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By the end of this section, you will be able to:
  • Model fraction addition
  • Add fractions with a common denominator
  • Model fraction subtraction
  • Subtract fractions with a common denominator

Before you get started, take this readiness quiz.

  1. Simplify: 2 x + 9 + 3 x 4 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .
  2. Draw a model of the fraction 3 4 .
    If you missed this problem, review Visualize Fractions .
  3. Simplify: 3 + 2 6 .
    If you missed this problem, review Multiply and Divide Mixed Numbers and Complex Fractions .

Model fraction addition

How many quarters are pictured? One quarter plus 2 quarters equals 3 quarters.

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

1 4 2 4 3 4 one quarter + two quarters = three quarters

Let’s use fraction circles to model the same example, 1 4 + 2 4 .

Start with one 1 4 piece. . .
Add two more 1 4 pieces. . .
The result is 3 4 . . .

So again, we see that

1 4 + 2 4 = 3 4
Doing the Manipulative Mathematics activity "Model Fraction Addition" will help you develop a better understanding of adding fractions

Use a model to find the sum 3 8 + 2 8 .

Solution

Start with three 1 8 pieces. . .
Add two 1 8 pieces. . .
How many 1 8 pieces are there? . .

There are five 1 8 pieces, or five-eighths. The model shows that 3 8 + 2 8 = 5 8 .

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Use a model to find each sum. Show a diagram to illustrate your model.

1 8 + 4 8

5 8

A circle divided into 8 sections, 5 of which are shaded.

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Use a model to find each sum. Show a diagram to illustrate your model.

1 6 + 4 6

5 6

A circle divided into 6 sections, 5 of which are shaded.

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Add fractions with a common denominator

[link] shows that to add the same-size pieces—meaning that the fractions have the same denominator —we just add the number of pieces.

Fraction addition

If a , b , and c are numbers where c 0 , then

a c + b c = a + b c

To add fractions with a common denominators, add the numerators and place the sum over the common denominator.

Find the sum: 3 5 + 1 5 .

Solution

3 5 + 1 5
Add the numerators and place the sum over the common denominator. 3 + 1 5
Simplify. 4 5
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Find each sum: 3 6 + 2 6 .

5 6

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Find each sum: 3 10 + 7 10 .

1

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Find the sum: x 3 + 2 3 .

Solution

x 3 + 2 3
Add the numerators and place the sum over the common denominator. x + 2 3

Note that we cannot simplify this fraction any more. Since x and 2 are not like terms, we cannot combine them.

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Find the sum: x 4 + 3 4 .

x + 3 4

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Find the sum: y 8 + 5 8 .

y + 5 8

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Find the sum: 9 d + 3 d .

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

a b = a b

9 d + 3 d
Rewrite the first fraction with the negative in the numerator. −9 d + 9 d
Add the numerators and place the sum over the common denominator. −9 + 3 d
Simplify the numerator. −6 d
Rewrite with negative sign in front of the fraction. 6 d
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Find the sum: 7 d + 8 d .

1 d

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Find the sum: 6 m + 9 m .

3 m

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Find the sum: 2 n 11 + 5 n 11 .

Solution

2 n 11 + 5 n 11
Add the numerators and place the sum over the common denominator. 2 n + 5 n 11
Combine like terms. 7 n 11
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Find the sum: 3 p 8 + 6 p 8 .

9 p 8

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Find the sum: 2 q 5 + 7 q 5 .

9 q 5

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Find the sum: 3 12 + ( 5 12 ) .

Solution

3 12 + ( 5 12 )
Add the numerators and place the sum over the common denominator. −3 + ( −5 ) 12
Add. −8 12
Simplify the fraction. 2 3
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Find each sum: 4 15 + ( 6 15 ) .

2 3

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Find each sum: 5 21 + ( 9 21 ) .

2 3

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Model fraction subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 12 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 7 12 of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or 2 12 of the pizza), how much is left? There would be 5 pieces left (or 5 12 of the pizza).

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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what is biological synthesis of nanoparticles
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
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AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
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Hello
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I'm interested in Nanotube
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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