Section 6.1: The Set of Fractions
Definition
a
b
(or a b ), where b ≠ 0 . The number a is called the numerator and the number b is called
the denominator.
A fraction is a number that can be represented by an ordered pair of whole numbers
In set notation, the set of fractions is
⎧a
⎫
F = ⎨ a and b are whole numbers, b ≠ 0⎬ .
⎩b
⎭
The term fraction is used in two distinct ways in elementary mathematics.
•
Initially, fractions are used as numerals to indicate the number of equivalent parts
of a whole to be considered. The use of a fraction as a numeral in this way is
commonly called the “part-to-whole” model. The term equivalent parts means
equivalent in some attribute, such as length, area, volume, number, or weight,
depending on the composition of the whole and appropriate parts.
•
Then, fractions are used as numbers to indicate the relative amount of a whole to
be considered. For example, the fraction ¼ can be used to indicate ¼ of a
teaspoon, or ¼ of a cup, etc. So, when considering a fraction as a number, one
must focus on the relative amount, without regard to size, shape, arrangement,
orientation, number of equivalent parts, and so on.
The notion of fraction as a numeral displaying a part-to-whole relationship can be merged
with the concept of fraction as a number. For example, we can show that ¼, 2/8, 4/16,
etc. represent the same relative amount if we “cut” the pie differently.
Two fractions that represent the same relative amount are said to be equivalent
fractions.
Definition
a
c
a c
be any fractions. Then = if and only if ad = bc .
Let and
b
d
b d
In other words, two fractions are equal fractions if and only if their cross-products (the
products ad and bc obtained by cross-multiplication) are equal.
Every fraction has an infinite number of representations, which is illustrated by the
following theorem.
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Theorem 1:
a
a an na
=
.
Let be any fraction and n a nonzero whole number. Then =
b
b bn nb
This theorem can be used in two ways.
a
an
• To replace the fraction with
, which finds an equivalent fraction, and
b
bn
an
a
• To replace the fraction
with , a process called simplifying.
bn
b
A fraction is written in its simplest form or lowest terms when its numerator and
denominator have no common prime factors.
Fractional Forms
Every whole number can be expressed in terms of a fraction; therefore, every whole
number has an infinite number of fractional representations.
Fractions whose numerator is less than its denominator are called proper fractions and
represent a quantity less than the whole number one.
Fractions whose numerator and denominator are equal represent the whole number one.
Fractions whose numerator is greater than its denominator are called improper fractions
and represent a quantity greater than the whole number one.
We can rewrite an improper fraction as a mixed number, which is a whole number with
a proper fraction juxtaposed to the right of the whole number.
Ordering Fractions
The concepts of less than and greater than in fractions are extensions of the respective
whole-number concepts. To reiterate, the smaller of two fractions occurs to the left of the
larger fraction on a fractional number line.
When the denominators of two fractions are equal, we can easily determine which
fraction is smaller (or larger) by comparing their numerators. To be more succinct,
consider the following definition.
Definition
a
b
a b
Let and
be any fractions. Then < if and only if a < b .
c
c
c c
Similarly,
if a ≥ b .
a b
a b
a b
≤ if and only if a ≤ b , > if and only if a > b , and ≥ if and only
c c
c c
c c
2
When the denominators of two fractions are not equal, ordering them can become more
difficult. However, we have two options available to us.
•
•
We can find equivalent forms of the fractions so that we obtain equal
denominators, and then apply the last definition, or
We can apply the following theorem.
Theorem 2
a
c
a c
Let and
be any fractions. Then < if and only if ad < bc .
b
d
b d
a c
a c
a c
≤ if and only if ad ≤ bc , > if and only if ad > bc , and ≥ if and
b d
b d
b d
only if ad ≥ bc .
Similarly,
Often fractions can be ordered mentally using your “fraction sense.” For example to order
7
2
7
1
2
2 7
and I notice that
is close to and that is closer to zero. Hence < .
15
7
15
2
7
7 15
There are an infinite number of fractions between any two different fractions. This fact is
called the density property of fractions. The following theorem is used to find one of the
infinite numbers of fractions between two different fractions
Theorem 3
a
c
a c
a a+c c
Let and
be any fractions, where < . Then <
< .
b
d
b d
b b+d d
Proof:
a c
Let < . Then
b d
ad < bc
ad + ab < bc + ab
a (b + d ) < b(a + c )
a a+c
.
<
b b+d
Similarly, since
ad < bc
ad + cd < bc + cd
d (a + c ) < c(b + d )
a+c c
<
b+d d
Thus
a a+c c
<
<
b b+d d
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Section 6.2: Fractions: Addition and Subtraction
Addition
The sum of two fractions with the same denominator can be found by adding the
numerators. This fact is summarized in the following definition.
Definition
a
c
a c a+c
Let and be any fractions. Then
+ =
.
b
b
b b
b
This fact can easily be illustrated by two concrete models: the region model and the
fraction number-line model.
When we want to add two fractions with unlike denominators, we must find equivalent
fractions with common denominators. Then the sum will be represented by the sum of the
numerators over the common denominator. This fact is summarized next.
Theorem
a
c
a c ad + bc
Let and
be any fractions. Then
.
