একটি ভগ্নাংশকে অপর একটি ভগ্নাংশ দ্বারা ভাগ করার অর্থ প্রথমটিকে দ্বিতীয়টির গুণাত্মক বিপরীত ভগ্নাংশ দ্বারা গুণ করা।

উদাহরণস্বরূপ, $\frac{x}{y}$ ও $\frac{z}{y}$ দ্বারা ভাগ করতে হবে,

তাহলে $\frac{x}{y}÷\frac{z}{y}$

            $=\frac{x}{y}\times \frac{z}{y}$    [এখানে $\frac{y}{z}$ হলো $\frac{z}{y}$ এর গুণাত্মক বিপরীত ভগ্নাংশ]

            $=\frac{x}{z}$

উদাহরণ ৯। ভাগ কর :

(ক) $\frac{a^3{b}^2}{c^{2}d}$ কে $\frac{a^2{b}^3}{c{d}^3}$ দ্বারা 

(খ) $\frac{12a^4{x}^3{y}^2}{10x^4{y}^3{z}^2}$ কে $\frac{6a^3{b}^{2}c}{5x^2{y}^2{z}^2}$ দ্বারা 

(গ) $\frac{a^2-b^2}{a^2+ab+b^2}$ কে $\frac{a+b}{a^3-b^3}$ কে 

(ঘ) $\frac{x^3-27}{x^2-7x+6}$ কে $\frac{x^2-9}{x^2-36}$ দ্বারা 

(ঙ) $\frac{x^3-y^3}{x^3+y^3}$ কে $\frac{x^2-y^2}{{\left(x+y\right)}^2}$ দ্বারা 

সমাধান :

(ক) ১ম ভগ্নাংশ $=\frac{a^3{b}^2}{c^{2}d}$

        ২য়    "       $=\frac{a^2{b}^3}{c{d}^3}$

২য় ভগ্নাংশের গুণাত্মক বিপরীত হলো $\frac{c{d}^3}{a^2{d}^3}$

                                                           $\frac{a^3{b}^2}{c^{2}d}÷\frac{a^2{b}^3}{c{d}^3}$

                                                         $=\frac{a^3{b}^2}{c^{2}d}\times \frac{c{d}^3}{a^2{b}^3}$

$\therefore$ নির্ণেয় ভাগফল $=\frac{a^3{b}^{2}c{d}^3}{a^2{b}^3{c}^{2}d}=\frac{a{d}^2}{bc}$

(খ) $\frac{12a^4{x}^3{y}^2}{10x^4{y}^3{z}^2}÷\frac{6a^3{b}^{2}c}{5x^2{y}^2{z}^2}$

  $=\frac{12a^4{x}^3{y}^2}{10x^4{y}^3{z}^2}\times \frac{5x^2{y}^2{z}^2}{6a^3{b}^{2}c}$

$\therefore$ নির্ণেয় ভাগফল $\frac{axy}{b^{2}c}$

(গ) $\frac{a^2-b^2}{a^2+ab+b^2}÷\frac{a+b}{a^3-b^3}$

 $=\frac{\left(a+b\right)\left(a-b\right)}{\left(a^2+ab+b^2\right)}\times \frac{\left(a-b\right)\left(a^2+ab+b^2\right)}{a+b}$

 $=\left(a-b\right)\left(a-b\right)$

$\therefore$ নির্ণেয় ভাগফল $={\left(a-b\right)}^2$

(ঘ) $\frac{x^3-27}{x^2-7x+6}÷\frac{x^2-9}{x^2-36}$

 $=\frac{x^3-3^3}{x^2-6x-x+6}\times \frac{x^2-6^2}{x^2-3^2}$

 $=\frac{\left(x-3\right)\left(x^2+3x+3^2\right)}{x^2-6x-x+6}\times \frac{\left(x+6\right)\left(x-6\right)}{\left(x+3\right)\left(x-3\right)}$

$\therefore$ নির্ণেয় ভাগফল $=\frac{\left(x^2+3x+9\right)\left(x+6\right)}{\left(x-1\right)\left(x+3\right)}$

(ঙ) $\frac{x^3-y^3}{x^3+y^3}÷\frac{x^2-y^2}{{\left(x+y\right)}^2}$

$\therefore$ নির্ণেয় ভাগফল $=\frac{x^2+xy+y^2}{x^2-xy+y^2}$

উদাহরণ ১০। সরল কর :

