Question

Perform the indicated operation and simplify the result.

Answer 1

Given:

$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}$

Solution:

$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}$

Apply the fraction rule:

$\frac{w}{x} \div \frac{y}{z}=\frac{w}{x} \times \frac{z}{y}$

Using this rule,

$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}=\frac{a-a b}{a^{2}} \times \frac{a^{3}}{a-1}$

Factor out common term a in first fraction.

                       $=\frac{a(1-b)}{a^{2}}\times \frac{a^{3}}{a-1}$

Cancel the common factor a.

                        $=\frac{1-b}{a} \times \frac{a^{3}}{a-1}$

Apply the multiplication of fraction rule:

$\frac{w}{x} \times \frac{y}{z}=\frac{w \times y}{x \times z}$

Using this rule, we get

                         $=\frac{(1-b) a^{3}}{a(a-1)}$

Cancel the common factor a.

                         $=\frac{a^{2}(1-b)}{a-1}$

Therefore,

$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}=\frac{a^{2}(1-b)}{a-1}$