8.2 Reducing rational expressions

$\frac{16}{40}$ can be reduced to $\frac{2}{5}$ .   Process:

$\frac{16}{40}=\frac{2·2·2·2}{2·2·2·5}$

Remove the three factors of 1; $\frac{2}{2}·\frac{2}{2}·\frac{2}{2}.$

$\frac{\cancel{2}·\cancel{2}·\cancel{2}·2}{\cancel{2}·\cancel{2}·\cancel{2}·5}=\frac{2}{5}$

Notice that in $\frac{2}{5}$ , there is no factor common to the numerator and denominator.

$\frac{111}{148}$ can be reduced to $\frac{3}{4}$ .   Process:

$\frac{111}{148}=\frac{3·37}{4·37}$

Remove the factor of 1; $\frac{37}{37}$ .

$\frac{3·\cancel{37}}{4·\cancel{37}}$

$\frac{3}{4}$

Notice that in $\frac{3}{4}$ , there is no factor common to the numerator and denominator.

$\frac{3}{9}$ can be reduced to $\frac{1}{3}$ .   Process:

$\frac{3}{9}=\frac{3·1}{3·3}$

Remove the factor of 1; $\frac{3}{3}$ .

$\frac{\cancel{3}·1}{\cancel{3}·3}=\frac{1}{3}$

Notice that in $\frac{1}{3}$ there is no factor common to the numerator and denominator.

$\frac{5}{7}$ cannot be reduced since there are no factors common to the numerator and denominator.

Problems 1, 2, and 3 shown above could all be reduced. The process in each reduction included the following steps:

1. Both the numerator and denominator were factored.
2. Factors that were common to both the numerator and denominator were noted and removed by dividing them out.