
Chapter 3—Fractions 33 Reducing Fractions
When you complete the work for this section, you should be able to:  Identify fractions that can be reduced.
 Determine the largest common factor for a fraction that can be reduced.
 Demonstrate you ability to reduce any proper fraction as necessary.
 Reducing fractions is the opposite of raising fractions to higher terms. Instead of multiplying the numerator and denominator by the same number, reducing fractions is a matter of dividing the numerator and denominator by the same number. A fraction such as ^{12}/_{16} might look a lot different from ^{3}/_{4}, but it represents exactly the same portion. Stated mathematically: ^{12}/_{16} = ^{3}/_{4} In fact, ^{6}/_{8} is also the same as ^{3}/_{4}, and so is ^{15}/_{20}. You can express the same portions in an unlimited number of ways. However, there is always one version that is the simplest and most direct. This is the reduced version of the fraction—where there is no integer (other than 1) that can divide evenly into both the numerator and denominator. This is called a reduced fraction.  Twelve sixteenth is the same portion as three fourths.  Definition A reduced fraction is one where there is no integer (except 1) that divides evenly into both the numerator and denominator.  Examples  The fraction ^{6}/_{8} is not reduced because the numerator and denominator can both be divided by 2.
 The fraction ^{12}/_{16} is not reduced because the numerator and denominator can both be divided by 2 or by 4.
 The fraction ^{15}/_{20} is not reduced because the numerator and denominator can both be divided by 5.
 The fraction ^{3}/_{4} is reduced because the numerator and denominator cannot be evenly divided by any integer except 1.
Examples and Exercises Identifying Reduced Fractions Use these interactive examples and exercises to strengthen your understanding and build your skills:   Keeping All Things Equal Dividing the numerator and denominator by the same number does not change the actual value of the fraction. Examples  After dividing the numerator and denominator of ^{2}/_{4} by 2,
we see that ^{2}/_{4} = ^{1}/_{2}  After dividing the numerator and denominator of ^{3}/_{12} by 3,
we see that ^{3}/_{12} = ^{1}/_{4}  After dividing the numerator and denominator of ^{10}/_{18} by 2,
we see that ^{10}/_{18} = ^{5}/_{9}  After dividing the numerator and denominator of ^{5}/_{15} by 5,
we see that ^{5}/_{15} = ^{1}/_{3} Thinking Mathematically  Dividing any number by 1 does not change the value of that number:
Examples:  2 ÷ 1 = 2 5 ÷ 1 = 5 120 ÷ 1 = 120   When the numerator and denominator of a fraction have the same value, the fraction is equal to 1:
Examples:  ^{3}/_{3} = 1 ^{12}/_{12} = 1 ^{8}/_{8} = 1   Therefore, dividing the numerator and denominator of a fraction by the same number does not change the actual value of the fraction.
 Reducing: The BruteForce Method When working with fractions—adding, subtracting, multiplying, or dividing them, for instance—the result is often not in its reduced form. Instead of simple and tidy results such as ^{1}/_{2}, ^{1}/_{3}, and ^{3}/_{4}, we often get messy results such as ^{8}/_{16}, ^{40}_{60}, ^{75}/_{100}. The "messy results" should be cleaned up. Fractions that can be reduced, should be reduced. Why? Mainly because we can get a better sense of their proportional value. For example, people tend to have a better understanding of the fraction ^{1}/_{2} than one of its expanded (or unreduced) versions such as ^{256}/_{512}. Procedure The bruteforce method for reducing fractions isn't very elegant, but it gets the job done.  Find any integer greater than 1 that can be divided evenly into both the numerator and denominator.
 Divide the numerator and denominator by the integer from Step 1.
 Repeat Steps 1 and 2 until the fraction is completely reduced.
 Example 1 Problem Reduce ^{6}/_{8}.   Procedure    Find any integer greater than 1 that can be divided evenly into both the numerator and denominator.
 2 divides into both.   Divide the numerator and denominator by the integer from Step 1.
 Numerator: 6 ÷ 2 = 3 Denominator: 8 ÷ 2 = 4 ^{6}/_{8} = ^{3}/_{4}  Can the result be reduced?  Can ^{3}/_{4} be reduced? No.  Solution ^{6}/_{8}.. = ^{3}/_{4}   Example 2 Problem Reduce ^{120}/_{350}.   Procedure    Find any integer greater than 1 that can be divided evenly into both the numerator and denominator.
 5 divides into both.   Divide the numerator and denominator by the integer from Step 1.
 Numerator: 120 ÷ 5 = 24 Denominator: 350 ÷ 5 = 70 ^{120}/_{350} =^{ 24}/_{70}  Can the result be reduced?  Can ^{24}/_{70} be reduced? Yes.   Repeat Steps 1 and 2 until the result cannot be further reduced.
 Numerator: 24 ÷ 2 = 12 Denominator: 70 ÷ 2 = 35 ^{24}/_{70} = ^{12}/_{35}  Can the result be reduced?  Can ^{12}/_{35} be further reduced? No  Solution ^{120}/_{350}. = ^{12}/_{35}   Examples and Exercises Reducing Proper Fractions Use these interactive examples and exercises to strengthen your understanding and build your skills:   Reducing by the Largest Common Factor (LCF) When you are doing arithmetic with fractions, it is often necessary to reduce fractions to their simplest and most direct form. A fraction is reduced by dividing both the numerator and denominator by the largest common factor (LCF). Definition The largest common factor is the largest integer that can be divided evenly into both the numerator and denominator.  Question:  Are there any integers, besides 1, that can be divided evenly into both the numerator and denominator of the fraction ^{8}/_{12}?  Answer:  Yes, 4 divides evenly into both the numerator and denominator. 8 ÷ 4 = 2 and 12 ÷ 4 = 3.    Question:  Are there any integers, besides 1, that can be divided evenly into both the numerator and denominator of the fraction ^{16}/_{64}?  Answer:  Yes, 16 divides evenly into both the numerator and denominator. 16 ÷ 16 = 1 and 64 ÷ 16 = 4.    Question:  Are there any integers, besides 1, that can be divided evenly into both the numerator and denominator of the fraction ^{3}/_{4}?  Answer:  No. There are no integers (other than 1) that divide evenly into both the numerator and denominator. So a fraction such as ^{3}/_{4} is already a reduced fraction.  More Examples  The LCF for ^{2}/_{10} is 2, because 2 is the largest integer that divides evenly into both 2 and 10.
 The LCF for ^{24}/_{32} is 8, because 8 is the largest integer that divides evenly into both 24 and 32.
 The LCF for ^{12}/_{64} is 4, because 4 is the largest integer that divides evenly into both 12 and 64.
Examples and Exercises Determining the Largest Common Factor (LCF) Use these interactive examples and exercises to strengthen your understanding and build your skills:  
