Now the fractions we have been working with are called *proper* fractions: these are fractions where the numerator is smaller than the denominator. These types of fractions are smaller than one, which is why they are called proper as they are a fractional part of one. This implies there are things called *improper* fractions, and there are. Not improper in the sense of your drunk uncle at a wedding, but improper because they are greater than or equal to one. These will be fractions with a numerator equal to or greater than the denominator.

For example \[

\frac{5}{3}

\] is an improper fraction. Now they can be added, subtracted, multiplied, and simplified like any other fraction. But when the result of an operation with fractions results in an improper fraction, you are expected to convert this to a *mixed* fraction – a whole number plus a proper fraction. So \[

\frac{5}{3}

\] is equal to \[

1\frac{2}{3}

\]

So how do you convert an improper fraction to a mixed one? Just take out the whole parts and leave the resulting proper fraction. You do this by dividing: \[

\frac{5}{3}\hspace{0.33em}{=}\hspace{0.33em}{5}\hspace{0.33em}\div\hspace{0.33em}{3}\hspace{0.33em}{=}\hspace{0.33em}{1}

\] plus a reminder of 2. So in this case, the improper fraction is one whole plus two thirds left over or \[

1\frac{2}{3}

\].

Now try \[

\frac{24}{11}

\]:

\[

\frac{24}{11}\hspace{0.33em}{=}\hspace{0.33em}{24}\hspace{0.33em}\div\hspace{0.33em}{11}\hspace{0.33em}{=}\hspace{0.33em}{2}

\] with a remainder of 2. So \[

\frac{24}{11}\hspace{0.33em}{=}\hspace{0.33em}{2}\frac{2}{11}

\].

One more example:

\[\frac{66}{33}\hspace{0.33em}{=}\hspace{0.33em}{66}\hspace{0.33em}\div\hspace{0.33em}{33}\hspace{0.33em}{=}\hspace{0.33em}{2}

\].

Sometime there is no remainder and you just have a whole number. By the way, this result could also have been obtained by factoring:

\[\frac{66}{33}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{3}\rlap{/}{3}\hspace{0.33em}\times\hspace{0.33em}{2}}{\rlap{/}{3}\rlap{/}{3}\hspace{0.33em}\times\hspace{0.33em}{1}}\hspace{0.33em}{=}\hspace{0.33em}\frac{2}{1}\hspace{0.33em}{=}\hspace{0.33em}{2}

\]