Using the Fundamental Theorem of Fractions






The Fundamental Theorem of Fractions:

a/b = ac/bc,  where a and c are not equal to zero.

The Fundamental Theorem of Fractions is used in 
    
   
  • Reducing Fractions 
  • Building Fractions 


Reducing Fractions

Note if the numerator and denominator of a fraction have a common factor we can divide both
numerator and denominator of the fraction by that common factor and reduce the fraction.

Examples: Reduce each of the following: 
    
  
  • 10/14 = [2(5)]/[2(7)] = 5/7     Divide both numerator and denominator by 2. 
  • [3x]/[7x] = 3/7                        Divide both numerator and denominator by x. 
  • [12ab³ ]/[18a² b² ]  =  [(2² )(3)(a)(b³ )]/[(2)(3² )(a² )(b² )] =  [2b]/[3a]

                                                Divide numerator and denominator by 2(3)(a)(b² ). 
     

 If numerator and denominator of a fraction are polynomials we must factor numerator
 and denominator and  then reduce. 
  • [x²  +  5x  +  6]/[x²  -  4]  =  [(x + 3)(x + 2)]/[(x + 2)(x - 2)] =  (x + 3)/(x - 2)

                                                Divide numerator and denominator by x + 2. 
 


Building Fractions


To build a fraction we multiply both numerator and denominator of the fraction by the same
nonzero number to obtain an equivalent fraction.

To write the fraction 3/4 as 6/8 we we multiply the numerator and demonimator by the building
factor of 2.  To find the building factor divide the desired denominator 8 by the original
denominator 4 to obtain the building factor 2.

Examples:  Write each of the following fractions as an equivalent fraction with specified
               denominator. 
    

  • 3/5 = ?/35                                       Building Factor:  35/5 = 7

3/5 = [(3)(7)]/[(5)(7)] = 21/35 
  • [7a]/[5b] = ?/[10b² ]                        Building Factor:  [10b² ]/[5b] = 2b

[7a]/[5b] = [7a(2b)]/[5b(2b)] = [14ab]/[10b² ] 
  • [x - 1]/[x + 5] = ?/[(x + 5)(x - 2)]    Building Factor: [(x + 5)(x - 2)]/[x + 5] = (x - 2)

[x - 1]/[x + 5] = [(x - 1)(x - 2)]/[(x + 5)(x - 2)] 
  • 2/[x - 4] = ?/[x² + x - 20]                 To find the building factor first factor denominator.

2/[x - 4] = ?/[(x - 4)(x + 5)]            Building Factor:  [(x - 4)(x + 5)]/[x - 4] = (x + 5)

2/[x - 4] = [2(x + 5)]/[(x - 4)(x + 5)] = [2x + 10]/[x²  + x - 20]

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