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# Proper, improper fractions and mixed fractions

A fraction is proper when its numerator is smaller than its denominator in absolute value. The fraction is then a rational number between $-1$ and $1$

When a fraction is not proper, it is improper or top-heavy.

$1/2$ and $-18/19$ are proper; $19/18$ and $-3/2$ are improper.

A mixed fraction is a whole number combined with a proper fraction. It is easier to compare two mixed fractions than two improper fractions.

Negative fractions are treated just the same way as positive fractions.

$\Torange{1} \frac{1}{2}$, $\Torange{8} \frac{5}{17}$ and $\Torange{-3} \frac{9}{20}$ are mixed fractions.

$\Torange{2} \frac{1}{4}$ and $\Torange{1} \frac{83}{91}$ are easier to compare than $\frac{9}{4}$ and $\frac{174}{91}$ although the values are the same.

It is easy to convert mixed and improper fractions.

• Convert to mixed: $\displaystyle \frac{19}{\Tred{5}} = \frac{\Torange{3}\times \Tred{5} + \Tblue{4}}{\Tred{5}} = \frac{\Torange{3}\times \Tred{5}}{\Tred{5}} + \frac{\Tblue{4}}{\Tred{5}} = \Torange{3} \frac{\Tblue{4}}{\Tred{5}}$
• To improper: $\displaystyle -\Torange{2}\frac{\Tblue{1}}{\Tred{2}} = -\left( \frac{\Torange{2}\times\Tred{2}}{\Tred{2}} + \frac{\Tblue{1}}{\Tred{2}}\right) = -\left( \frac{5}{\Tred{2}}\right) = -\frac{5}{\Tred{2}}$
Equivalent improper and mixed fractions