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1-3 Fractions
In a fraction, the number that is written above the fraction bar is called the numerator.
The number below the fraction bar is called the denominator.
2
1
9
5
3
10
The numerators and denominators of these fractions are natural or counting numbers.
The set of natural numbers is {1, 2, 3, 4, 5, …}
(The three dots following the 5 indicates that the list continues in the same pattern
without end.)
----------------------------------------------------------------------------------------------------------Mixed Numbers and Improper Fractions
A mixed number consists of a natural number and a fraction. It indicates a sum of the
two types of numbers.
4
4
5
5
3 =3+
An improper fraction is a fraction whose numerator is greater than or equal to its
denominator.
21
3
19
8
3
5
----------------------------------------------------------------------------------------------------------Converting a Mixed Number to an Improper Fraction
1. Multiply the natural number and the denominator of the fraction.
2. Add the numerator to the product from step 1.
3. Write the result from step 2 over the denominator.
Example: Convert 3
4
to an improper fraction.
5
1. 3 ∙ 5 = 15
2. 15 + 4 = 19
3.
19
5
Example: Convert 21
7
8
to an improper fraction.
1. 21 ∙ 8 = 168
2. 168 + 7 = 175
3.
175
8
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Converting an Improper Fraction to a Mixed Number
1. Divide the numerator by the denominator.
2. Write the remainder as a fraction (the divisor is the denominator)
quotient
Example: Convert
remainder
original denominator
42
5
to a mixed number.
8
5 42
8 25
5 42
 40
2
 40
2
Example: Convert
175
8
to a mixed number.
21
8 175
217 8
8 175
 16
 16
15
8
7
15
8
7
----------------------------------------------------------------------------------------------------------Factors and Prime Factorization (p. 22)
To factor a number means to write it as the product of two or more natural numbers.
Example: 21 can be factored as 7 ∙ 3 or 21 ∙ 1
The factors of a number are the numbers that can be multiplied to equal that number.
The factors of 21 are 1, 3, 7 and 21.
Some natural numbers have only two factors: the number itself and 1. These numbers are
called prime numbers.
Definition: A prime number is a natural number that has exactly two different factors –
itself and 1.
The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
Definition: A composite number is a natural number greater than 1 that is not
a prime number. (It has more than two factors.)
Every composite number can be expressed as the product of prime numbers. Expressing a
composite number as the product of primes is called prime factorization.
Example: Find the prime factorization of 45.
 45 is not even so 2 is not a factor.
 The next prime is 3. 45 = 3 ∙ 15
 Now factor 15 ( the composite factor).
 15 = 3 ∙ 5 (Both 3 and 5 are prime.)
 Therefore, 45 = 3 ∙ 3 ∙ 5
Example: Find the prime factorization of 72.
 72 is even so 2 is a factor. 72 = 2 ∙ 36
 Now factor 36 ( the composite factor).
 36 is even so 2 is a factor. 36 = 2 ∙ 18
 Now factor 18 ( the composite factor).
 18 is even so 2 is a factor. 18 = 2 ∙ 9
 Now factor 9 ( the composite factor).
 9 is not even so 2 is not a factor.
 The next prime is 3. 9 = 3 ∙ 3
 Therefore, 72 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3
----------------------------------------------------------------------------------------------------------Reducing Fractions
Writing a fraction as an equivalent fraction with a smaller denominator is called
reducing the fraction. A fraction is said to be reduced to lowest terms when the
numerator and denominator have no common factor other than 1.
NOTE: Equivalent fractions are two different fractions that represent the
same value.
To reduce a fraction , we use the
Identity Property of 1
𝑎 ∙ 1 = 1 ∙ 𝑎 = 𝑎 Multiplying a number by 1 equals the number.
1 is called the multiplicative identity.
Fraction Notation for 1
𝑐
𝑐
=1
𝑐≠0
Any nonzero number divided by itself is 1.
Fundamental Principle of Fractions
The value of a fraction does not change if both the numerator and the denominator
are multiplied or divided by the same nonzero number (multiplied or divided by 1).
𝑎 𝑎∙𝑐
=
𝑏 𝑏∙𝑐
Example: Reduce
4
6
to lowest terms.
 Find the prime factorization of 4 and 6.
4
6
=
2∙2
3∙2
 Divide both the numerator and denominator by each common prime
factor. (This is also called canceling common factors.)
2
 Write the resulting fraction.
Example: Reduce
15
75
3
to lowest terms.
 Find the prime factorization of 15 and 75.
15
75
=
3∙5
3∙5∙5
 Divide both the numerator and denominator by each common prime
factor: 3 and 5.
 Write the resulting fraction.
1
5
NOTE: Since all factors in the numerator divided out, the factor left is 1.
----------------------------------------------------------------------------------------------------------Multiplying Fractions (p. 24)
The product of two or more fractions is the product of their numerators over the product
of their denominators.
