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Chapter 3—Fractions
34 Converting Between Improper Fractions and Mixed Numbers
When you complete the work for this section, you should be able to:  Convert any improper fraction to a mixed number and reduce the fraction where necessary.
 Convert any mixed number to an improper fraction.

You are getting ready to do some basic arithmetic operationsaddition, subtraction, multiplication, and divisionwith fractions and mixed numbers. These operations often require you to convert between improper fractions and mixed numbers. To begin, here is a review of the basic definition.
Definitions  A proper fraction is one where the numerator is smaller than the denominator.
Examples: ^{1}/_{2}, ^{1}/_{3}, ^{2}/_{3}, ^{5}/_{8}  An improper fraction is one where the numerator is greater than, or equal to, the denominator.
Examples: ^{3}/_{2}, ^{8}/_{3}, ^{16}/_{5}, ^{7}/_{7}  A mixed fraction is one that includes an integer as well as a fractional part.
Examples: 1^{1}/_{2}, 2 ^{3}/_{4}, 6 ^{5}/_{8}, 4^{ 1}/_{4} 
Examples and Exercises
Identifying Improper Fractions Use these interactive examples and exercises to strengthen your understanding and build your skills:  
Converting Improper Fractions to Mixed Numbers
Arithmetic operations with fractions often result in fractions in an improper form. You should finish the work be converting this answer to a proper proper fractions (and reducing if possible). Suppose an addition operation results in an improper fraction such as ^{11}/_{3}. Converting to a mixed number, the answer becomes 3^{ 2}/_{3}. This section describes how to make this important kind of conversion.
Procedure Converting improper fractions to mixed numbers Step 1: Divide the denominator into the numerator. Use ordinary wholenumber division that produces a quotient and a remainder. Step 2: Assemble the mixed number.  The wholenumber part of the mixed number is the wholenumber part of the quotient from Step 1.
 The numerator of the fraction part of the mixed number is the remainder from the quotient in Step 1.
 The denominator of the fraction part of the mixed number is the denominator of the original improper fraction.
 Note: If there is no remainder in Step 1, then the mixednumber part is a whole number. There is no fraction part. Example: ^{10}/_{5} = 2  Steps for converting an improper fraction to a proper mixed number. 
Examples
Sometimes the fractional part of these conversions needs to be reduced.
Example: Convert ^{12}/_{8} to a mixed number.
 Doing the division: ^{12}/_{8} = 1 R 4
 Assembling the mixed number: ^{12}/_{8} = 1 ^{4}/_{8}
 Reducing the fraction: 1 ^{4}/_{8} = 1 ^{1}/_{2}
Examples and Exercises
Converting Improper Fractions to Mixed Numbers Use these interactive examples and exercises to strengthen your understanding and build your skills:  
Converting Mixed Numbers to Improper Fractions
Converting mixed numbers to improper fractions is often necessary for setting up arithmetic operations with fractions.
Procedure Converting mixed numbers to improper fractions. Step 1:  Multiply the whole number by the denominator and add the numerator. This becomes the numerator of the improper fraction.  Step 2:  Set the denominator of the improper fraction equal to the denominator of the fraction in the mixed number.   Note: A whole number can be expressed as an improper fraction by putting that number over 1. Example: 14 = ^{14}/_{1}  
Example
Problem Convert the mixed number 3 ^{1}/_{2} to an improper fraction  
Procedure  
 Multiply the whole number times the denominator of the fraction, and assign the result to the numerator of the improper fraction and use the original denominator.
 3 x 2 + 1 = 7 3 ^{1}/_{2} = ^{7}/_{?} 
 Set the denominator of the improper fraction equal to the denominator of the fraction in the mixed number.
 3 ^{1}/_{2} = ^{7}/_{2} 
Solution 3 ^{1}/_{2} = ^{7}/_{2}  
Examples and Exercises
Converting Mixed Numbers to Improper Fractions Use these interactive examples and exercises to strengthen your understanding and build your skills:  