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Mathematics LibreTexts

4.1: Visualize Fractions (Part 1)

  • Page ID
    4991
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    Skills to Develop

    • Understand the meaning of fractions
    • Model improper fractions and mixed numbers
    • Convert between improper fractions and mixed numbers
    • Model equivalent fractions
    • Find equivalent fractions
    • Locate fractions and mixed numbers on the number line
    • Order fractions and mixed numbers

    be prepared!

    Before you get started, take this readiness quiz.

    1. Simplify: \(5 • 2 + 1\). If you missed this problem, review Example 2.1.8.
    2. Fill in the blank with \(<\) or \(>\): \(−2\)__\(−5\). If you missed this problem, review Example 3.1.2.

    Understand the Meaning of Fractions

    Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts. In math, we write \(\frac{1}{2}\) to mean one out of two parts.

    An image of a round pizza sliced vertically down the center, creating two equal pieces. Each piece is labeled as one half.

    Figure \(\PageIndex{1}\)

    On Tuesday, 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 \(\frac{1}{4}\) of the pizza.

    An image of a round pizza sliced vertically and horizontally, creating four equal pieces. Each piece is labeled as one fourth.

    Figure \(\PageIndex{2}\)

    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 \(\frac{1}{12}\) of the pizza.

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

    Figure \(\PageIndex{3}\)

    Definition: Fractions

    A fraction is written \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b ≠ 0\). 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 Figure \(\PageIndex{4}\), the circle has been divided into three parts of equal size. Each part represents \(\frac{1}{3}\) 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.

    Figure \(\PageIndex{4}\)

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

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

    Figure \(\PageIndex{5}\)

    Example \(\PageIndex{1}\): name the fraction

    Name the fraction of the shape that is shaded in each of the figures.

    1. In part “a”, a circle is divided into eight equal wedges. Five of the wedges are shaded. In part “b”, a square is divided into nine equal pieces. Two of the pieces are shaded.
    2. In part “a”, a circle is divided into eight equal wedges. Five of the wedges are shaded. In part “b”, a square is divided into nine equal pieces. Two of the pieces are shaded.

    Solution

    We need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator.

    How many equal parts are there? There are eight equal parts.
    How many are shaded? Five parts are shaded.

    Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is \(\frac{5}{8}\).

    How many equal parts are there? There are nineequal parts.
    How many are shaded? Two parts are shaded.

    Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is \(\frac{2}{9}\).

    Exercise \(\PageIndex{1}\)

    Name the fraction of the shape that is shaded in each figure:

    In part “a”, a circle is divided into eight equal wedges. Three of the wedges are shaded. In part “b”, a square is divided into nine equal pieces. Four of the pieces are shaded.

    Answer a

    \(\frac{3}{8}\)

    Answer b

    \(\frac{4}{9}\)

    Exercise \(\PageIndex{2}\)

    Name the fraction of the shape that is shaded in each figure:

    In part “a”, a circle is divided into five equal wedges. Three of the wedges are shaded. In part “b”, a square is divided into four equal pieces. Three of the pieces are shaded.

    Answer a

    \(\frac{3}{5}\)

    Answer b

    \(\frac{3}{4}\)

    Example \(\PageIndex{2}\):

    Shade \(\frac{3}{4}\) of the circle.

    An image of a circle.

    Solution

    The denominator is \(4\), so we divide the circle into four equal parts (a). The numerator is \(3\), so we shade three of the four parts (b).

    In “a”, a circle is shown divided into four equal pieces. An arrow points from “a” to “b”. In “b”, the same image is shown with three of the pieces shaded.

    \(\frac{3}{4}\) of the circle is shaded.

    Exercise \(\PageIndex{3}\)

    Shade \(\frac{6}{8}\) of the circle.

    A circle is divided into eight equal pieces.

    Answer

    Exercise 4.1.3.png

    Exercise \(\PageIndex{4}\)

    Shade \(\frac{2}{5}\) of the rectangle.

    A rectangle is divided vertically into five equal pieces.

    Answer

    Exercise 4.1.4.png

    In Example \(\PageIndex{1}\) and Example \(\PageIndex{2}\), we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in Figure \(\PageIndex{6}\). Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.

