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In this section we will help you understand how to multiply fractions that contain rational expressions (expressions containing variables).  If you still need help with fractions that only contain numbers,
When fractions are multiplied, they are multiplied by multiplying the numerators by each other and the denominators by each other.  No cross-multiplication is involved!  Always remember that variables stand for numbers!  Because of that fact, nothing changes when dealing with variables in a multiplication problem with fractions that have rational expressions in them. 

               _          _
1. Expand: x^2| x^2   3y^3 |
           ---| --- - ---- |
           y^2|_ y      m _|
     

 Solution: Two multiplications are supposed to be done.  
           
                                      x^2     x^2
           You have to first multiply --- by --- and then
                                      y^2     y
           
                                x^2    -3y^3
           you have to multiply --- by -----.
                                y^2      m            
           
           
           That gives you:
  
           (x^2)(x^2)   (x^2)(3y^3)
           ---------- - -----------.
             (y^2)y        (y^2)m  
  
           Lastly, you simplify both expressions since 
           answers are not considered correct in Algebra
           unless they are simplified. After 
           simplification (multiply terms together and 
           cancel things out if possible), you get the 
           answer -
  
           x4   3yx2
           -- - ----
           y3     m

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When adding or subtracting fractions from other numbers, and especially other fractions, you must have the denominators of each fraction be the same expression.  Addition and subtraction are discussed further down on this page, but we will help you understand the Lowest Common Multiple in this section.

Specifically, we will help you understand how to find LCMs of algebraic expressions.

1. Find the LCM of 15a2b and 10ab3.

 
   Solution: Write the expressions as products 
             of prime and literal factors.
  
             15a2b                 10ab3
             3 * 5 * a * a * b     2 * 5 * a * b * b * b
  
             From the listing of prime and literal factors, 
             take the groups of factors with the most 
             instances of that factor.  For example, there 
             are three bs in 10ab3's factors, therefore, 
             three bs are listed in the LCM.
  
             The LCM is the following:
  
             2 * 3 * 5 * a * a * b * b * b = 30a2b3
    

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The addition or subtraction of fractions is complicated by the fact that the denominators must be the same before the fractions can be added or subtracted.  As we move toward more complex Algebra, you will come across fractions that have polynomials in them.  If the denominator is a polynomial, this polynomial must be a factor in the least common multiple.

1. Add:      a      b   
             - + -------
             x   (x + y)
     
   Solution: First, find the LCM of the denominators, 
             which will become the new denominator.  
  
             -------- + --------
             x(x + y)   x(x + y)
  
             So that the problem does not change, the 
             numerator of each term has to be multiplied 
             by the same quantity that its respective 
             denominator was.  The original denominator 
             of the first term was x, and it has 
             been multiplied by (x + y), so the
             original numerator, a, must be
             multiplied by (x + y), too.
  
             a(x + y)           
             -------- + --------
             x(x + y)   x(x + y)
  
             The original denominator of the second term 
             was (x + y), but it was multiplied by 
             x, so the original numerator must also
             be multiplied by x.  Now, the fractions
             can be added together.
  
             a(x + y)      xb      a(x + y) + xb
             -------- + -------- = -------------
             x(x + y)   x(x + y)     x(x + y)   
  
             This is one of the rare times in Algebra that 
             there are multiple forms of the correct answer.  
             In this case, you can multiply out the numerator 
             and/or denominator if you want, and since doing 
             that does not help you simplify the answer any 
             further, they are also correct forms of the 
             answer.
    

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Another name for a fraction divided by a fraction is a complex fraction.  Complex fractions are typically shown as follows:

 a
 -
 b
---
 c
 -
 d

where b, c, and d do not equal 0.  This is the same as (a/b)/(c/d).  To solve these fractions, you will need to multiply the numerator, or the first term of the problem by the reciprocal of the denominator, or the second term of the problem.

1. Simplify:  a 
              - 
              b 
             ---
              c  (b, c do not equal 0)


   Solution:  Multiply the numerator (a/b) 
              by the reciprocal of the denominator, 
              which is (1/c).


               a   1   a 
               - * - = --
               b   c   bc