Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Simplifying Complex Fractions

There are some fractions whose numerator and/or denominator themselves are expressions containing other fractions. We call these fractions complex fractions.

We already know one meaning of "simplifying fractions". That is, taking a simple fraction (i.e. a fraction that contains no other fractions neither in its numerator nor in its denominator) and manipulate it in order to get an equivalent algebraic form which has less terms in its numerator and denominator. However, when we talk about simplifying a complex fraction what we want to do is to rearrange the given complex fraction into an equivalent simple fraction which is in simplest form. (Note that by the phrase simple fraction, we mean a fraction which does not contain any other fractions in its numerator or in its denominator.)

There is one case where simplifying a complex fraction is very straightforward:

when the numerator and denominator of a complex fraction are just single simple fractions themselves.

Therefore, for the first step in simplifying the complex fraction, we just use the wellknown “invert and multiply” rule: multiply the fraction in the numerator by the reciprocal of the fraction in the denominator:

As you can see, the initial complex fraction on the left has been turned into a single simple fraction on the right. This step is justified only if the numerator and denominator of the original complex fraction are both single simple fractions. When the pattern in the box above is valid, all that is left to do in simplifying the original complex fraction is to use methods to check whether the simple fraction on the right can be simplified any further.


Example 1:



This is a complex fraction in which both numerator and denominator contain fractions. So we begin by simplifying each part into fully factored single simple fractions. For the numerator, we get

For the denominator, we get

So now, getting back to the original fraction, we have

Since neither numerator nor denominator of this last form can be factored further, and since there are no common factors that can be cancelled here between the numerator and the denominator, this is as much as the original fraction can be simplified and so this last form must be the required final answer.


Example 5:



This is a complex fraction similar in form to the one in Example 1. The fact that the denominators in the fractions found in the numerator and denominator of this complex fraction are binomials does not change the overall strategy of simplification here. We still start by reducing the expressions in the numerator and denominator of the complex fraction to fully factored single simple fractions. For the numerator this gives

For the denominator, we get

Substituting these two simplified forms for the numerator and denominator, respectively, of the original complex fraction then gives

as the final simplified form. We know we are done here because:

  • this is a single simple fraction (no other fractions are present in either its numerator or its denominator)
  • this simple fraction cannot be simplified further because the numerator and denominator cannot be factored further and there are no common factors present in the two parts which can be cancelled.
All Right Reserved. Copyright 2005-2007