Arithmetic Review: Equivalent Fractions *

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1 OpenStax-CNX module: m2 Arithmetic Review: Equivalent Fractions * Wade Ellis Denny Burzynski This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License.0 Abstract This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student. Overview Equivalent Fractions Reducing Fractions To Lowest Terms Raising Fractions To Higher Terms 2 Equivalent Fractions Equivalent Fractions Fractions that have the same value are called equivalent fractions. For example, 2 and represent the same part of a whole quantity and are therefore equivalent. Several more collections of equivalent fractions are listed below. Example 2, 2 20, Example 2, 2, 9, 2 Example, 2, 2, 2 2, 0 * Version.: May 2, :09 pm

2 OpenStax-CNX module: m2 2 Reducing Fractions To Lowest Terms Reduced to Lowest Terms It is often useful to convert one fraction to an equivalent fraction that has reduced values in the numerator and denominator. When a fraction is converted to an equivalent fraction that has the smallest numerator and denominator in the collection of equivalent fractions, it is said to be reduced to lowest terms. The conversion process is called reducing a fraction. We can reduce a fraction to lowest terms by. Expressing the numerator and denominator as a product of prime numbers. (Find the prime factorization of the numerator and denominator. See Section () for this technique.) 2. Divide the numerator and denominator by all common factors. (This technique is commonly called cancelling.) Sample Set A Reduce each fraction to lowest terms. Example 2 2 )2 ) )2 ) 2 and are common factors. Example )2 )2 2 2 )2 )2 2 is the only common factor. Example )2 ) )2 ) 2 and are common factors. Example Thus, There are no common factors. is reduced to lowest terms.

3 OpenStax-CNX module: m2 Raising a Fraction to Higher Terms Equally important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator. The higher, equivalent fraction is constructed by multiplying the original fraction by. Notice that and 9 are equivalent, that is 9. Also, This observation helps us suggest the following method for raising a fraction to higher terms. Raising a Fraction to Higher Terms A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same nonzero number. For example, 2 can be raised to 2 by multiplying both the numerator and denominator by, that is, multiplying by in the form. 2 2 How did we know to choose as the proper factor Since we wish to convert to 2 by multiplying it by some number, we know that must be a factor of 2. This means that divides into 2. In fact, 2. We divided the original denominator into the new, specied denominator to obtain the proper factor for the multiplication. Sample Set B Determine the missing numerator or denominator. Example. Divide the original denominator,, into the new denominator,.. Multiply the original numerator by. Example 9. Divide the original numerator,, into the new numerator,. 9. Multiply the original denominator by Exercises For the following problems, reduce, if possible, each fraction lowest terms.

4 OpenStax-CNX module: m2 Exercise (Solution on p..) Exercise 2 0 Exercise (Solution on p..) Exercise Exercise (Solution on p..) 2 Exercise 20 Exercise (Solution on p..) 0 Exercise Exercise 9 (Solution on p..) 0 2 Exercise Exercise (Solution on p..) 0 Exercise Exercise (Solution on p..) 2 Exercise 2 Exercise (Solution on p..) 2 Exercise 2 0 Exercise (Solution on p..) 22 Exercise Exercise 9 (Solution on p..) 2 Exercise 20 2 Exercise 2 (Solution on p..) 9 Exercise 22

5 OpenStax-CNX module: m2 Exercise 2 (Solution on p..) 2 2 Exercise Exercise 2 (Solution on p..) 0 For the following problems, determine the missing numerator or denominator. Exercise 2 2 Exercise 2 (Solution on p..) 0 Exercise 2 9 Exercise 29 (Solution on p..) Exercise 0 Exercise (Solution on p..) 2 Exercise 2 2 Exercise (Solution on p..) Exercise 2 Exercise (Solution on p..) 0

6 OpenStax-CNX module: m2 Solutions to Exercises in this Module Solution to Exercise (p. ) Solution to Exercise (p. ) 2 9 Solution to Exercise (p. ) Solution to Exercise (p. ) 2 Solution to Exercise (p. ) 20 Solution to Exercise (p. ) Solution to Exercise (p. )