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