    One long, undivided rectangular tile is shown, labeled “1”. Below it is a rectangular tile of the same size and shape that has been divided vertically into two equal pieces, each labeled as one half. Below that is another rectangular tile that has been divided into three equal pieces, each labeled as one third. Below that is another rectangular tile that has been divided into four equal pieces, each labeled as one fourth. Below that is another rectangular tile that has been divided into six pieces, each labeled as one sixth.

    Figure \(\PageIndex{6}\)

    We’ll be using fraction tiles to discover some basic facts about fractions. Refer to Figure \(\PageIndex{6}\) to answer the following questions:

    How many \(\frac{1}{2}\) tiles does it take to make one whole tile? It takes two halves to make a whole, so two out of two is \(\frac{2}{2}\) = 1.
    How many \(\frac{1}{3}\) tiles does it take to make one whole tile? It takes three thirds, so three out of three is \(\frac{3}{3}\) = 1.
    How many \(\frac{1}{4}\) tiles does it take to make one whole tile? It takes four fourths, so four out of four is \(\frac{4}{4}\) = 1.
    How many \(\frac{1}{5}\) tiles does it take to make one whole tile? It takes six sixths, so six out of six is \(\frac{6}{6}\) = 1.
    What if the whole were divided into 24 equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many \(\frac{1}{24}\) tiles does it take to make one whole tile? It takes 24 twenty-fourths, so \(\frac{24}{24}\) = 1.

    It takes \(24\) twenty-fourths, so \(\frac{24}{24} = 1\). This leads us to the Property of One.

    Definition: Property of One

    Any number, except zero, divided by itself is one.

    \[\frac{a}{a} = 1 \qquad \qquad (a \neq 0)\]

    Example \(\PageIndex{3}\): fraction circles to form wholes

    Use fraction circles to make wholes using the following pieces:

    1. \(4\) fourths
    2. \(5\) fifths
    3. \(6\) sixths

    Solution

    Three circles are shown. The circle on the left is divided into four equal pieces. The circle in the middle is divided into five equal pieces. The circle on the right is divided into six equal pieces. Each circle says “Form 1 whole” beneath it.

    Exercise \(\PageIndex{5}\)

    Use fraction circles to make wholes with the following pieces: \(3\) thirds.

    Answer

    Exercise 4.1.5.png

    Exercise \(\PageIndex{6}\)

    Use fraction circles to make wholes with the following pieces: \(8\) eighths.

    Answer

    Exercise 4.1.6.png

    What if we have more fraction pieces than we need for \(1\) whole? We’ll look at this in the next example.

    Example \(\PageIndex{4}\): fraction circles to form whole

    Use fraction circles to make wholes using the following pieces:

    1. \(3\) halves
    2. \(8\) fifths
    3. \(7\) thirds

    Solution

    1. \(3\) halves make \(1\) whole with \(1\) half left over.

    Two circles are shown, both divided into two equal pieces. The circle on the left has both pieces shaded and is labeled as “1”. The circle on the right has one piece shaded and is labeled as one half.

    1. \(8\) fifths make \(1\) whole with \(3\) fifths left over.

    Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as “1”. The circle on the right has three pieces shaded and is labeled as three fifths.

    1. \(7\) thirds make \(2\) wholes with \(1\) third left over.

    Three circles are shown, all divided into three equal pieces. The two circles on the left have all three pieces shaded and are labeled with ones. The circle on the right has one piece shaded and is labeled as one third.

    Exercise \(\PageIndex{7}\)

    Use fraction circles to make wholes with the following pieces: \(5\) thirds.

    Answer

    Exercise 4.1.7.png

    Exercise \(\PageIndex{8}\)

    Use fraction circles to make wholes with the following pieces: \(5\) halves.

    Answer

    Exercise 4.1.8.png

    Model Improper Fractions and Mixed Numbers

    In Example \(\PageIndex{4b}\), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, \(\frac{1}{5}\), so altogether you had eight fifths, which we can write as \(\frac{8}{5}\). The fraction \(\frac{8}{5}\) is one whole, \(1\), plus three fifths, \(\frac{3}{5}\), or \(1 \frac{3}{5}\), which is read as one and three-fifths.

