In my last post, I showed how to convert an improper fraction to a mixed one. Today, let’s do the opposite: convert a mixed fraction to an improper one. Why would one want to do this apart from the pure joy and ecstasy of doing maths? Because, this would be the first step in adding, subtracting, or multiplying mixed fractions together. Notice that I have been avoiding division of fractions. This will be the topic of another post – we are well and truly going to beat the number of Rocky movies!

Let’s first show how to convert a mixed fraction to an improper one then we will use this skill eventually in a problem. Consider

\[2\frac{3}{4}

\]

Now please realise that this is maths shorthand for \[

{2}\hspace{0.33em}{+}\hspace{0.33em}\frac{3}{4}

\]. The denominator in the fraction part of this is indicating a whole of something divided into 4 pieces. This will be the denominator in our eventual improper fraction as well.

So to convert the 2 whole pieces into the number of total pieces if each whole was divided into 4 pieces is to multiply the 2 by the 4 to get 8 pieces. That is, if you divided 2 whole things into 4 pieces each, you get 8 total pieces.

Now we also have to remember that we have this fractional part of \[

\frac{3}{4}

\]. Remember that this fraction means that we have 3 pieces of a whole that was divided into 4 pieces. So we need to add these 3 pieces to the 8 pieces we have so far to get 11. So we have 11 pieces of some items that were each divided into 4 pieces. In fractional form, this is \[

\frac{11}{4}\]. This is the answer. So to convert a mixed fraction to an improper one, you multiply the denominator part by the whole number part, then add the numerator part. Let’s do another one:

{5}\frac{3}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{{(}{8}\hspace{0.33em}\times\hspace{0.33em}{5}{)}\hspace{0.33em}{+}\hspace{0.33em}{3}}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{{40}\hspace{0.33em}{+}\hspace{0.33em}{3}}{8}\hspace{0.33em}{=}\hspace{0.33em}\frac{43}{8}

\]

In a picture, you

and then set the result over the denominator.

In my next post, we will use this to do some math on mixed fractions.