Download Introduction to Fractions and Multiplication and

OpenStax-CNX module: m34931
1
Introduction to Fractions and
Multiplication and Division of
Fractions: Summary of Key
Concepts
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This
module reviews the key concepts from the chapter "Introduction to Fractions and Multiplication and
Division of Fractions."
1 Summary of Key Concepts
Fraction ( here1 )
The idea of breaking up a whole quantity into equal parts gives us the word fraction.
Fraction Bar, Denominator, Numerator ( here2 )
A fraction has three parts:
1. The fraction bar
2. The nonzero whole number below the fraction bar is the denominator.
3. The whole number above the fraction bar is the numerator.
Proper Fraction ( here3 )
Proper fractions are fractions
∗ Version
in which the numerator is strictly less than the denominator.
1.2: Aug 18, 2010 8:36 pm +0000
† http://creativecommons.org/licenses/by/3.0/
1 "Introduction to Fractions and Multiplication and Division of Fractions: Fractions of Whole Numbers"
<http://legacy.cnx.org/content/m34908/latest/>
2 "Introduction to Fractions and Multiplication and Division of Fractions: Fractions of Whole Numbers"
<http://legacy.cnx.org/content/m34908/latest/>
3 "Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed
Numbers" <http://legacy.cnx.org/content/m34912/latest/>
http://legacy.cnx.org/content/m34931/1.2/
OpenStax-CNX module: m34931
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5
2
is a proper fraction
Improper Fraction ( here4 )
Improper fractions are fractions
in which the numerator is greater than or equal to the denominator.
Also, any nonzero number placed over 1 is an improper fraction.
5
5 5
4 , 5 , and 1 are improper fractions
Mixed Number ( here5 )
A mixed number is a number
that is
the sum of a whole number and a proper fraction.
1 51 is a mixed number 1 15 = 1 +
1
5
Correspondence Between Improper Fractions and Mixed Numbers ( here6 )
Each improper fraction corresponds to a particular mixed number, and each mixed number corresponds to
a particular improper fraction.
Converting an Improper Fraction to a Mixed Number ( here7 )
A method, based on division, converts an improper fraction to an equivalent mixed number.
1
5
4 can be converted to 1 4
Converting a Mixed Number to an Improper Fraction ( here8 )
A method, based on multiplication, converts a mixed number to an equivalent improper fraction.
5 87 can be converted to 47
8
Equivalent Fractions ( here9 )
Fractions that represent the same quantity are equivalent
3
6
4 and 8 are equivalent fractions
Test for Equivalent Fractions ( here10 )
If the cross products of two fractions are equal,
Thus,
3
4
and
6
8
fractions.
then the two fractions are equivalent.
are equivalent.
Relatively Prime ( here11 )
Two whole numbers are relatively prime
3 and 4 are relatively prime
when 1 is the only number that divides both of them.
Reduced to Lowest Terms ( here12 )
A fraction is reduced to lowest terms
The number
The number
3
4
6
8
if its numerator and denominator are relatively prime.
is reduced to lowest terms, since 3 and 4 are relatively prime.
is
reduced to lowest terms since 6 and 8 are not relatively prime.
not
4 "Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed
Numbers" <http://legacy.cnx.org/content/m34912/latest/>
5 "Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed
Numbers" <http://legacy.cnx.org/content/m34912/latest/>
6 "Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed
Numbers" <http://legacy.cnx.org/content/m34912/latest/>
7 "Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed
Numbers" <http://legacy.cnx.org/content/m34912/latest/>
8 "Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed
Numbers" <http://legacy.cnx.org/content/m34912/latest/>
9 "Introduction to Fractions and Multiplication and Division of Fractions: Equivalent Fractions, Reducing Fractions to
Lowest Terms, and Raising Fractions to Higher Terms" <http://legacy.cnx.org/content/m34927/latest/>
10 "Introduction to Fractions and Multiplication and Division of Fractions: Equivalent Fractions, Reducing Fractions to
Lowest Terms, and Raising Fractions to Higher Terms" <http://legacy.cnx.org/content/m34927/latest/>
11 "Introduction to Fractions and Multiplication and Division of Fractions: Equivalent Fractions, Reducing Fractions to
Lowest Terms, and Raising Fractions to Higher Terms" <http://legacy.cnx.org/content/m34927/latest/>
12 "Introduction to Fractions and Multiplication and Division of Fractions: Equivalent Fractions, Reducing Fractions to
Lowest Terms, and Raising Fractions to Higher Terms" <http://legacy.cnx.org/content/m34927/latest/>
http://legacy.cnx.org/content/m34931/1.2/
OpenStax-CNX module: m34931
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Reducing Fractions to Lowest Terms ( here13 )
Two methods, one based on dividing out common primes and one based on dividing out any common factors,
are available for reducing a fraction to lowest terms.
