On this Page:
1. Complex Number Fractions
2. Complex Variable Fractions
3. Simplify Complex Fractions
4. Examples
1. Complex Number Fractions
2. Complex Variable Fractions
3. Simplify Complex Fractions
4. Examples
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 convert the numerator and denominator so we have a single fraction form.Such as going from \frac{\space \frac{a}{b} \space}{\frac{c}{d}} to \frac{\space s \space}{t}.
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}
Examples
(1.1)
Simplify the following, \frac{1 \space {\text{--}} \space \frac{3}{{2\scriptsize{x}}}}{6 \space {\text{--}} \space \frac{11}{{\scriptsize{x}}^{\tt{2}}}}.
Solution
\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}
(1.2)
Simplify the fololwing, \frac{\frac{5x}{3}}{{\tiny{7}x + 4}}.
Solution
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}