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Fractions Review (page 2 of 5)

Sections: Reducing fractions, Mixed numbers and improper fractions, Multiplying and dividing fractions, Adding and subtracting fractions, Adding polynomial fractions

If you have a big pizza party and have one pineapple pizza and half an anchovy pizza left over afterward, you would say that you have "one and a half" pizzas. "One and a half" is the standard spoken-English way of expressing this number, and it is written as "11/2". This symbol, "11/2", is called a "mixed number", because it combines the "regular" number "1" with the fraction "1/2".

While mixed numbers are the natural choice for spoken English (and are therefore well-suited to the answers of word problems), they aren't generally the easiest fractions to compute with. In algebra, you will almost always prefer that your fractions not be mixed numbers. Instead, you will use "improper fractions", or fractions where the top number is bigger than the bottom number. The standard way to convert a mixed number to an improper fraction is to multiply the bottom number by the "regular" number, add in the top number, and then put this on top of the bottom number as a new fraction. For instance, to convert 11/2 to an improper fraction, you do the following:

    1 + 1/2 = 3/2

I multiplied the bottom 2 by the "regular" 1, and then added in the 1 from on top, getting 3. Then I put this 3 on top of the 2 from underneath.   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

  • Convert 2 + 3/16 to an improper fraction.
    • 2 + 3/16 = 35/16

  • Convert 6 + 2/5 to an improper fraction.
    • 6 + 2/5 = 32/5

To go from an improper fraction to a mixed number, you do the long division. Remember that a fraction is just division. Divide the top number by the bottom number. Whatever you get on top of the division symbol is your "regular" number. Whatever your remainder is, put that number on top of the number you divided by. (To convert to mixed numbers, don't use decimals. Just find the quotient and the remainder. Then stop.)




  • Convert 45/4 to a mixed number.

    First, I do the long division to find the whole-number part (being the quotient) and the remainder:

      45 ÷ 4 = 11, with remainder 1

    Since the remainder is 1 and I'm dividing by 4, the fractional part will be 1/4.

      45/4 = 11 + 1/4

This procedure works perfectly well on rational expressions (polynomial fractions):

  • Convert (x^2+3x+1)/(x+2) to a mixed number.

    First, do the long division to find the regular polynomial part and the remainder:

      ( x^2+3x+1 ) ÷ ( x+2 ) = x + 1, with remainder -1

    The polynomial on top is "x + 1" and the remainder is –1. Since you're dividing by "x + 2", the fractional part will be "(–1)/(x + 2)":

      (x^2+3x+1)/(x+2) = x + 1 + (-1)/(x+2)

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Cite this article as:

Stapel, Elizabeth. "Fractions Review: Mixed Numbers and Improper Fractions." Purplemath. Available from Accessed



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