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Locate and label each of the given fractions on a number line:

2 3 , 2 3 , 2 1 4 , −2 1 4 , 3 2 , 3 2


A number line is shown. The integers from negative 5 to 5 are labeled. Between negative 3 and negative 2, negative 2 and 1 fourth is labeled and marked with a red dot. Between negative 2 and negative 1, negative 3 halves is labeled and marked with a red dot. Between negative 1 and 0, negative 2 thirds is labeled and marked with a red dot. Between 0 and 1, 2 thirds is labeled and marked with a red dot. Between 1 and 2, 3 halves is labeled and marked with a red dot. Between 2 and 3, 2 and 1 fourth is labeled and marked with a red dot.

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Locate and label each of the given fractions on a number line:

3 4 , 3 4 , 1 1 2 , −1 1 2 , 7 3 , 7 3


A number line is shown. The integers from negative 5 to 5 are labeled. Between negative 3 and negative 2, negative 7 thirds is labeled and marked with a red dot. Between negative 2 and negative 1, negative 1 and 1 half is labeled and marked with a red dot. Between negative 1 and 0, negative 3 fourths is labeled and marked with a red dot. Between 0 and 1, 3 fourths is labeled and marked with a red dot. Between 1 and 2, 1 and 1 half is labeled and marked with a red dot. Between 2 and 3, 7 thirds is labeled and marked with a red dot.

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Order fractions and mixed numbers

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

Order each of the following pairs of numbers, using < or >:

  1. 2 3 ____ −1
  2. −3 1 2 ____ −3
  3. 3 7 ____ 3 8
  4. −2 ____ −16 9

Solution

2 3 > −1

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.

−3 1 2 < −3

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 7 < 3 8

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.

−2 < −16 9

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.
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Order each of the following pairs of numbers, using < or >:

  1. 1 3 __ 1
  2. −1 1 2 __ 2
  3. 2 3 __ 1 3
  4. −3 __ 7 3

  1. >
  2. >
  3. <
  4. <

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Order each of the following pairs of numbers, using < or >:

  1. −3 __ 17 5
  2. −2 1 4 __ −2
  3. 3 5 __ 4 5
  4. −4 __ 10 3

  1. >
  2. <
  3. >
  4. <

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Key concepts

  • Property of One
    • Any number, except zero, divided by itself is one.
      a a = 1 , where a 0 .
  • Mixed Numbers
    • A mixed number consists of a whole number a and a fraction b c where c 0 .
    • It is written as follows: a b c c 0
  • Proper and Improper Fractions
    • The fraction a b is a proper fraction if a < b and an improper fraction if a b .
  • Convert an improper fraction to a mixed number.
    1. Divide the denominator into the numerator.
    2. Identify the quotient, remainder, and divisor.
    3. Write the mixed number as quotient remainder divisor .
  • Convert a mixed number to an improper fraction.
    1. Multiply the whole number by the denominator.
    2. Add the numerator to the product found in Step 1.
    3. Write the final sum over the original denominator.
  • Equivalent Fractions Property
    • If a, b, and c are numbers where b 0 , c 0 , then a b = a c b c .

Practice makes perfect

In the following exercises, name the fraction of each figure that is shaded.

In part “a”, a circle is divided into 4 equal pieces. 1 piece is shaded. In part “b”, a circle is divided into 4 equal pieces. 3 pieces are shaded. In part “c”, a circle is divided into 8 equal pieces. 3 pieces are shaded. In part “d”, a circle is divided into 8 equal pieces. 5 pieces are shaded.

  1. 1 4
  2. 3 4
  3. 3 8
  4. 5 9

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In the following exercises, shade parts of circles or squares to model the following fractions.

In the following exercises, use fraction circles to make wholes, if possible, with the following pieces.

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

In part “a”, two circles are shown. Each is divided into 4 equal pieces. The circle on the left has all 4 pieces shaded. The circle on the right has 1 piece shaded. In part “b”, two circles are shown. Each is divided into 4 equal pieces. The circle on the left has all 4 pieces shaded. The circle on the right has 3 pieces shaded. In part “c”, two circles are shown. Each is divided into 8 equal pieces. The circle on the left has all 8 pieces shaded. The circle on the right has 3 pieces shaded.

  1. 5 4 = 1 1 4
  2. 7 4 = 1 3 4
  3. 11 8 = 1 3 8

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In part “a”, 3 circles are shown. Each is divided into 4 equal pieces. The first two circles have all 4 pieces shaded. The third circle has 3 pieces shaded. In part “b”, 3 circles are shown. Each is divided into 8 equal pieces. The first two circles have all 8 pieces shaded. The third circle has 3 pieces shaded.

  1. 11 4 = 2 3 4
  2. 19 8 = 2 3 8

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In the following exercises, draw fraction circles to model the given fraction.

In the following exercises, rewrite the improper fraction as a mixed number.

In the following exercises, rewrite the mixed number as an improper fraction.

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

How many sixths equal one-third?

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How many twelfths equal one-third?

4

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How many eighths equal three-fourths?

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How many twelfths equal three-fourths?

9

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How many fourths equal three-halves?

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How many sixths equal three-halves?

9

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In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1 3

Answers may vary. Correct answers include 2 6 , 3 9 , 4 12 .

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5 6

Answers may vary. Correct answers include 10 12 , 15 18 , 20 24 .

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5 9

Answers may vary. Correct answers include 10 18 , 15 27 , 20 36 .

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In the following exercises, plot the numbers on a number line.

3 4 , 3 4 , 1 2 3 , −1 2 3 , 5 2 , 5 2

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2 5 , 2 5 , 1 3 4 , −1 3 4 , 8 3 , 8 3


A number line is shown. The numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, and 4 are labeled. Between negative 3 and negative 2, negative 8 thirds is labeled and shown with a red dot. Between negative 2 and negative 1, negative 1 and 3 fourths is labeled and shown with a red dot. Between negative 1 and 0, negative 2 fifths is labeled and shown with a red dot. Between 0 and 1, 2 fifths is labeled and shown with a red dot. Between 1 and 2, 1 and 3 fourths is labeled and shown with a red dot. Between 2 and 3, 8 thirds is labeled and shown with a red dot.

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In the following exercises, order each of the following pairs of numbers, using < or >.

Everyday math

Music Measures A choreographed dance is broken into counts. A 1 1 count has one step in a count, a 1 2 count has two steps in a count and a 1 3 count has three steps in a count. How many steps would be in a 1 5 count? What type of count has four steps in it?

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Music Measures Fractions are used often in music. In 4 4 time, there are four quarter notes in one measure.

  1. How many measures would eight quarter notes make?
  2. The song “Happy Birthday to You” has 25 quarter notes. How many measures are there in “Happy Birthday to You?”

  1. 8
  2. 4

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Baking Nina is making five pans of fudge to serve after a music recital. For each pan, she needs 1 2 cup of walnuts.

  1. How many cups of walnuts does she need for five pans of fudge?
  2. Do you think it is easier to measure this amount when you use an improper fraction or a mixed number? Why?

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Writing exercises

Give an example from your life experience (outside of school) where it was important to understand fractions.

Answers will vary.

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Explain how you locate the improper fraction 21 4 on a number line on which only the whole numbers from 0 through 10 are marked.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

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Brian Reply
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Damian
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LITNING Reply
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
what king of growth are you checking .?
Renato
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Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
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Devang Reply
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s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Practice Key Terms 4

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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