Complex Fractions

One use of the LCD is to simplify complex fractions.  Complex fractions are fractions that contain other “minor” fractions within the numerators and/or denominators.

The first method is convenient if the original problem has a single fraction in the numerator and a single fraction in the denominator from the start of the problem.

Observe the following example involving a single fraction in the numerator and a single fraction in the denominator:
               ... Recognize that this complex fraction is actually a division problem
 

                               ... Change the problem to a multiplication of fractions by 
                                                           reciprocating ("flipping-over") the divisor

                               ... Using Rules for Exponents simplify the resulting fraction
 

For all other complex fractions, a second method may be easiest.  This method involves the examination of all of the “minor” fractions both in the numerator and in the denominator of the complex fraction to determine one single LCD for the entire complex fraction.  Once this single LCD is determined, all of the terms of the complex fraction are multiplied by the LCD.

     Example 1:
                                          … First determine the LCD = 4
                               … Next multiply all terms by the LCD = 4
 

                                          ... Combine like terms in the numerator

                                          … Problem is now fully simplified
 

      Example 2:
                                                          … First determine the LCD = (x – 1)

                                     … Next multiply all terms by the LCD = (x – 1)
 

                                       … Combine like terms

                              … Factor out the common factor of 2 and reduce