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Study Guides > Prealgebra

Locating and Ordering Fractions and Mixed Numbers on the Number Line

Learning Outcomes

  • Locate and label improper and proper fractions on a number line
  • Order fractions and mixed numbers on a number line
  • Use inequality symbols to compare fractions and mixed numbers
Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values. Doing the Manipulative Mathematics activity "Number Line Part [latex]3[/latex] " will help you develop a better understanding of the location of fractions on the number line. Let us locate [latex]\frac{1}{5},\frac{4}{5},3,3\frac{1}{3},\frac{7}{4},\frac{9}{2},5[/latex], and [latex]\frac{8}{3}[/latex] on the number line. We will start with the whole numbers [latex]3[/latex] and [latex]5[/latex] because they are the easiest to plot. A number line is shown with the numbers 3, 4, and 5. There are red dots at 3 and at 5. The proper fractions listed are [latex]\frac{1}{5}[/latex] and [latex]\frac{4}{5}[/latex]. We know proper fractions have values less than one, so [latex]\frac{1}{5}[/latex] and [latex]\frac{4}{5}[/latex] are located between the whole numbers [latex]0[/latex] and [latex]1[/latex]. The denominators are both [latex]5[/latex], so we need to divide the segment of the number line between [latex]0[/latex] and [latex]1[/latex] into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as [latex]\frac{1}{5},\frac{2}{5},\frac{3}{5}[/latex], and [latex]\frac{4}{5}[/latex]. Now plot points at [latex]\frac{1}{5}[/latex] and [latex]\frac{4}{5}[/latex]. A number line is shown. It shows 0, 1 fifth, 2 fifths, 3 fifths, 4 fifths, and 1. There are red dots at 1 fifth and at 4 fifths. The only mixed number to plot is [latex]3\frac{1}{3}[/latex]. Between what two whole numbers is [latex]3\frac{1}{3}?[/latex] Remember that a mixed number is a whole number plus a proper fraction, so [latex]3\frac{1}{3}>3[/latex]. Since it is greater than [latex]3[/latex], but not a whole unit greater, [latex]3\frac{1}{3}[/latex] is between [latex]3[/latex] and [latex]4[/latex]. We need to divide the portion of the number line between [latex]3[/latex] and [latex]4[/latex] into three equal pieces (thirds) and plot [latex]3\frac{1}{3}[/latex] at the first mark. A number line is shown with whole number 0 through 5. Between 3 and 4, 3 and 1 third and 3 and 2 thirds are labeled. There is a red dot at 3 and 1 third. Finally, look at the improper fractions [latex]\frac{7}{4},\frac{9}{2}[/latex], and [latex]\frac{8}{3}[/latex]. Locating these points will be easier if you change each of them to a mixed number. [latex-display]\frac{7}{4}=1\frac{3}{4},\frac{9}{2}=4\frac{1}{2},\frac{8}{3}=2\frac{2}{3}[/latex-display] Here is the number line with all the points plotted. A number line is shown with whole numbers 0 through 6. Between 0 and 1, 1 fifth and 4 fifths are labeled and shown with red dots. Between 1 and 2, 7 fourths is labeled and shown with a red dot. Between 2 and 3, 8 thirds is labeled and shown with a red dot. Between 3 and 4, 3 and 1 third is labeled and shown with a red dot. Between 4 and 5, 9 halves is labeled and shown with a red dot.

