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Simplifying Complex Fractions
with Variables

The dividing fractions page introduced what are known as ‘complex fractions’, where one fraction or more is present within a larger fraction.
Complex fractions can also be referred to as ‘stacked fractions’.

Complex Fractions with Numbers:

To recap, the following are examples of what a complex fraction can look like.

\bf{\frac{7}{\frac{4}{5}}}     ,     \bf{\frac{\frac{1}{2}}{15}}     ,     \bf{\frac{\space \frac{1}{2} \space}{\frac{4}{5}}}

Any of these complex fractions above could be simplified by being treated as a fractions division sum.

\bf{\frac{\space \frac{1}{2} \space}{15}}    =    \bf{\frac{1}{2}} ÷ 15    = ;   \bf{\frac{1}{2}} ÷ \bf{\frac{15}{1}},

\bf{\frac{1}{2}} ÷ \bf{\frac{15}{1}}    =    \bf{\frac{1}{2}} × \bf{\frac{1}{15}}    =    \bf{\frac{1}{30}}

This approach can also be used when we want to attempt simplifying complex fractions with variables in Math.

Variables in Complex Fractions:

In further branches of Math we can often encounter complex fractions that contain variables as well as numbers.
Complex fractions which could have a similar form to the following.

\frac{4}{\frac{3}{{\scriptsize{x}}^{\tt{2}}}}     ,     \frac{1 \space {\text{–}} \space x}{\frac{4}{{\scriptsize{x}}^{\tt{3}}}}     ,     \frac{1 \space + \space \frac{2}{{\scriptsize{x}}}}{4 \space {\text{–}} \space \frac{5}{{\scriptsize{x}}^{\tt{2}}}}

Simplifying Complex Fractions with Variables:

When looking to simplify a complex fraction and variables are involved, we aim to have both the numerator and denominator in single fraction form.

Such as    \frac{\space \frac{a}{b} \space}{\frac{c}{d}}.

The complex fraction    \frac{4}{\frac{3}{{\scriptsize{x}}^{\tt{2}}}}    is straightforward to make so.

\frac{\space 4 \space}{\frac{3}{{\scriptsize{x}}^{\tt{2}}}}   =   \frac{\space \frac{4}{1} \space}{\frac{3}{{\scriptsize{x}}^{\tt{2}}}}   =   \frac{4}{1} {\scriptsize{\div}} \frac{3}{x^{\tt{2}}}   =   \frac{4}{1} {\scriptsize{\times}} \frac{x^{\tt{2}}}{3}   =   \frac{4x^{\tt{2}}}{3}

A little more work is required however with a fraction such as    \frac{1 \space + \space \frac{2}{{\scriptsize{x}}}}{4 \space {\text{–}} \space \frac{5}{{\scriptsize{x}}^{\tt{2}}}}.

We can change the form of the  1  and the  4  to help us.
We can rewrite them according to the relevant denominator of the fraction they are connected with.

1  =  \frac{x}{x}     ,     4  =  \frac{4x^{\tt{2}}}{x^{\tt{2}}}

Now:     \frac{1 \space + \space \frac{2}{{\scriptsize{x}}}}{4 \space {\text{–}} \space \frac{5}{{\scriptsize{x}}^{\tt{2}}}}   =   \frac{\frac{{\scriptsize{x}}}{{\scriptsize{x}}} \space + \space \frac{2}{{\scriptsize{x}}}}{\frac{4{\scriptsize{x}}^{\tt{2}}}{{\scriptsize{x}}^{\tt{2}}} \space {\text{–}} \space \frac{5}{{\scriptsize{x}}^{\tt{2}}}}   =   \frac{\frac{{\scriptsize{x}} \space + \space 2}{{\scriptsize{x}}}}{\frac{4{\scriptsize{x}}^{\tt{2}} \space {\text{–}} \space 5}{{\scriptsize{x}}^{\tt{2}}}}

From which we can simplify further.

\frac{x \space + \space 2}{x} ÷ \frac{4x^{\tt{2}} \space {\text{–}} \space 5}{x^{\tt{2}}}    =    \frac{x \space + \space 2}{x} × \frac{x^{\tt{2}}}{4x^{\tt{2}} \space {\text{–}} \space 5}

=    \frac{x^{\tt{3}} \space + \space 2x^{\tt{2}}}{4x^{\tt{3}} \space {\text{–}} \space 5x}    =    \frac{x(x^{\tt{2}} \space + \space 2x)}{x(4x^{\tt{2}} \space {\text{–}} \space 5)}    =    \frac{x^{\tt{2}} \space + \space 2x}{4x^{\tt{2}} \space {\text{–}} \space 5}


Simplify the following,     \frac{1 \space {\text{–}} \space \frac{3}{{2\scriptsize{x}}}}{6 \space {\text{–}} \space \frac{11}{{\scriptsize{x}}^{\tt{2}}}}.


\frac{1 \space {\text{–}} \space \frac{3}{{2\scriptsize{x}}}}{6 \space {\text{–}} \space \frac{11}{{\scriptsize{x}}^{\tt{2}}}}   =   \frac{{\frac{2{\scriptsize{x}}}{2{\scriptsize{x}}}} \space {\text{–}} \space \frac{3}{{2\scriptsize{x}}}}{{\frac{6{\scriptsize{x}}^{\tt{2}}}{{\scriptsize{x}}^{\tt{2}}}} \space {\text{–}} \space \frac{11}{{\scriptsize{x}}^{\tt{2}}}}   =   \frac{{\frac{2{\scriptsize{x}} \space {\text{–}} \space 3}{2{\scriptsize{x}}}}}{{\frac{6{\scriptsize{x}}^{\tt{2}} \space {\text{–}} \space 11}{{\scriptsize{x}}^{\tt{2}}}}}

=>    \frac{2x \space {\text{–}} \space 3}{2x} ÷ \frac{6x^{\tt{2}} \space {\text{–}} \space 11}{x^{\tt{2}}}    =    \frac{2x \space {\text{–}} \space 3}{2x} × \frac{x^{\tt{2}}}{6x^{\tt{2}} \space {\text{–}} \space 11}

=    \frac{2x^{\tt{3}} \space {\text{–}} \space 3x^{\tt{2}}}{12x^{\tt{3}} \space {\text{–}} \space 22x}    =    \frac{x(2x^{\tt{2}} \space {\text{–}} \space 3x)}{x(12x^{\tt{2}} \space {\text{–}} \space 22)}    =    \frac{2x^{\tt{2}} \space {\text{–}} \space 3x}{12x^{\tt{2}} \space {\text{–}} \space 22}

Simplify the fololwing,     \frac{\frac{5x}{3}}{{\tiny{7}x + 4}}.


Here multiplying both the numerator on top and the denominator below by  3  will simplify the whole fraction and reduce it from being complex.

\frac{\frac{5x}{3}}{{\tiny{7}x + 4}} × \frac{3}{3}    =    \frac{\frac{5x}{3} {\tiny{\times \space 3}}}{{\tiny({7}x + 4) \times 3}}    =    \frac{5x}{21x \space + \space 12}

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