Identify and Apply Number
Properties in Fraction
Operations
Jen Kershaw
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2014 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: November 12, 2014
AUTHOR
Jen Kershaw
www.ck12.org
Chapter 1. Identify and Apply Number Properties in Fraction Operations
C HAPTER
1
Identify and Apply Number
Properties in Fraction Operations
Here you’ll learn to identify and apply number properties in fraction operations.
Do you know how to simplify an expression with fractions by using properties? Take a look at this dilemma.
Simplify: 23 × 27 × 23
To simplify this expression, you will need to know how to work with number properties and fractions. This Concept
will show you how to accomplish this task successfully.
Guidance
Now that we are working with fractions, you will have a chance to investigate how the different properties of addition
and subtraction can help us when we work with fractions in expressions.
Here are a few properties.
Additive Identity Property
The sum of any number and zero is that number:
3
11
+0 =
3
11
Additive Inverse Property
The sum of any number and its inverse is zero:
3
4
+ − 34 = 0
Which of the following shows the Additive Inverse Property?
a. xy + 0 = 0
b. xy + 0 = xy
x
c. y + − xy = 0
Consider choice a.
This equation states that a number added to zero is equal to zero. This is not necessarily correct, unless
equal to zero.
x
y
is also
Consider choice b.
This equation states that the sum of a number and zero is equal to that number. This is correct, but illustrates the
additive identity, not the additive inverse property.
Consider choice c.
This equation states that the sum of a number and its inverse is equal to zero. This illustrates the additive
inverse property, so this is the correct equation.
We can also use these properties to help us when we simplify a numerical expression. Remember that a
numerical expression is a group of numbers and operations. Because we are working with fractions, the
numerical expressions in this section will be made up of fractions.
Simplify: 81 + 14 + 83
You can use addition properties to reorganize this expression to make it easier to simplify.
First apply the commutative property: 18 + 41 + 38 = 18 + 83 + 14
Then apply the associate property: 18 + 83 + 41 = 18 + 83 + 14
1
www.ck12.org
Now you can easily simplify to find the sum.
1 4 1 2 1 3
1
3
8+8 +4 = 8+4 = 4+4 = 4
The answer is 34 .
Using properties to reorganize fractions can help us to work with these fractions. Notice that we reorganized
the common denominators together and this simplified our work.
Now let’s look at how the properties of multiplication and division can help us when working with fractions. You
have already learned the Commutative Property, the Associative Property, and the Distributive Property.
Multiplicative Identity
The product of any number and one is that number:
3
11
×1 =
3
11
Zero Property
The product of any number and zero is zero:
4
7
×0 = 0
Multiplicative Inverse
The product of any number and its reciprocal is one:
3
4
× 34 = 1
Which of the following shows the Multiplicative Inverse Property?
a. xy × 0 =
x
y
b. xy × xy = 0
c. xy × xy = 1
Consider choice a.
This equation states that the product of a number and zero is equal to that number. This is not correct.
Consider choice b.
This equation states that the product of a number and its reciprocal is equal to zero. This is not correct.
Consider choice c.
This equation states that the product of a number and its reciprocal is equal to one. This illustrates the
multiplicative inverse property, so this is the correct equation.
Take a few minutes to write these properties down in your notebooks. Be sure to include an example of each.
You can also use properties to help you simplify numerical expressions.
We could also use properties when working with a variable. Take a look at this one.
3
2
×
a
×
3
2
First, we can apply the commutative property:
2
3
× 23 × a
Now we apply the multiplication inverse property:
2
2
3
× 32 = 1
www.ck12.org
Chapter 1. Identify and Apply Number Properties in Fraction Operations
Our simplified expression is a.
If we had a value substituted in for a, then that would be our answer.
Example A
Name the property:
3
8
×0
Solution: Zero Property
Example B
Name the property:
5
6
× 65
Solution: Multiplicative Inverse
Example C
Simplify: 43 (b × 43 )
Solution: a
Now let’s go back to the dilemma from the beginning of the Concept.
Simplify: 23 × 27 × 23
You can use multiplication properties to reorganize this expression to make it easier to simplify.
First apply the commutative property: 23 × 72 × 32 = 23 × 23 × 27
Then apply the associate property: 23 × 23 × 72 = 23 × 23 × 27
Then apply the multiplicative inverse property: 23 × 32 × 72 = 1 × 27
Finally, apply the multiplicative identity property: 1 × 27 =
2
7
This is our answer.
Vocabulary
Additive Identity Property
any number plus zero is still that number.
Additive Inverse Property
any number plus it’s opposite or inverse is equal to 0.
Multiplicative Identity
any number times 1 is the same number.
Zero Property
any number times 0 is zero.
Multiplicative Inverse
any number times it’s reciprocal is 1.
3
www.ck12.org
Guided Practice
Here is one for you to try on your own.
Simplify:
4
5
+ 12 + x
Solution
First, we find a common denominator so that we can add the fractions. The lowest common denominator for 5 and
2 is 10. Let’s rename both fractions in terms of tenths.
8
10
5
+ 10
+x
Now we can add the fractions.
13
10
+x
Change the improper fraction to a mixed number.
3
+x
1 10
This is our simplified expression.
Video Review
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/65516
Explore More
Directions: Identify each property shown below.
1.
2.
3.
4.
5.
6.
7.
8.
3
4
3
4
6
7
5
8
+0 =
3
4
6
7
3
4
1
4
1
2
× 76 = 1
+ − 34 = 0
×0 = 0
×0 = 0
+ x = x + 34
+ y = y + 14
× (x + 3) = 21 x + 12 (3)
Directions: Simplify each expression.
9.
3
4
10.
+ 14 + x
6
7
× 13 × x
11. 2 × 84 x
12. 3x × 68
13.
4
4
5
6
+ 12 + 10
www.ck12.org
14.
15.
Chapter 1. Identify and Apply Number Properties in Fraction Operations
6
1
10 − 3
1
2 × 3x
5