    The number \(1 \frac{3}{5}\) is called a mixed number. A mixed number consists of a whole number and a fraction.

    Definition: Mixed Numbers

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

    \[a \frac{b}{c} \qquad \qquad c \neq 0\]

    Fractions such as \(\frac{5}{4}\), \(\frac{3}{2}\), \(\frac{5}{5}\), and \(\frac{7}{3}\) 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 \(\frac{1}{2}\), \(\frac{3}{7}\), and \(\frac{11}{18}\) are proper fractions.

    Definition: Proper and Improper Fractions

    The fraction \(\frac{a}{b}\) is a proper fraction if \(a < b\) and an improper fraction if \(a ≥ b\).

    Example \(\PageIndex{5}\): improper fraction

    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.

    Solution

    Each circle is divided into three pieces, so each piece is \(\frac{1}{3}\) of the circle. There are four pieces shaded, so there are four thirds or \(\frac{4}{3}\). The figure shows that we also have one whole circle and one third, which is \(1 \frac{1}{3}\). So, \(\frac{4}{3} = 1 \frac{1}{3}\).

    Exercise \(\PageIndex{9}\)

    Name the improper fraction. Then write it 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 two pieces shaded.

    Answer

    \(\frac{5}{3}=1\frac{2}{3}\)

    Exercise \(\PageIndex{10}\)

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

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

    Answer

    \(\frac{13}{8}=1\frac{5}{8}\)

    Example \(\PageIndex{6}\): model a fraction

    Draw a figure to model \(\frac{11}{8}\).

    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, \(\frac{11}{8} = 1 \frac{3}{8}\).

    Exercise\(\PageIndex{11}\)

    Draw a figure to model \(\frac{7}{6}\).

    Answer

    Exercise 4.1.11.png

    Exercise \(\PageIndex{12}\)

    Draw a figure to model \(\frac{6}{5}\).

    Answer

    Exercise 4.1.12.png

    Example \(\PageIndex{7}\): model a fraction

    Use a model to rewrite the improper fraction \(\frac{11}{6}\) as a mixed number.

    Solution

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

    \[\frac{6}{6} = 1 \nonumber \]

    That leaves us with five more sixths, which is \(\frac{5}{6}\) (11 sixths minus 6 sixths is 5 sixths). So, \(\frac{11}{6} = 1 \frac{5}{6}\).

    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.

    Exercise \(\PageIndex{13}\)

    Use a model to rewrite the improper fraction as a mixed number: \(\frac{9}{7}\).

    Answer

    \(1\frac{2}{7}\)

    Exercise \(\PageIndex{14}\)

    Use a model to rewrite the improper fraction as a mixed number: \(\frac{7}{4}\).

    Answer

    \(1\frac{3}{4}\)

    Example \(\PageIndex{8}\): model a fraction

    Use a model to rewrite the mixed number \(1 \frac{4}{5}\) as an improper fraction.

    Solution

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

    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.

    Exercise \(\PageIndex{15}\)

    Use a model to rewrite the mixed number as an improper fraction: \(1 \frac{3}{8}\).

    Answer

    \(\frac{11}{8}\)

    Exercise \(\PageIndex{16}\)

    Use a model to rewrite the mixed number as an improper fraction: \(1 \frac{5}{6}\).

    Answer

    \(\frac{11}{6}\)

    Convert between Improper Fractions and Mixed Numbers

    In Example \(\PageIndex{7}\), we converted the improper fraction \(\frac{11}{6}\) to the mixed number \(1 \frac{5}{6}\) using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the \(11\) pieces were left. We saw that \(\frac{11}{6}\) made one whole group of six sixths plus five more sixths, showing that \(\frac{11}{6} = \frac{15}{6}\).

    The division expression \(\frac{11}{6}\) (which can also be written as \(6 \overline{\smash{)}11}\)) tells us to find how many groups of \(6\) are in \(11\). To convert an improper fraction to a mixed number without fraction circles, we divide.

