fractions chapter 1 section 3 mth 11203 introductory algebra
TRANSCRIPT
FRACTIONS
CHAPTER 1 SECTION 3
MTH 11203Introductory Algebra
Algebra
Arithmetic ndash all quantities are known
Algebra ndash one or more of the quantities are unknown
Variables ndash lettersmybols that represent the numbers that are unknown x y and z are the most often letters used
Multiplication Symbols
If a and b represent any two mathematical quantities then ldquoa times brdquo can be written as follows
Example Example Exampleab 5x xy
a bull b 5 bull 2 5 bull x x bull y
a(b) 5(2) 5(x) x(y)
(a)b (5)2 (5) x (x) y
(a)(b) (5)(2) (5)(x) (x)(y)
Factors
Factors are numbers or variables that are multiplied in a multiplication problem
If a bull b = c then a and b are factors of c
Examples
2 bull 5 = 10 the numbers 2 and 5 are factors of the product 10
2x means ldquo2 times xrdquo both the number 2 and the variable x are factors
Fractions
Numerator ndash top number of a fraction
Denominator ndash bottom number of a fraction
Fractions in general are written
= a divide b =
Example 25 = 2 divide 5 =
b a
5 2
a
b
Simplified Fractions
To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1
Simplify1 Find the greatest common factor (GCF) which
is the largest number that will divide both the numerator and denominator without a remainder (Appendix B)
2 Divide both the numerator and the denominator by the GCF 2 1 2
4
2 2
1
2
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Algebra
Arithmetic ndash all quantities are known
Algebra ndash one or more of the quantities are unknown
Variables ndash lettersmybols that represent the numbers that are unknown x y and z are the most often letters used
Multiplication Symbols
If a and b represent any two mathematical quantities then ldquoa times brdquo can be written as follows
Example Example Exampleab 5x xy
a bull b 5 bull 2 5 bull x x bull y
a(b) 5(2) 5(x) x(y)
(a)b (5)2 (5) x (x) y
(a)(b) (5)(2) (5)(x) (x)(y)
Factors
Factors are numbers or variables that are multiplied in a multiplication problem
If a bull b = c then a and b are factors of c
Examples
2 bull 5 = 10 the numbers 2 and 5 are factors of the product 10
2x means ldquo2 times xrdquo both the number 2 and the variable x are factors
Fractions
Numerator ndash top number of a fraction
Denominator ndash bottom number of a fraction
Fractions in general are written
= a divide b =
Example 25 = 2 divide 5 =
b a
5 2
a
b
Simplified Fractions
To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1
Simplify1 Find the greatest common factor (GCF) which
is the largest number that will divide both the numerator and denominator without a remainder (Appendix B)
2 Divide both the numerator and the denominator by the GCF 2 1 2
4
2 2
1
2
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Multiplication Symbols
If a and b represent any two mathematical quantities then ldquoa times brdquo can be written as follows
Example Example Exampleab 5x xy
a bull b 5 bull 2 5 bull x x bull y
a(b) 5(2) 5(x) x(y)
(a)b (5)2 (5) x (x) y
(a)(b) (5)(2) (5)(x) (x)(y)
Factors
Factors are numbers or variables that are multiplied in a multiplication problem
If a bull b = c then a and b are factors of c
Examples
2 bull 5 = 10 the numbers 2 and 5 are factors of the product 10
2x means ldquo2 times xrdquo both the number 2 and the variable x are factors
Fractions
Numerator ndash top number of a fraction
Denominator ndash bottom number of a fraction
Fractions in general are written
= a divide b =
Example 25 = 2 divide 5 =
b a
5 2
a
b
Simplified Fractions
To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1
Simplify1 Find the greatest common factor (GCF) which
is the largest number that will divide both the numerator and denominator without a remainder (Appendix B)
2 Divide both the numerator and the denominator by the GCF 2 1 2
4
2 2
1
2
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Factors
Factors are numbers or variables that are multiplied in a multiplication problem
If a bull b = c then a and b are factors of c
Examples
2 bull 5 = 10 the numbers 2 and 5 are factors of the product 10
2x means ldquo2 times xrdquo both the number 2 and the variable x are factors
Fractions
Numerator ndash top number of a fraction
Denominator ndash bottom number of a fraction
Fractions in general are written
= a divide b =
Example 25 = 2 divide 5 =
b a
5 2
a
b
Simplified Fractions
To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1
Simplify1 Find the greatest common factor (GCF) which
is the largest number that will divide both the numerator and denominator without a remainder (Appendix B)
2 Divide both the numerator and the denominator by the GCF 2 1 2
4
2 2
1
2
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Fractions
Numerator ndash top number of a fraction
Denominator ndash bottom number of a fraction
Fractions in general are written
= a divide b =
Example 25 = 2 divide 5 =
b a
5 2
a
b
Simplified Fractions
To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1
Simplify1 Find the greatest common factor (GCF) which
is the largest number that will divide both the numerator and denominator without a remainder (Appendix B)
2 Divide both the numerator and the denominator by the GCF 2 1 2
4
2 2
1
2
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Simplified Fractions
To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1
Simplify1 Find the greatest common factor (GCF) which
is the largest number that will divide both the numerator and denominator without a remainder (Appendix B)
2 Divide both the numerator and the denominator by the GCF 2 1 2
4
2 2
1
2
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Simplified Fractions
