Math Fractions Expanding and Reducing

# Expanding and Reducing Fractions

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### 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:

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

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

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### Method

1. Divide numerator and denominator into prime factors
2. 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:
$\frac{16}{40}=\frac{2\cdot\rlap{\backslash}8}{5\cdot\rlap{\backslash}8}=\frac25$

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

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