𝑎
𝑐
𝑎∙𝑐
∙ = 𝑏∙𝑑
𝑏 𝑑
NOTE: The product should be reduced to lowest terms.
Examples: Multiply and reduce to lowest terms.
2 9
18
18
3 4
12
12
∙ =
=
2∙3∙3
2∙2∙3
=
3
2
or 1
1
2
5∙
7
12
5
7
= ∙
=
1 12
35
or 2
12
11
2
1
11 5
55
3
4
3
12
3 ∙1 =
12
∙ =
4
or 4
7
12
Study Tip: You can divide numerators and denominators by common factors before
multiplying.
2 9
2 3∙3
3 4
3 2∙2
∙ = ∙
1 3
3
1 2
2
= ∙ =
1
or 1
2
***********************************************************************
Dividing Fractions (p. 26)
The reciprocal of a number is found by inverting the fraction form of the number.
2
Examples:
3
and
3
5
2
11
and
11
are reciprocals.
are reciprocals.
5
1
4 and
4
4
are reciprocals. (Remember: 4 = )
1
----------------------------------------------------------------------------------------------------------1
10 ÷ 2 = 5
10 ∙ = 5
2 and
2
1
2
are reciprocals.
You can divide fractions using multiplication. The quotient of two fractions equals the
first fraction multiplied by the reciprocal of the second fraction.
𝑎
𝑐
𝑎 𝑑
÷𝑑 = 𝑏∙𝑐 =
𝑏
𝑎∙𝑑
𝑏∙𝑐
NOTE: The quotient should be reduced to lowest terms.
Examples: Divide and reduce to lowest terms.
2
3
÷
7
15
2 15
= ∙
3
7
3
1
19
4
2
4
4 ÷1 =
2 3∙5
= ∙
2 5
10
1 7
7
= ∙ =
3
7
3
19 2
19
2
4
2∙2 3
÷ =
∙ =
3
2
∙ =
or 1
3
7
19 1
19
2
6
∙ =
3
or 3
1
6
-----------------------------------------------------------------------------------------------------------
Adding and Subtracting Fractions with the Same Denominator (p. 27)
The sum or difference of two fractions with the same denominator is the sum or
difference of their numerators over the common denominator.
𝑎
𝑐
+𝑏 =
𝑏
𝑎+𝑐
𝑎
𝑏
𝑐
−𝑏 =
𝑏
and
𝑎−𝑐
𝑏
NOTE: The sum or difference should be reduced to lowest terms.
Examples: Add or subtract and reduce to lowest terms.
3
11
11
12
+
−
4
11
5
12
=
=
7
11
6
6
12
12
1
3
21
4
4
4
5 −2 =
−
11
4
=
=
2∙3
2∙2∙3
10
4
=
=
2∙5
2∙2
1
2
=
5
2
or 2
1
2
----------------------------------------------------------------------------------------------------------Adding and Subtracting Fractions with Unlike Denominators (p. 27)
You cannot add or subtract fractions if they have unlike denominators.
You must
 find the least common denominator (LCD)
 rewrite the fractions so they are equivalent but have the LCD as their denominator
 add or subtract (the fractions now have the same denominator)
----------------------------------------------------------------------------------------------------------Finding the LCD
The least common denominator or LCD is the smallest number that all of the
denominators divide into.
Method 1: Using your knowledge of basic arithmetic facts.
If you have a good grasp of the basic facts from arithmetic, you may be able to find the
LCD mentally.
1
5
3
4
+
−
3
4
1
6
you know that the LCD is 20 – the smallest number divisible by 5 and 4.
you know that the LCD is 12 – the smallest number divisible by 4 and 6.
Method 2: Using multiples.
Start with the smallest denominator and count by multiples of that number. When you
come to a multiple that is also a multiple of the other denominator, you have found the
LCD.
1
5
+
3
count by multiples of 4: 4, 8, 12, 16, 20
4
20 is also a multiple of 5. 20 is the LCD.
3
4
−
1
count by multiples of 4: 4, 8, 12
6
12 is also a multiple of 6. 12 is the LCD.
Method 3: Prime Factorization
1
15
+
7
24
these denominators may present difficulties finding the LCD by
Methods 1 or 2.
Find the prime factorization of each denominator.
15 = 3 ∙ 5
24 = 2 ∙ 2 ∙ 2 ∙ 3
The prime factors are 2, 3 and 5. The LCD contains each of these factors the greatest
number of times it appears in any one denominator.
LCD = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 5 = 120
----------------------------------------------------------------------------------------------------------Rewriting as Equivalent Fractions having the LCD as Denominator
The same 3 principles used to reduce fractions allow us to change fractions to equivalent
form having the LCD as the new denominator.
1
15
=
?
7
120
24
Since 15 ∙ 8 = 120,
8
120
+
35
120
1∙8
15∙8
=
=
8
120
=
?
120
Since 24 ∙ 5 = 120,
7∙5
24∙5
=
35
120
43
120
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