Prime Numbers Every integer that can only be divided by itself and 1 is prime but 1 is not a prime number There are 25 prime numbers between 1 and 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 and 97
GCF ndash Greatest Common Factor 1 Write both the numerator and denominator as a
product of primes2 Determine all the prime factors that are common to
both prime factorizations3 Multiply the common prime factors to obtain the GCF
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Simplified Fractions
GCF
40 10
2 20 2 5
2 10
2 5
GCF 2 5 = 10
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Simplified Fractions
GCF
60 105
2 30 5 21
2 15 3 7
3 5
GCF 3 5 = 15
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Simplified Fractions
Examples pg 27
40 4 1024
10
1 104
41
60 4 1530
105
7 154
7
40 40 10 424 4
10 10 10 1
60 60 15 430
105 105 15 7
oror
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Multiply Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 bull 2 mean multiply 2 ndash 2 means
subtract
Whole number are 1 2 3 4 5 6 hellip
hellip is called ellipsis meaning continues indefinitely
Multiple Fractionsa c ac
b d bd
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Examples of Multiplying Fractions
Find the product (pg 27)
6 7 6 7 4252
13 17 13 17 221
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Examples of Multiplying Fractions
Find the product (pg 27)
36 16 3654
48 45
4
161
483
455
4 1 4
3 5 15
36 2 2 3 3 48 2 2 2 2 3
2 2 3 12
36 12 3 48 12 4
common factors are
16 divides 16 and 48
9 divides 36 and 45
36 16 354
48 45
1
41
16
4
4515
4
15
OR
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Examples of Multiplying Fractions
Find the product (pg 27)
3 10 3 1058
8 11
5
84
3 5 15
4 11 4411
2 divides 10 and 8
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Divide Fractions
Evaluate means to answer the problem using the given operation
2 + 2 means add 2divide2 mean divide2 ∙ 2 mean multiply 2 ndash 2 means subtract
Divide Fractions - turn it into what you already know
a c a d ad
b d b c bc
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Examples of Dividing Fractions
Find the quotient (pg 27)
3 3 357
8 4
1
41
82
31
1 1 1
2 1 2
Change to multiplication4 divides 8 and 41 divides 3 and 3
5 5 30 560 30
9 9 1
1
1
9 30
6
1 1 1
9 6 54
Put the whole number over 1Change to multiplication5 divides 5 and 30
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Add and Subtract Fractions
Fractions with a common denominator can easily be added and subtracted
Add or subtract the numerator and keep the same denominator
Add
Subtract
a b a b
c c c
a b a b
c c c
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Examples of Adding and Subtracting Fractions
Add with common denominator (pg 28)
Subtract with common denominator (pg 28)18 1 18 1 17
68 36 36 36 36
1 3 1 3 470 1
4 4 4 4
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding and Subtracting Fractions
To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD)
LCD ndash Write each denominator as a product of prime factors Determine the maximum number of times that prime number appears in the factorization Multiply these prime numbers
Help with finding LCD is in Appendix B
GCF ndash Write each number as a product of prime factors Determine the prime factors common to all numbers Multiply the common factors
Prime Numbers are numbers that has only 2 factors 1 and itself The first 7 prime numbers are 2 3 5 7 11 13 17
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding and Subtracting Fractions
ExampleFind the GCF and the LCD of 108 and 156
108 156 2 54 2 78
2 27 2 39 3 9 3
13 3 3
LCD = 2 2 3 3 3 13 = 1404GCF = 2 2 3 = 12
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding and Subtracting Fractions
Add with unlike terms (pg 28)8 2 9
73 17 34 17
LCD of 17 and 34 = 1 2 17 = 34 17 = 1 17 34 = 2 17
8 2 16 16 2 16 2 18
17 2 34 34 34 34
9
3417
9
17
2 1 2
34 1 34
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 17 3274
7 35 35
LCD of 7 and 35 = 1 7 5 35 7 = 1 7 35 = 5 7
3 5 15 15 17 15 17 32
7 5 35 35 35 35 35
17 1
35
17
1 35
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding and Subtracting Fractions
Add with unlike terms (pg 28)
3 5 7178
7 12 84
LCD of 7 and 12 = 1 2 2 3 7 84 7 = 1 7 12 =2 2 3
3 12 36 36 35 36 35 71
7 12 84 84 84 84 84
5 7 35
12 7 84
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding and Subtracting Fractions
Subtract with unlike terms (pg 28)
1 1 176
6 18 9
LCD of 6 and 18 = 2 3 3 = 18 6 = 2 3 18 = 2 3 3
1 3 3 3 1 3 1 2 1
6 3 18 18 18 18 18 9
1 1
18
1
1 18
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Common Error
Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions
You cannot divide out common factors when adding and subtracting
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Mixed Numbers
Mixed Number is a whole numbered followed by a fraction 2 frac12 5 frac14 3 ⅝
Changing a mixed number to a fraction
(Whole number times the denominator plus the numerator) divide by the denominator
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Mixed Numbers
1 5 3 1 1636 (pg 27) 5
3 3 3
3 4 4 3 1938 (pg 27) 4
4 4 4
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Mixed Numbers
Changing a fraction to a mixed number
218 444 (pg 27) = 7 18 = 2
7 7144
5110 10 148 (pg 27) 20 110 5 5
20 20 210010
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Adding Mixed Numbers
3 3 3 382 (pg 28) 2 3 2 2
8 4 8 8
3 6 + 3 3
4 8
9 9 5 5 5 1
8 8
1 1
68 8
49 1 or 6
8 8
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
Subtracting Mixed Numbers
2 1 2 6 2786 (pg 28) 8 - 3 8 8 7
7 3 7 21 21
1 7 7 - 3 3 3
3 21 21
20 104 4 or
21 21
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89
HOMEWORK 13
Page 27 - 2825 27 43 51 53 55 64 66 67 69 71 72 81 83 89