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−1 −3 = 1 3 negative negative = positive

Placement of negative sign in a fraction

For any positive numbers a and b ,

a b = a b = a b

Which of the following fractions are equivalent to 7 −8 ?

−7 −8 , −7 8 , 7 8 , 7 8

Solution

The quotient of a positive and a negative is a negative, so 7 −8 is negative. Of the fractions listed, −7 8 and 7 8 are also negative.

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Which of the following fractions are equivalent to −3 5 ?

−3 −5 , 3 5 , 3 5 , 3 −5

3 5 , 3 −5

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Which of the following fractions are equivalent to 2 7 ?

−2 −7 , −2 7 , 2 7 , 2 −7

−2 7 , 2 −7

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Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, 4 + 8 5 3 means ( 4 + 8 ) ÷ ( 5 3 ) . The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.

We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

Grouping symbols


Parentheses, brackets, braces, an absolute value sign, and a fraction bar are shown.

Simplify an expression with a fraction bar.

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Simplify the fraction.

Simplify: 4 + 8 5 3 .

Solution

4 + 8 5 3
Simplify the expression in the numerator. 12 5 3
Simplify the expression in the denominator. 12 2
Simplify the fraction. 6
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Simplify: 4 + 6 11 2 .

10 9

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Simplify: 3 + 5 18 2 .

1 2

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Simplify: 4 2 ( 3 ) 2 2 + 2 .

Solution

4 2 ( 3 ) 2 2 + 2
Use the order of operations. Multiply in the numerator and use the exponent in the denominator. 4 6 4 + 2
Simplify the numerator and the denominator. −2 6
Simplify the fraction. - 1 3
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Simplify: 6 3 ( 5 ) 3 2 + 3 .

−3 4

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Simplify: 4 4 ( 6 ) 3 3 + 3 .

2 3

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Simplify: ( 8 4 ) 2 8 2 4 2 .

Solution

( 8 4 ) 2 8 2 4 2
Use the order of operations (parentheses first, then exponents). ( 4 ) 2 64 16
Simplify the numerator and denominator. 16 48
Simplify the fraction. 1 3
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Simplify: ( 11 7 ) 2 11 2 7 2 .

2 9

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Simplify: ( 6 + 2 ) 2 6 2 + 2 2 .

8 5

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Simplify: 4 ( −3 ) + 6 ( −2 ) −3 ( 2 ) −2 .

Solution

4 ( −3 ) + 6 ( −2 ) −3 ( 2 ) −2
Multiply. −12 + ( −12 ) −6 2
Simplify. −24 −8
Divide. 3
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Simplify: 8 ( −2 ) + 4 ( −3 ) −5 ( 2 ) + 3 .

4

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Simplify: 7 ( −1 ) + 9 ( −3 ) −5 ( 3 ) −2 .

2

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Key concepts

  • Multiply or divide mixed numbers.
    1. Convert the mixed numbers to improper fractions.
    2. Follow the rules for fraction multiplication or division.
    3. Simplify if possible.
  • Simplify a complex fraction.
    1. Rewrite the complex fraction as a division problem.
    2. Follow the rules for dividing fractions.
    3. Simplify if possible.
  • Placement of negative sign in a fraction.
    • For any positive numbers a and b , - a b = a - b = - a b .
  • Simplify an expression with a fraction bar.
    1. Simplify the numerator.
    2. Simplify the denominator.
    3. Simplify the fraction.

Practice makes perfect

Multiply and Divide Mixed Numbers

In the following exercises, multiply and write the answer in simplified form.

2 2 5 ( −2 2 9 )

16 3

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−1 7 20 · 2 11 12

63 16

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In the following exercises, divide, and write your answer in simplified form.

−9 3 5 ÷ ( −1 3 5 )

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−18 3 4 ÷ ( −3 3 4 )

5

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Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

the quotient of 5 u and 11

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the quotient of 7 v and 13

7 v 13

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the quotient of a and b

a b

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the quotient of r and the sum of s and 10

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the quotient of A and the difference of 3 and B

A 3 B

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Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

9 16 33 40

15 22

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Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to 5 −11 ?

−5 −11 , −5 11 , 5 11 , 5 11

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Which of the following fractions are equivalent to −4 9 ?

−4 −9 , −4 9 , 4 9 , 4 9

−4 9 , 4 9

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Which of the following fractions are equivalent to 11 3 ?

−11 3 , 11 3 , −11 −3 , 11 −3

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Which of the following fractions are equivalent to 13 6 ?

13 6 , 13 −6 , −13 −6 , −13 6

13 −6 , −13 6

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In the following exercises, simplify.

−6 + 3 17 8

1 3

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8 3 + 2 9 14 + 3

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9 6 4 7 22 + 3

26 25

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15 5 5 2 2 10

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12 9 3 2 3 18

11 6

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5 6 3 4 4 5 2 3

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8 9 7 6 5 6 9 2

5 2

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6 2 4 2 4 6

−10

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2 + 4 ( 3 ) −3 2 2

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7 + 3 ( 5 ) −2 3 2

−2

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7 4 2 ( 8 5 ) 9.3 3.5

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9 7 3 ( 12 8 ) 8.7 6.6

51 20

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9 ( 8 2 ) −3 ( 15 7 ) 6 ( 7 1 ) −3 ( 17 9 )

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8 ( 9 2 ) −4 ( 14 9 ) 7 ( 8 3 ) −3 ( 16 9 )

18 7

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Everyday math

Baking A recipe for chocolate chip cookies calls for 2 1 4 cups of flour. Graciela wants to double the recipe.

  1. How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number.
  2. Measuring cups usually come in sets with cups for 1 8 , 1 4 , 1 3 , 1 2 , and 1 cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe.

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Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains 2 2 3 cups of chocolate chips per pound.

  1. How many cups of chocolate chips are in a half-pound of the fudge?
  2. The owners of the booth make the fudge in 10 -pound batches. How many chocolate chips do they need to make a 10 -pound batch? Write your results as improper fractions and as a mixed numbers.

  1. 4 3 = 1 1 3 cups
  2. 80 3 = 26 2 3 cups

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Writing exercises

Explain how to find the reciprocal of a mixed number.

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Explain how to multiply mixed numbers.

Answers will vary.

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Randy thinks that 3 1 2 · 5 1 4 is 15 1 8 . Explain what is wrong with Randy’s thinking.

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Explain why 1 2 , −1 2 , and 1 −2 are equivalent.

Answers will vary.

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Self check

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

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Practice Key Terms 1

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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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