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1.19: Convert Between Types of Fractions

  • Page ID
    45757
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    Learning Outcomes

    • Identify different types of fractions and convert between them

    Andy and Bobby love pizza. On Tuesday night, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has \Large{\frac{1}{4}} of the pizza.

    An image of a round pizza sliced vertically and horizontally, creating four equal pieces. Each piece is labeled as one fourth.
    On Wednesday, the family invites some friends over for a pizza dinner. There are a total of 12 people. If they share the pizza equally, each person would get \Large{\frac{1}{12}} of the pizza.

    An image of a round pizza sliced into twelve equal wedges. Each piece is labeled as one twelfth.

    Fractions

    A fraction is written \Large{\frac{a}{b}}, where a and b are integers and b\ne 0. In a fraction, a is called the numerator and b is called the denominator.

    A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined.

    In the image below, the circle has been divided into three parts of equal size. Each part represents \Large{\frac{1}{3}} of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

    A circle is divided into three equal wedges. Each piece is labeled as one third.

    What does the fraction \Large{\frac{2}{3}} represent? The fraction \Large{\frac{2}{3}} means two of three equal parts.

    A circle is divided into three equal wedges. Two of the wedges are shaded.

    Watch the following video to see more examples of how to write fractions given a model.

    Thumbnail for the embedded element "Ex: Determine the Fraction Modeled"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=56

    Mixed Numbers and Improper Fractions

    What would happen if you have eight equal fifth pieces. You can use five of them to make one whole, but you’ll have three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, {\Large\frac{1}{5}}, so altogether you had eight fifths, which we can write as {\Large\frac{8}{5}}. The fraction {\Large\frac{8}{5}} is one whole, 1, plus three fifths, {\Large\frac{3}{5}}, or 1{\Large\frac{3}{5}}, which is read as one and three-fifths.

    The number 1{\Large\frac{3}{5}} is called a mixed number.

    Mixed Numbers

    A mixed number consists of a whole number a and a fraction {\Large\frac{b}{c}} where c\ne 0. It is written as follows.

    a{\Large\frac{b}{c}}\text{, }c\ne 0

    The number {\Large\frac{8}{5}} is called an improper fraction.

    Proper and Improper Fractions

    The fraction {\Large\frac{a}{b}} is a proper fraction if a<b and an improper fraction if a\ge b.

    Fractions such as {\Large\frac{5}{4}},{\Large\frac{3}{2}},{\Large\frac{5}{5}}, and {\Large\frac{7}{3}} are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as {\Large\frac{1}{2}},{\Large\frac{3}{7}}, and {\Large\frac{11}{18}} are proper fractions.

    Example

    Draw a figure to model {\Large\frac{11}{8}}.
    [reveal-answer q=”992194″]Show Answer[/reveal-answer]
    [hidden-answer a=”992194″]

    Solution:
    The denominator of the improper fraction is 8. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have 11 eighths. We must shade three of the eight parts of another circle.

    Two circles are shown, both divided into eight equal pieces. The circle on the left has all eight pieces shaded and is labeled as eight eighths. The circle on the right has three pieces shaded and is labeled as three eighths. The diagram indicates that eight eighths plus three eighths is one plus three eighths.
    So, {\Large\frac{11}{8}}=1{\Large\frac{3}{8}}.

    [/hidden-answer]

     

    Try it

    Draw a figure to model {\Large\frac{7}{6}}
    [reveal-answer q=”924546″]Show Answer[/reveal-answer]
    [hidden-answer a=”924546″]

    Two circles are shown. Each is divided into six sections. All of the first circle is shaded and one section of the second circle is shaded.

    [/hidden-answer]

     

    Draw a figure to model {\Large\frac{6}{5}}
    [reveal-answer q=”203648″]Show Answer[/reveal-answer]
    [hidden-answer a=”203648″]

    Two circles are shown. Each is divided into five sections. All of the first circle is shaded and one section of the second circle is shaded.

    [/hidden-answer]

     

    Example

    Name the improper fraction modeled. Then write the improper fraction as a mixed number.

    Two circles are shown, both divided into three equal pieces. The circle on the left has all three pieces shaded. The circle on the right has one piece shaded.

    [reveal-answer q=”670905″]Show Answer[/reveal-answer]
    [hidden-answer a=”670905″]

    Solution:
    Each circle is divided into three pieces, so each piece is {\Large\frac{1}{3}} of the circle. There are four pieces shaded, so there are four thirds or {\Large\frac{4}{3}}. The figure shows that we also have one whole circle and one third, which is 1{\Large\frac{1}{3}}. So, {\Large\frac{4}{3}}=1{\Large\frac{1}{3}}.[/hidden-answer]

    try it

    [ohm_question]145976[/ohm_question]

    [ohm_question]145977[/ohm_question]

    Example

    Use a model to rewrite the improper fraction {\Large\frac{11}{6}} as a mixed number.
    [reveal-answer q=”121702″]Show Answer[/reveal-answer]
    [hidden-answer a=”121702″]

    Solution:
    We start with 11 sixths \left({\Large\frac{11}{6}}\right). We know that six sixths makes one whole.

    {\Large\frac{6}{6}}=1

    That leaves us with five more sixths, which is {\Large\frac{5}{6}} (11 sixths minus 6 sixths is 5 sixths).

    So, {\Large\frac{11}{6}}=1{\Large\frac{5}{6}}

    Two circles are shown, both divided into six equal pieces. The circle on the left has all six pieces shaded and is labeled as six sixths. The circle on the right has five pieces shaded and is labeled as five sixths. Below the circles, it says one plus five sixths, then six sixths plus five sixths equals eleven sixths, and one plus five sixths equals one and five sixths. It then says that eleven sixths equals one and five sixths.

    [/hidden-answer]

    Try it

    [ohm_question]145982[/ohm_question]

    In the next video we show another way to draw a model that represents a fraction.  You will see example of both proper and improper fractions shown.

    Thumbnail for the embedded element "Draw Models of Fractions and Explain the Meaning of the Fraction"

    A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=56

    Example

    Use a model to rewrite the mixed number 1{\Large\frac{4}{5}} as an improper fraction.
    [reveal-answer q=”852331″]Show Answer[/reveal-answer]
    [hidden-answer a=”852331″]

    Solution:
    The mixed number 1{\Large\frac{4}{5}} means one whole plus four fifths. The denominator is 5, so the whole is {\Large\frac{5}{5}}. Together five fifths and four fifths equals nine fifths.
    So, 1{\Large\frac{4}{5}}={\Large\frac{9}{5}}

    Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as 5 fifths. The circle on the right has four pieces shaded and is labeled as 4 fifths. It then says that 5 fifths plus 4 fifths equals 9 fifths and that 9 fifths is equal to one plus 4 fifths.

    [/hidden-answer]

    Try it

    [ohm_question]145981[/ohm_question]

    There is another method to turning a mixed number into an improper fraction — it’s just a shortcut to what you’ve been practicing above.

    Mixed Numbers to Improper Fractions

    1. Multiply the whole number by the denominator
    2. Add that value to the numerator (this becomes the numerator of the improper fraction)
    3. Place the denominator of the mixed number in the denominator of the improper fraction

    EXAMPLE

    Convert 5\frac{2}{3} into an improper fraction using the shortcut

    1. Multiply the whole number by the denomimator 5\cdot{3}=15
    2. Add that value to the numerator 15+2=17
    new numerator for improper fraction \Large\frac{17}{?}
    3. Place the denominator of the mixed number in the denominator of the improper fraction \Large\frac{17}{3}

    TRY IT

    [ohm_question]156957[/ohm_question]

     

     

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