Example

Locate and label the following on a number line: [latex]\frac{3}{4},\frac{4}{3},\frac{5}{3},4\frac{1}{5}[/latex], and [latex]\frac{7}{2}[/latex]. Solution: Start by locating the proper fraction [latex]\frac{3}{4}[/latex]. It is between [latex]0[/latex] and [latex]1[/latex]. To do this, divide the distance between [latex]0[/latex] and [latex]1[/latex] into four equal parts. Then plot [latex]\frac{3}{4}[/latex]. A number line is shown. It shows 0, 1 fourth, 2 fourths, 3 fourths, and 1. There is a red dot at 3 fourths. Next, locate the mixed number [latex]4\frac{1}{5}[/latex]. It is between [latex]4[/latex] and [latex]5[/latex] on the number line. Divide the number line between [latex]4[/latex] and [latex]5[/latex] into five equal parts, and then plot [latex]4\frac{1}{5}[/latex] one-fifth of the way between [latex]4[/latex] and [latex]5[/latex] . A number line is shown. It shows 4, 4 and 1 fifth, 4 and 2 fifths, 4 and 3 fifths, 4 and 4 fifths, and 5. There is a red dot at 4 and 1 fifth. Now locate the improper fractions [latex]\frac{4}{3}[/latex] and [latex]\frac{5}{3}[/latex] . It is easier to plot them if we convert them to mixed numbers first. [latex-display]\frac{4}{3}=1\frac{1}{3},\frac{5}{3}=1\frac{2}{3}[/latex-display] Divide the distance between [latex]1[/latex] and [latex]2[/latex] into thirds. A number line is shown. It shows 1, 1 and 1 third, 1 and 2 thirds, and 2. Below 1 it says 3 thirds. Below 1 and 1 third it says 4 thirds. Below 1 and 2 thirds it says 5 thirds. Below 2 it says 6 thirds. There are red dots at 1 and 1 third and 1 and 2 thirds. Next let us plot [latex]\frac{7}{2}[/latex]. We write it as a mixed number, [latex]\frac{7}{2}=3\frac{1}{2}[/latex] . Plot it between [latex]3[/latex] and [latex]4[/latex]. A number line is shown. It shows 3, 3 and 1 half, and 4. Below 3 it says 6 halves. Below 3 and 1 half it says 7 halves. Below 4 it says 8 halves. There is a red dot at 3 and 1 half. The number line shows all the numbers located on the number line. A number line is shown. It shows the whole numbers 0 through 5. Between any 2 numbers are 10 tick marks. Between 0 and 1, between the 7th and 8th tick mark, 3 fourths is labeled and shown with a red dot. Between 1 and 2, 4 thirds and 5 thirds are labeled and shown with red dots. Between 3 and 4, 7 halves is labeled and shown with a red dot. Between 4 and 5, 4 and 1 fifth is labeled and shown with a red dot.
 

try it

#146007 [ohm_question height="270"]146007[/ohm_question] #146008 [ohm_question height="270"]146008[/ohm_question]
Watch the following video to see more examples of how to locate fractions on a number line. https://youtu.be/EIdmdTRQWTE In Introduction to Integers, we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of [latex]7[/latex] is [latex]-7[/latex] and the opposite of [latex]-7[/latex] is [latex]7[/latex]. A number line is shown. It shows the numbers negative 7, 0 and 7. There are red dots at negative 7 and 7. The space between negative 7 and 0 is labeled as 7 units. The space between 0 and 7 is labeled as 7 units. Fractions have opposites, too. The opposite of [latex]\frac{3}{4}[/latex] is [latex]-\frac{3}{4}[/latex]. It is the same distance from [latex]0[/latex] on the number line, but on the opposite side of [latex]0[/latex]. A number line is shown. It shows the numbers negative 1, negative 3 fourths, 0, 3 fourths, and 1. There are red dots at negative 3 fourths and 3 fourths. The space between negative 3 fourths and 0 is labeled as 3 fourths of a unit. The space between 0 and 3 fourths is labeled as 3 fourths of a unit. Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate [latex]-\frac{15}{8}[/latex] on the number line, first think of where [latex]\frac{15}{8}[/latex] is located. It is an improper fraction, so we first convert it to the mixed number [latex]1\frac{7}{8}[/latex] and see that it will be between [latex]1[/latex] and [latex]2[/latex] on the number line. So its opposite, [latex]-\frac{15}{8}[/latex], will be between [latex]-1[/latex] and [latex]-2[/latex] on the number line. A number line is shown. It shows the numbers negative 2, negative 1, 0, 1, and 2. Between negative 2 and negative 1, negative 1 and 7 eighths is labeled and marked with a red dot. The distance between negative 1 and 7 eighths and 0 is marked as 15 eighths units. Between 1 and 2, 1 and 7 eighths is labeled and marked with a red dot. The distance between 0 and 1 and 7 eighths is marked as 15 eighths units.

Example

Locate and label the following on the number line: [latex]\frac{1}{4},-\frac{1}{4},1\frac{1}{3},-1\frac{1}{3},\frac{5}{2}[/latex], and [latex]-\frac{5}{2}[/latex].

