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# Complex Fractions

A fraction is defined by the ratio of two numbers and is represented in the form of $\frac{a}{b}$, where variable 'a' denotes the value known as numerator and variable 'b' denotes the value known as denominator. Here b must not be equal to zero.
Thus, the fractions are classified as

• Proper Fraction
• Improper Fraction
• Complex Fraction
• Mixed Fraction
Here, we shall discuss about complex fractions.

Complex Fractions:

Complex fraction is a fraction in which that either numerator or denominator or both contain fractions. This is also referred as rational expression.
For example:
• $\frac{\frac{2}{5}-7}{\frac{6}{11}+1}$
• $\frac{x^{2}+\frac{1}{2}}{\frac{x}{y}}$
Rules for Simplifying Complex Fractions
Step 1:
Simplify numerator as well as denominator into single fractions, if there are any algebraic operations.
Step 2:
Reciprocate denominator and multiply it with numerator.
Step 3:
Finally reduce the fraction.

Let us understand this process with the help of an example:

Example: Simplify the following complex fraction

$\frac{\frac{1}{2}-\frac{1}{3}}{1-\frac{1}{12}}$

Solution:
Given fraction is

$\frac{\frac{1}{2}-\frac{1}{3}}{1-\frac{1}{12}}$

Step 1:
Simplifying numerator

$\frac{1}{2}-\frac{1}{3}$

= $\frac{1}{6}$

Step 2:
Simplifying denominator

$1-$$\frac{1}{12}$

= $\frac{11}{12}$

Step 3:
Substituting above value back in given fraction

$\frac{\frac{1}{6}}{\frac{11}{12}}$

= $\frac{1}{6}\times \frac{12}{11}$

= $\frac{2}{11}$

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