Transcript PowerPoint
Slide 1
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 2
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 3
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 4
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 5
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 6
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 7
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 8
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 9
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 10
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 11
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 12
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 13
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 14
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 15
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 16
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 17
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 18
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 19
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 20
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 21
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 22
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 23
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 24
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 2
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 3
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 4
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 5
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 6
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 7
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 8
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 9
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 10
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 11
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 12
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 13
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 14
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 15
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 16
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 17
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 18
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 19
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 20
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 21
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 22
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 23
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29
Slide 24
Columbus State Community College
Chapter 4 Section 3
Multiplying and Dividing Signed Fractions
Ch 4 Sec 3: Slide #1
Multiplying and Dividing Signed Fractions
1. Multiply signed fractions.
2. Multiply fractions that involve variables.
3. Divide signed fractions.
4. Divide fractions that involve variables.
5. Solve application problems involving multiplying and
dividing fractions.
Ch 4 Sec 3: Slide #2
“ of ”
NOTE
When used with fractions, the word of indicates multiplication.
For example,
1
3
of
1
4
means
1
3
•
1
4
Ch 4 Sec 3: Slide #3
Multiplying Fractions
Multiplying Fractions
If a, b, c, and d are numbers (but b and d are not 0), then
a
b
•
c
d
=
a • c
b • d
Ch 4 Sec 3: Slide #4
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(a)
–
3
4
• –
5
7
–
3
4
• –
5
7
=
3•5
4•7
=
15
28
The answer is in lowest
terms because
and 28 have no common
The product
of two 15
negative
factor other than 1.
numbers is positive.
Ch 4 Sec 3: Slide #5
Multiplying Signed Fractions
EXAMPLE 1
Multiplying Signed Fractions
Multiply.
(b)
1
2
•
3
5
1
2
•
3
5
=
1•3
2•5
=
3
10
The answer is in lowest terms because 3 and 10 have no common
factor other than 1.
Ch 4 Sec 3: Slide #6
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(a)
6
7
Using Prime Factorization to Multiply Fractions
• –
6
7
14
18
• –
14
18
=
1 1
1
2•3•2•7
–
7•2•3•3
1 1 1
=
– 2
3
Multiplying a positive number times a negative
number gives a negative product.
Ch 4 Sec 3: Slide #7
Using Prime Factorization to Multiply Fractions
EXAMPLE 2
(b)
5
9
Using Prime Factorization to Multiply Fractions
27
35
of
5
9
•
27
35
=
1 1 1
5•3•3•3
3•3•5•7
1 1 1
=
3
7
Ch 4 Sec 3: Slide #8
Multiplying a Fraction and an Integer
EXAMPLE 3
Multiplying a Fraction and an Integer
2
Find
of 45.
3
2
3
•
45
1
=
1
2•3•3•5
3•1
1
=
30
1
=
30
Ch 4 Sec 3: Slide #14
Multiplying Fractions with Variables
EXAMPLE 4
(a)
4 m2 n3
5
4 m2 n3
5
•
Multiplying Fractions with Variables
•
15
12 m n4
15
12 m n4
1 1 1
1 1 1 1 1
2•2•m•m•n•n•n•3•5
5•2•2•3•m•n•n•n•n
1 1 1 1 1 1 1 1
=
=
m
n
Ch 4 Sec 3: Slide #15
Multiplying Fractions with Variables
EXAMPLE 4
(b)
5a
6b
5a
6b
•
•
Multiplying Fractions with Variables
12 b2
35 a
12 b2
35 a
=
1 1 1
1 1
5•a•2•2•3•b•b
2•3•b•5•7•a
1 1 1 1
1
=
2b
7
Ch 4 Sec 3: Slide #16
Reciprocal of a Fraction
Reciprocal of a Fraction
Two numbers are reciprocals of each other if their product is 1.
a
The reciprocal of the fraction
b
a
b
•
b
a
=
b
is
a
1 1
a • b
b • a
1 1
because
=
1
1
=
1
Ch 4 Sec 3: Slide #17
Reciprocals
Find the reciprocal of each number.
Number
Reciprocal
Reason
1.
4
7
7
4
4
7
2.
2
9
9
2
2
9
3.
8
1
8
8
1
7
4
=
9
2
1
8
28
28
18
= 1
18
=
=
= 1
8
8
= 1
Ch 4 Sec 3: Slide #18
Reciprocal
NOTE
Every number has a reciprocal except 0. Why not 0?
0 • (reciprocal) ≠ 1
Put any number here. When you
multiply it by 0, you get 0, never 1.
Ch 4 Sec 3: Slide #19
Dividing Fractions
Dividing Fractions
If a, b, c, and d are numbers (but b, c, and d are not 0), then we
have the following.
a
c
÷
b
d
=
a
b
•
d
c
In other words, change division to multiplying by the reciprocal
of the divisor.
Ch 4 Sec 3: Slide #20
Dividing Signed Fractions
EXAMPLE 5
(a)
2
÷
5
2
÷
5
Dividing Signed Fractions
–
–
8
15
8
15
=
1
2
5
1
•
3
15
–
8
4
=
– 3
4
Change division toReciprocals
multiplication
Ch 4 Sec 3: Slide #21
Dividing Signed Fractions
EXAMPLE 5
(b)
5 ÷
Dividing Signed Fractions
1
4
5 ÷
1
4
=
5
1
•
4
1
=
20
Change divisionReciprocals
to multiplication
Ch 4 Sec 3: Slide #22
Dividing Signed Fractions
EXAMPLE 5
(c)
4
9
÷
Dividing Signed Fractions
0
Division by zero is undefined.
Ch 4 Sec 3: Slide #23
Dividing Fractions with Variables
EXAMPLE 6
(a)
a2
b3
a2
b3
÷
•
Dividing Fractions with Variables
a
2 b2
2 b2
a
=
1
1 1
a•a•2•b•b
b•b•b•a
1 1
1
=
2a
b
Ch 4 Sec 3: Slide #24
Dividing Fractions with Variables
EXAMPLE 6
(b)
n3
6
n3
6
÷
•
Dividing Fractions with Variables
n4
1
n4
=
1 1 1
n•n•n•1
2•3•n•n•n•n
1 1 1
=
1
6n
Ch 4 Sec 3: Slide #25
Indicator Words
Indicator Words for
Multiplication
Indicator Words for
Division
product
per
double
each
triple
goes into
times
divided by
twice
divided into
of (when of follows a fraction)
divided equally
Ch 4 Sec 3: Slide #26
Using Indicator Words to Solve Application Problems
EXAMPLE 7
(a)
Using Indicator Words – Solving Applications
Alberto will spend
1
of his paycheck on his bills. If
6
Alberto receives a paycheck for $480, how much will he
spend on his bills?
1
of 480
6
=
80
1
480
•
6
1
1
=
80
Alberto will spend $80 on his bills.
Ch 4 Sec 3: Slide #27
Using a Sketch to Solve Application Problems
EXAMPLE 7
(b)
Using a Sketch – Solving Applications
Bina purchased a spool containing 36 yd of ribbon. She wants
2
to make awards for a banquet. If each award requires
yd
3
of ribbon, how many awards can she make?
36 yd of ribbon
2
yd
3
2
36 ÷
3
=
18
36
•
1
3
2
1
=
54
Bina can make 54 awards.
Ch 4 Sec 3: Slide #28
Multiplying and Dividing Signed Fractions
Chapter 4 Section 3 – Completed
Written by John T. Wallace
Ch 4 Sec 3: Slide #29