Answer: Solution: Draw a number line. Mark [latex]0[/latex] in the middle and then mark several units to the left and right. To locate [latex]\frac{1}{4}[/latex], divide the interval between [latex]0[/latex] and [latex]1[/latex] into four equal parts. Each part represents one-quarter of the distance. So plot [latex]\frac{1}{4}[/latex] at the first mark. A number line is shown. It shows the numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4. There are 4 tick marks between negative 1 and 0. There are 4 tick marks between 0 and 1. The first tick mark between 0 and 1 is labeled as 1 fourth and marked with a red dot. To locate [latex]-\frac{1}{4}[/latex], divide the interval between [latex]0[/latex] and [latex]-1[/latex] into four equal parts. Plot [latex]-\frac{1}{4}[/latex] at the first mark to the left of [latex]0[/latex]. A number line is shown. It shows the numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4. There are 4 tick marks between negative 1 and 0. There are 4 tick marks between 0 and 1. The first tick mark between 0 and 1 is labeled as 1 fourth and marked with a red dot. The first tick mark between 0 and negative 1 is labeled as negative 1 fourth and marked with a red dot. Since [latex]1\frac{1}{3}[/latex] is between [latex]1[/latex] and [latex]2[/latex], divide the interval between [latex]1[/latex] and [latex]2[/latex] into three equal parts. Plot [latex]1\frac{1}{3}[/latex] at the first mark to the right of [latex]1[/latex]. Then since [latex]-1\frac{1}{3}[/latex] is the opposite of [latex]1\frac{1}{3}[/latex] it is between [latex]-1[/latex] and [latex]-2[/latex]. Divide the interval between [latex]-1[/latex] and [latex]-2[/latex] into three equal parts. Plot [latex]-1\frac{1}{3}[/latex] at the first mark to the left of [latex]-1[/latex]. A number line is shown. The integers from negative 2 to 2 are labeled. Between negative 2 and negative 1, negative 1 and 1 third is labeled and marked with a red dot. Between 1 and 2, 1 and 1 third is labeled and marked with a red dot. To locate [latex]\frac{5}{2}[/latex] and [latex]-\frac{5}{2}[/latex], it may be helpful to rewrite them as the mixed numbers [latex]2\frac{1}{2}[/latex] and [latex]-2\frac{1}{2}[/latex]. Since [latex]2\frac{1}{2}[/latex] is between [latex]2[/latex] and [latex]3[/latex], divide the interval between [latex]2[/latex] and [latex]3[/latex] into two equal parts. Plot [latex]\frac{5}{2}[/latex] at the mark. Then since [latex]-2\frac{1}{2}[/latex] is between [latex]-2[/latex] and [latex]-3[/latex], divide the interval between [latex]-2[/latex] and [latex]-3[/latex] into two equal parts. Plot [latex]-\frac{5}{2}[/latex] at the mark. A number line is shown. The integers from negative 4 to 4 are labeled. Between negative 3 and negative 2, negative 5 halves is labeled and marked with a red dot. Between 2 and 3, 5 halves is labeled and marked with a red dot.

 

Try it

#146009 [ohm_question height="270"]146009[/ohm_question] #146011 [ohm_question height="270"]146011[/ohm_question]
In teh next video we give more examples of how to locate negative and positive fractions on a number line. https://youtu.be/nbSlAAZVQV4

Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that [latex]a>b[/latex] means that [latex]a[/latex] is to the right of [latex]b[/latex] on the number line. As we move from left to right on a number line, the values increase.

Example

Order each of the following pairs of numbers, using [latex]<[/latex]; or [latex]>:[/latex]
  1. [latex]-\frac{2}{3}[/latex] __ [latex]- 1[/latex]
  2. [latex]-3\frac{1}{2}[/latex] __ [latex]- 3[/latex]
  3. [latex]-\frac{3}{7}-[/latex] __ [latex]\frac{3}{8}[/latex]
  4. [latex]-2[/latex] __ [latex]\frac{-16}{9}[/latex]

Answer: Solution: 1. [latex]-\frac{2}{3}>-1[/latex] A number line is shown. The integers from negative 3 to 3 are labeled. Negative 1 is marked with a red dot. Between negative 1 and 0, negative 2 thirds is labeled and marked with a red dot. 2. [latex]-3\frac{1}{2}<-3[/latex] A number line is shown. The integers from negative 4 to 4 are labeled. There is a red dot at negative 3. Between negative 4 and negative 3, negative 3 and one half is labeled and marked with a red dot. 3. [latex]-\frac{3}{7}\text{<}-\frac{3}{8}[/latex] A number line is shown. The numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3 are labeled. Between negative 1 and 0, negative 3 sevenths and negative 3 eighths are labeled and marked with red dots. 4. [latex]-2<\frac{-16}{9}[/latex] A number line is shown. The numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3 are labeled. There is a red dot at negative 2. Between negative 2 and negative 1, negative 16 over 9 is labeled and marked with a red dot.

Try it

#146013 [ohm_question height="270"]146013[/ohm_question] #146012 [ohm_question height="270"]146012[/ohm_question]
In the following video we show another example of how to order integers, fractions and mixed numbers using inequality symbols. https://youtu.be/Phsf_fJgerc

Licenses & Attributions

CC licensed content, Original

  • Question ID: 146007, 146008, 146009, 146011, 146012, 146013. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

CC licensed content, Shared previously

  • Ex: Identify a Fraction on a Number Line. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Determine Negative Proper and Improper Fractions on the Number Line. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.

CC licensed content, Specific attribution