{2}] \\ = \frac{1}{4 a} [\frac{2 (2 a + 3) - (a - 2)}{2}] \\ = \frac{1}{4 a} [\frac{4 a + 6 - a + 2}{2}] \\ = \frac{1}{4 a} [\frac{3 a + 8}{2}] \\ = \frac{3 a + 8}{8 a} \end{matrix}$
$\begin{matrix} \frac{y - 2 x}{20 x} - \frac{x - 3 y}{24 y} & = \frac{1}{4} [\frac{y - 2 x}{5 x} - \frac{x - 3 y}{6 y}] \\ = \frac{1}{4} [\frac{6 y (y - 2 x) - 5 x (x - 3 y)}{30 x y}] \\ = \frac{1}{4} [\frac{6 y^{2} - 12 x y - 5 x^{2} + 15 x y}{30 x y}] \\ = \frac{1}{4} [\frac{6 y^{2} + 3 x y - 5 x^{2}}{30 x y}] \\ = \frac{6 y^{2} + 3 x y - 5 x^{2}}{120 x y} \end{matrix}$
$\begin{matrix} \frac{x - 1}{3} - \frac{x - 2}{4} - \frac{x + 3}{6} & = \frac{4 (x - 1) - 3 (x - 2) - 2 (x + 3)}{12} \\ = \frac{4 x - 4 - 3 x + 6 - 2 x - 6}{12} \\ = \frac{- x - 4}{12} \\ = - \frac{x + 4}{12} \end{matrix}$
$\begin{matrix} \frac{3}{5} - \frac{2 a + 1}{10 a} - \frac{4 a^{2} + 1}{20 a^{2}} & = \frac{1}{5} [3 - \frac{2 a + 1}{2 a} - \frac{4 a^{2} + 1}{4 a^{2}}] \\ = \frac{1}{5} [\frac{12 a^{2} - 2 a (2 a + 1) - (4 a^{2} + 1)}{4 a^{2}}] \\ = \frac{1}{5} [\frac{12 a^{2} - 4 a^{2} - 2 a - 4 a^{2} - 1}{4 a^{2}}] \\ = \frac{1}{5} [\frac{4 a^{2} - 2 a - 1}{4 a^{2}}] \\ = \frac{4 a^{2} - 2 a - 1}{20 a^{2}} \end{matrix}$
$\begin{matrix} \frac{3}{5 x} - \frac{x - 1}{3 x^{2}} - \frac{x^{2} + 2 x + 3}{15 x^{3}} & = \frac{1}{x} [\frac{3}{5} - \frac{x - 1}{3 x} - \frac{x^{2} + 2 x + 3}{15 x^{2}}] \\ = \frac{1}{x} [\frac{9 x^{2} - 5 x (x - 1) - (x^{2} + 2 x + 3)}{15 x^{2}}] \\ = \frac{1}{x} [\frac{9 x^{2} - 5 x^{2} + 5 x - x^{2} - 2 x - 3}{15 x^{2}}] \\ = \frac{1}{x} [\frac{3 x^{2} + 3 x - 3}{15 x^{2}}] \\ = \frac{3 (x^{2} + x - 1)}{x^{3}} \\ = \frac{x^{2} + x - 1}{5 x^{3}} \end{matrix}$
$\begin{matrix} \frac{1}{2 a} - \frac{2 + b}{3 a b} - \frac{5}{6 a^{2} b^{3}} & = \frac{1}{a} [\frac{1}{2} - \frac{2 + b}{3 b} - \frac{5}{6 a b^{3}}] \\ = \frac{1}{a} [\frac{3 a b^{3} - 2 a b^{2} (2 + b) - 5}{6 a b^{3}}] \\ = \frac{1}{a} [\frac{3 a b^{3} - 4 a b^{2} - 2 a b^{3} - 5}{6 a b^{3}}] \\ = \frac{1}{a} [\frac{a b^{3} - 4 a b^{2} - 5}{6 a b^{3}}] \\ = \frac{a b^{3} - 4 a b^{2} - 5}{6 a^{2} b^{3}} \end{matrix}$