(ক) $\left(1+\frac{1}{x}\right)÷\left(1-\frac{1}{x^2}\right)$

(খ) $\left(\frac{x}{x+y}+\frac{y}{x-y}\right)÷\left(\frac{x}{x-y}-\frac{y}{x+y}\right)$

(গ) $\frac{a^3+b^3}{{\left(a-b\right)}^2+3ab}÷\frac{{\left(a+b\right)}^2-3ab}{a^3+b^3}\times \frac{a+b}{a-b}$

(ঘ) $\frac{x^2+3x-4}{x^2-7x+12}÷\frac{x^2-16}{x^2-9}\times \frac{{\left(x-4\right)}^2}{{\left(x-1\right)}^2}$

(ঙ) $\frac{x^3+y^3+3xy\left(x+y\right)}{{\left(x+y\right)}^2-4xy}÷\frac{{\left(x-y\right)}^2+4xy}{x^3-y^3-3xy\left(x-y\right)}$

সমাধান : (ক) $\left(1+\frac{1}{x}\right)÷\left(1-\frac{1}{x^2}\right)$

                       $=\frac{\left(x+1\right)}{x}÷\frac{x^2-1}{x^2}$

                       $=\frac{\left(x+1\right)}{x}\times \frac{x^2}{\left(x+1\right)\left(x-1\right)}$

(খ) $\left(\frac{x}{x+y}+\frac{y}{x-y}\right)÷\left(\frac{x}{x-y}-\frac{y}{x+y}\right)$

       $=\frac{x^2-xy+xy+y^2}{\left(x+y\right)\left(x-y\right)}÷\frac{x^2+xy-xy+y^2}{\left(x-y\right)\left(x+y\right)}$

       $=\frac{x^2+y^2}{x^2-y^2}÷\frac{x^2+y^2}{x^2-y^2}$

       $=\frac{x^2+y^2}{x^2-y^2}\times \frac{x^2-y^2}{x^2+y^2}$

       $=1$

(গ) $\frac{a^3+b^3}{{\left(a-b\right)}^2+3ab}÷\frac{{\left(a+b\right)}^2-3ab}{a^3+b^3}\times \frac{a+b}{a-b}$

       $=\frac{\left(a+b\right)\left(a^2-ab+b^2\right)}{a^2-2ab+b^2+3ab}÷\frac{a^2+2ab+b^2-3ab}{\left(a+b\right)\left(a^2+ab+b^2\right)}\times \frac{a+b}{a-b}$

       $=\frac{\left(a+b\right)\left(a^2-ab+b^2\right)}{\left(a^2+ab+b^2\right)}\times \frac{\left(a-b\right)\left(a^2+ab+b^2\right)}{\left(a^2-ab+b^2\right)}\times \frac{a+b}{a-b}$

       $=\left(a+b\right)\left(a+b\right)$

       $={\left(a+b\right)}^2$

(ঘ) $\frac{x^2+3x-4}{x^2-7x+12}÷\frac{x^2-16}{x^2-9}\times \frac{{\left(x-4\right)}^2}{{\left(x-1\right)}^2}$

      $=\frac{x^2+4x-x-4}{x^2-3x-4x+12}\times \frac{x^2-3^2}{x^2-4^2}\times \frac{{\left(x-4\right)}^2}{{\left(x-1\right)}^2}$

      $=\frac{\left(x+4\right)\left(x-1\right)}{\left(x-3\right)\left(x-4\right)}\times \frac{\left(x+3\right)\left(x-3\right)}{\left(x+4\right)\left(x-4\right)}\times \frac{{\left(x-4\right)}^2}{{\left(x-1\right)}^2}$

      $=\frac{x+3}{x-1}$

(ঙ) $\frac{x^3+y^3+3xy\left(x+y\right)}{{\left(x+y\right)}^2-4xy}÷\frac{{\left(x-y\right)}^2+4xy}{x^3-y^3-3xy\left(x-y\right)}$

       $=\frac{{\left(x+y\right)}^3}{{\left(x-y\right)}^2}÷\frac{{\left(x+y\right)}^2}{{\left(x-y\right)}^3}$

       $=\frac{{\left(x+y\right)}^3}{{\left(x-y\right)}^2}\times \frac{{\left(x-y\right)}^3}{{\left(x+y\right)}^2}$

        $=\left(x+y\right)\left(x-y\right)$

        $=x^2-y^2$