    Example \(\PageIndex{9}\):

    Convert \(\frac{11}{6}\) to a mixed number.

    Solution

    Divide the denominator into the numerator. Remember \(\frac{11}{6}\) means 11 ÷ 6.
    Identify the quotient, remainder and divisor.
    Write the mixed number as \(quotient \frac{remainder}{divisor}\). \(1 \frac{5}{6}\)

    So, \(\frac{11}{6} = 1 \frac{5}{6}\).

    Exercise \(\PageIndex{17}\)

    Convert the improper fraction to a mixed number: \(\frac{13}{7}\).

    Answer

    \(1\frac{6}{7}\)

    Exercise \(\PageIndex{18}\)

    Convert the improper fraction to a mixed number: \(\frac{14}{9}\).

    Answer

    \(1\frac{5}{9}\)

    HOW TO: CONVERT AN IMPROPER FRACTION TO A MIXED NUMBER

    Step 1. Divide the denominator into the numerator.

    Step 2. Identify the quotient, remainder, and divisor.

    Step 3. Write the mixed number as \(quotient \frac{remainder}{divisor}\).

    Example \(\PageIndex{10}\):

    Convert the improper fraction \(\frac{33}{8}\) to a mixed number.

    Solution

    Divide the denominator into the numerator. Remember, \(\frac{33}{8}\) means \(8 \overline{\smash{)}33}\).
    Identify the quotient, remainder, and divisor.
    Write the mixed number as \(quotient \frac{remainder}{divisor}\). \(4 \frac{1}{8}\)

    So, \(\frac{33}{8} = 4 \frac{1}{8}\).

    Exercise \(\PageIndex{19}\)

    Convert the improper fraction to a mixed number: \(\frac{23}{7}\).

    Answer

    \(3\frac{2}{7}\)

    Exercise \(\PageIndex{20}\)

    Convert the improper fraction to a mixed number: \(\frac{48}{11}\).

    Answer

    \(4\frac{4}{11}\)

    In Example \(\PageIndex{8}\), we changed \(1 \frac{4}{5}\) to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.

    \[\frac{5}{5} + \frac{4}{5} = \frac{9}{5} \nonumber \]

    Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction.

    Example \(\PageIndex{11}\): convert

    Convert the mixed number \(4 \frac{2}{3}\) to an improper fraction.

    Multiply the whole number by the denominator. \(4 \frac{2}{3}\)
    The whole number is 4 and the denominator is 3.
    Simplify.
    Add the numerator to the product.
    The numerator of the mixed number is 2.
    Simplify.
    Write the final sum over the original denominator.
    The denominator is 3. \(\frac{14}{3}\)

    Exercise \(\PageIndex{21}\)

    Convert the mixed number to an improper fraction: \(3 \frac{5}{7}\).

    Answer

    \(\frac{26}{7}\)

    Exercise \(\PageIndex{22}\)

    Convert the mixed number to an improper fraction: \(2 \frac{7}{8}\).

    Answer

    \(\frac{23}{8}\)

    HOW TO: CONVERT A MIXED NUMBER TO AN IMPROPER FRACTION

    Step 1. Multiply the whole number by the denominator.

    Step 2. Add the numerator to the product found in Step 1.

    Step 3. Write the final sum over the original denominator.

    Example \(\PageIndex{12}\):

    Convert the mixed number \(10 \frac{2}{7}\) to an improper fraction.

    Multiply the whole number by the denominator. \(10 \frac{2}{7}\)
    The whole number is 10 and the denominator is 7.
    Simplify.
    Add the numerator to the product.
    The numerator of the mixed number is 2.
    Simplify.
    Write the final sum over the original denominator.
    The denominator is 7. \(\frac{72}{7}\)

    Exercise \(\PageIndex{23}\)

    Convert the mixed number to an improper fraction: \(4 \frac{6}{11}\).

    Answer

    \(\frac{50}{11}\)

    Exercise \(\PageIndex{24}\)

    Convert the mixed number to an improper fraction: \(11 \frac{1}{3}\).

    Answer

    \(\frac{34}{3}\)

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