+ =
b
d
b d
bd
Application of this theorem usually does not result in the LCM of the denominators
(called the least common denominator). Finding and using the LCD simplifies paperand-pencil calculations. However, using the LCD does not mean that the answer will
always be in its simplest form. Whenever we add, subtract, multiply, or divide fractions,
we should always express the final answer in its simplest form.
Properties of Fraction Addition
(For simplicity, all properties are stated using common denominators, since any two
fractions can be expressed with the same denominator.)
•
•
•
•
Closure Property – The sum of two fractions is a fraction.
a
c
a c c a
Commutative Property – Let and be any fractions. Then
+ = + .
b
b
b b b b
a c
e
Associative Property – Let , , and be any fractions. Then
b b
b
a
c
e
a
c
e
⎡
⎤
⎡
⎤
⎢ b + b ⎥ + b = b + ⎢b + b ⎥ .
⎣
⎦
⎣
⎦
a
0
Additive Identity – Let be any fraction. There is a unique fraction, , such that
b
b
a 0 0 a a
+ = + = .
b b b b b
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Subtraction
The difference of two fractions with the same denominator can be found by subtracting
the numerators. This fact is summarized in the following definition.
Definition
a
c
a c a−c
Let and be any fractions with a ≥ c . Then − =
.
b
b
b b
b
Subtraction of fractions can be viewed in two ways – either as (1) take-away or (2) using
the missing-addend approach. Both of these approaches can be illustrated using the
region or the fraction number-line models.
When we want to subtract two fractions with unlike denominators, we must find
equivalent fractions with common denominators. Then the difference will be represented
by the difference of the numerators over the common denominator. This fact is
summarized next.
Theorem
a
c
a c
a c ad − bc
Let and
be any fractions where ≥ . Then
.
− =
b
d
b d
b d
bd
Mental math and estimation techniques similar to those used with whole numbers can be
used with fractions.
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Section 6.3: Fractions: Multiplication and Division
Multiplication
The product of two fractions is the fraction whose numerator is the product of the two
fractions numerator, and whose denominator is the product of the two fractions
denominators. This fact is summarized in the following definition.
Definition
a
c
a c ac
Let and
be any fractions. Then ⋅ =
.
b
d
b d bd
Multiplication of fractions can be viewed in two ways – (1) repeated addition, when
multiplying a fraction and a whole number, and (2) a fraction of a fraction, when
multiplying two fractions together.
Properties of Fraction Multiplication
a c
e
be any fractions.
Let , , and
b d
f
• Closure Property – The product of any two fractions is a fraction.
a c c a
• Commutative Property – ⋅ = ⋅ .
b d d b
⎛a c ⎞ e a ⎛c e ⎞
• Associative Property – ⎜ ⋅ ⎟ ⋅ = ⋅ ⎜⎜ ⋅ ⎟⎟ .
⎝b d ⎠ f b ⎝d f ⎠
a
a a
m
• Multiplicative Identity – ⋅1 = 1 ⋅ = , where 1 = , m ≠ 0 .
b
b b
m
a
• Multiplicative Inverse – For every nonzero fraction , there is a unique fraction
b
b
a b
b
such that ⋅ = 1 . The fraction is called the multiplicative inverse or
a
b a
a
a
reciprocal of .
b
a ⎛c e⎞ a c a e
• Distributive Properties – Over addition: ⋅ ⎜⎜ + ⎟⎟ = ⋅ + ⋅ ;
b ⎝d f ⎠ b d b f
Over subtraction:
a ⎛c e⎞ a c a e
⋅⎜ − ⎟ = ⋅ − ⋅ .
b ⎜⎝ d f ⎟⎠ b d b f
Division
Method 1: When we divide two fractions with common denominators, the quotient is
found by dividing the numerators of the two fractions. Hence, we can reduce the division
of fractions to division of whole numbers. This fact is summarized next in the following
definition.
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Definition
a
c
a c a
Let and be any fractions with c ≠ 0 . Then ÷ = .
b
b
b b c
Method 2: To divide fractions with different denominators, we can rewrite the fractions
a c ad bc ad a d
so that they have the same denominator. Thus we see that ÷ =
÷
=
= ⋅ .
b d bd bd bc b c
This fact leads to the following theorem.
Theorem
a
c
a c a d
be any fractions with c ≠ 0 . Then ÷ = ⋅ .
Let and
b
d
b d b c
a c a ÷c
÷ =
.
b d b÷d
This method is convenient only when c divides a evenly and d divides b evenly.
Method 3: This theorem leads to a third method for division of fractions:
Method 4: One can also find the quotient by using the missing-factor approach. In other
a c e
a c e
e
words, to find ÷ = , rewrite as = ⋅ , and then determine .
b d f
b d f
f
Models of Division
1. The division of fractions in which the divisor is not a whole number can be viewed as
a measurement division problem.
c
a
(how many groups of size can be found in )
b
b
2. On the other hand, if the division problem has a divisor that is a whole number, say n,
then it should be viewed as a partitive division problem.
(how much of the fraction can be distributed into n groups)
Now, through the division of fractions, we can perform the division of any whole
numbers without having to use remainders. This fact is summarized below in the
following definition.
Definition
For any whole numbers a and b, b ≠ 0 , a ÷ b =
a
b
Expressing a division problem as a fraction is a useful idea. For example, complex
fractions can be used to divide fractions, using the reciprocal.
Mental math and estimation techniques similar to those used with whole numbers can be
used with fractions.
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