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Beginning _ Intermediate Algebra_ 4ed

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Beginning _ Intermediate Algebra_ 4ed Powered By Docstoc
					§ 1.3


        Fractions
Numerators and Denominators

A quotient of two numbers is called a fraction.

                         The fraction 1 represents the
                                       4
                         shaded part of the circle. 1 out of
                                             1
                         4 pieces is shaded. 4 is read “one-
                         fourth.”

                              1             numerator
                              4             denominator


            Martin-Gay, Beginning and Intermediate Algebra, 4ed   2
          Simplifying Fractions
To simplify fractions we can simplify the numerator and
the denominator.
                       2 · 5 = 10
                      factors           product

A fraction is said to be simplified or in lowest terms
when the numerator and denominator have no factors in
common other than 1.
                           2        17           1
                           3        23           9
              Martin-Gay, Beginning and Intermediate Algebra, 4ed   3
Prime and Composite Numbers
A prime number is a natural number, other than 1,
whose only factors are 1 and itself.

          2, 3, 5, 7, 11, 13, 17, 19, 23, 29
            The first 10 prime numbers

A natural number, other than 1, that is not a prime
number is called a composite number. Every
composite number can be written as a product of
prime numbers

            Martin-Gay, Beginning and Intermediate Algebra, 4ed   4
           Product of Primes
Example:
Write the number 24 as a product of primes.
 24 = 4  6                    Write 24 as the product of any two
                               whole numbers.

     22 23                   If the factors are not prime, they
                               must be factored.

24 = 2  2  2  3             When all of the factors are prime, the
                               number has been completely factored.




           Martin-Gay, Beginning and Intermediate Algebra, 4ed          5
The Fundamental Principal of Fractions
  The Fundamental Principal of Fractions
  If a is a fraction and c is a nonzero real number, then
    b
                              ac a
                                 
                              bc b

  Example:
                     25
  Write the fraction    in lowest terms.
                     40
            25     55  5              5
                                     
            40 2  2  2  5 2  2  2 8

               Martin-Gay, Beginning and Intermediate Algebra, 4ed   6
          Multiplying Fractions
To multiply two fractions, multiply numerator times
numerator to obtain the numerator of the product.
Multiply denominator times denominator to obtain the
denominator of the product.

 Multiplying Fractions
           a c ac
                 , if b  0 and d  0
           b d bd

          3 2   6                         3 2  6
             
          7 5   35                        7  5  35
              Martin-Gay, Beginning and Intermediate Algebra, 4ed   7
         Multiplying Fractions
                      12 3
   Example: Multiply.   
                      17 24
       12 3   12  3   36                     Multiply numerators.
                   
       17 24 17  24 408                      Multiply denominators.



Simplify the product by dividing the numerator and
the denominator by any common factors.
          36    2  2  3 3     3
                              
          408 2  2  2  3 17 34

             Martin-Gay, Beginning and Intermediate Algebra, 4ed       8
           Dividing Fractions
 Two fractions are reciprocals of each other if their
 product is 1.
           3 4                    3    4
             1                  4
                                    and are reciprocals.
                                       3
           4 3

Dividing Fractions
          a c a d
              , if b  0 and d  0
          b d b c



            Martin-Gay, Beginning and Intermediate Algebra, 4ed   9
       Dividing Fractions
                 3 1
Example: Divide.  
                 4 4

    3   1   3 4   12  3
             
    4   4   4 1    4




        Martin-Gay, Beginning and Intermediate Algebra, 4ed   10
Fractions with the Same Denominator

 To add or subtract fractions with the same denominator,
 combine numerators and place the sum or difference
 over the common denominator.
                       2   1   3
                            
                       4   4   4
 Adding and Subtracting Fractions with the Same Denominator
                  a c ac
                        , if b  0
                  b b  b
                  a c ac
                        , if b  0
                  b b  b

               Martin-Gay, Beginning and Intermediate Algebra, 4ed   11
           Equivalent Fractions
Equivalent fractions are fractions that represent the
same quantity.




          3 is shaded.                       1 is shaded.
          6                                  2
               Equivalent fractions



              Martin-Gay, Beginning and Intermediate Algebra, 4ed   12
           Equivalent Fractions
              3
Example: Write as an equivalent fraction with a
              4
denominator of 20.

     Since 4 · 5 = 20, multiply the fraction by 5 .
                                                                     5
                     3 3 5 3  5 15
                             
                     4 4 5 4  5 20

             5
  Multiply by or 1.
             5
               Martin-Gay, Beginning and Intermediate Algebra, 4ed       13
Fractions without the Same Denominator
To add or subtract fractions without the same denominator,
first write the fractions as equivalent fractions with a
common denominator
The least common denominator (LCD) is the smallest
number both denominators will divide evenly into.

                  3   1
 Example: Add.                         3 3
                                          
                                             9                       1 4
                                                                       
                                                                          4
                  8   6                 8 3  24                      6 4  24
               LCD = 24
                                         9     4   13
                                                
                                         24   24   24

               Martin-Gay, Beginning and Intermediate Algebra, 4ed             14
Fractions without the Same Denominator

                         5    7
Example: Subtract.         
                        12   30
                        LCD = 60

            5 5   25                         7   2   14
                                                
           12 5   60                         30 2    60

                          25   14   11
                                 
                          60   60   60


             Martin-Gay, Beginning and Intermediate Algebra, 4ed   15

				
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