Raising Fractions to Higher Terms ( here14 )
A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same
nonzero number.
3
4
3·2
4·2
=
=
6
8
The Word OF Means Multiplication ( here15 )
In many mathematical applications, the word "of" means multiplication.
Multiplication of Fractions ( here16 )
To multiply two or more fractions, multiply the numerators together and multiply the denominators together.
Reduce if possible.
5
8
4
·
15
=
5·4
8·15
=
20
120
=
1
6
Multiplying Fractions by Dividing Out Common Factors ( here17 )
Two or more fractions can be multiplied by rst dividing out common factors and then using the rule for
multiplying fractions.
1
)5
)8
1
·
)4
)15
2
=
1·1
2·3
=
1
6
3
Multiplication of Mixed Numbers ( here18 )
To perform a multiplication in which there are mixed numbers, rst convert each mixed number to an
improper fraction, then multiply. This idea also applies to division of mixed numbers.
Reciprocals ( here19 )
Two numbers whose product is 1 are reciprocals.
7 and 71 are reciprocals
Division of Fractions ( here20 )
To divide one fraction by another fraction, multiply the dividend by the reciprocal of the divisor.
4
5
÷
2
15
=
4
5
·
15
2
Dividing 1 by a Fraction ( here21 )
When dividing 1 by a fraction, the quotient is the reciprocal of the fraction.
1
3
7
=
7
3
Multiplication Statements ( here22 )
A mathematical statement of the form
product = (factor 1) (factor 2)
13 "Introduction to Fractions and Multiplication and Division of Fractions: Equivalent Fractions, Reducing Fractions to
Lowest Terms, and Raising Fractions to Higher Terms" <http://legacy.cnx.org/content/m34927/latest/>
14 "Introduction to Fractions and Multiplication and Division of Fractions: Equivalent Fractions, Reducing Fractions to
Lowest Terms, and Raising Fractions to Higher Terms" <http://legacy.cnx.org/content/m34927/latest/>
15 "Introduction to Fractions and Multiplication and Division of Fractions: Multiplication of Fractions"
<http://legacy.cnx.org/content/m34928/latest/>
16 "Introduction to Fractions and Multiplication and Division of Fractions: Multiplication of Fractions"
<http://legacy.cnx.org/content/m34928/latest/>
17 "Introduction to Fractions and Multiplication and Division of Fractions: Multiplication of Fractions"
<http://legacy.cnx.org/content/m34928/latest/>
18 "Introduction to Fractions and Multiplication and Division of Fractions: Multiplication of Fractions"
<http://legacy.cnx.org/content/m34928/latest/>
19 "Introduction to Fractions and Multiplication and Division of Fractions: Division of Fractions"
<http://legacy.cnx.org/content/m34929/latest/>
20 "Introduction to Fractions and Multiplication and Division of Fractions: Division of Fractions"
<http://legacy.cnx.org/content/m34929/latest/>
21 "Introduction to Fractions and Multiplication and Division of Fractions: Division of Fractions"
<http://legacy.cnx.org/content/m34929/latest/>
22 "Introduction to Fractions and Multiplication and Division of Fractions: Applications Involving Fractions"
<http://legacy.cnx.org/content/m34930/latest/>
http://legacy.cnx.org/content/m34931/1.2/
OpenStax-CNX module: m34931
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is a multiplication statement.
By omitting one of the three numbers, one of three following problems result:
M
1.
= (factor 1) · (factor 2) Missing product statement.
2. product = (factor 1) ·
Missing factor statement.
3. product =
· (factor 2) Missing factor statement.
M
M
Missing products are determined by simply multiplying the known factors. Missing factors are determined
by
missing factor = (product) ÷ (known factor)
http://legacy.cnx.org/content/m34931/1.2/