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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
Equivalent improper and mixed fractions