Expanding and Reducing Fractions

Remember

If you are expanding or reducing, the value of the fraction does not change. You only change the shape.

Expanding

A fraction is extended by multiplying numerator and denominator with the same number (except 0). In general:

$\frac{a}{b}=\frac{a\cdot c}{b\cdot c}\quad$ ($c\neq 0$)

Note

For sums in the numerator or denominator, all summands must be multiplied with the same expansion factor c.

Example: $\frac{2+3}{8}=\frac{(2+3)\cdot\color{red}{5}}{8\cdot\color{red}{5}}=\frac{10+15}{40}=\frac{25}{40}$

Examples

Expand the fraction with $4$
$\frac12=\frac{1\cdot4}{2\cdot4}=\frac48$

Expand the fraction with $3$
$\frac{12-4}{3+1}=\frac{(12-4)\cdot3}{(3+1)\cdot3}=\frac{36-12}{9+3}=\frac{24}{12}$

Expand the algebraic fraction with $2x$
$\frac{2y}{4x}=\frac{2y\cdot2x}{4x\cdot2x}=\frac{4xy}{8x^2}$

Reducing

A fraction is shortened by dividing numerator and denominator with the same number (except 0). In general:

$\frac{a\cdot \rlap{\backslash}c}{b\cdot \rlap{\backslash}c}=\frac{a}{b}$

Note

Reducing sums is not allowed.

Example: $\frac{2+3}{8+3}$ Reducing would be wrong here!

In mathematics, it is customary to reduce fractions as much as possible. It is said that the fractions are completely shortened.

Method

Divide numerator and denominator into prime factors
Delete factors that are in numerator and denominator

Examples

Shorten this fraction/ algebraic fraction as far as possible.

$\frac{16}{40}=\frac{2\cdot\rlap{\backslash}2\cdot\rlap{\backslash}2\cdot\rlap{\backslash}2}{5\cdot\rlap{\backslash}2\cdot\rlap{\backslash}2\cdot\rlap{\backslash}2}=\frac25$

However, anyone who sees immediately that numerator and denominator are divisible by 8 can also use the following variant, for example:
$\frac{16}{40}=\frac{2\cdot\rlap{\backslash}8}{5\cdot\rlap{\backslash}8}=\frac25$

$\frac{4+3}{1+3}=\frac{7}{4}=\text{Reducing is not possible!}$

$\frac{4x}{6x}=\frac{2\cdot\rlap{\backslash}2\cdot\rlap{\backslash}x}{3\cdot\rlap{\backslash}2\cdot\rlap{\backslash}x}=\frac23$