Section 2.3 – 2.4Section 2.3 – 2.4Multiplying and Dividing Multiplying and Dividing

Rational NumbersRational Numbers

• The term Rational Numbers refers to any number that can be written as a fraction.

• This includes fractions that are reduced, fractions that can be reduced, mixed numbers, improper fractions, and even integers and whole numbers.

• An integer, like 4, can be written as a fraction by putting the number 1 under it.

Rational NumbersRational Numbers

4=41

• When multiplying fractions, they do NOT need to have a common denominator.

• To multiply two (or more) fractions, multiply across, numerator by numerator and denominator by denominator.

• If the answer can be simplified, then simplify it.

• Example:

• Example:

Multiplying FractionsMultiplying Fractions

25

⋅92

=2⋅95⋅2

=1810

34

⋅52

=3⋅54⋅2

=158

÷2÷2

=95

• When multiplying fractions, we can simplify the fractions and also simplify diagonally. This isn’t necessary, but it can make the numbers smaller and keep you from simplifying at the end.

• From the last slide:

• An alternative:

Simplifying DiagonallySimplifying Diagonally

25

⋅92

=2⋅95⋅2

=1810

÷2÷2

=95

25

⋅92

1

1

=1⋅95⋅1

=95

You do not have to simplify diagonally, it is just an option. If you are more comfortable, multiply across and simplify at the end.

• To multiply mixed numbers, convert them to improper fractions first.

Mixed NumbersMixed Numbers

325

⎛ ⎝

⎞ ⎠ 1

14

⎛ ⎝

⎞ ⎠ =

3⋅5+25

⎛ ⎝

⎞ ⎠

1⋅4+14

⎛ ⎝

⎞ ⎠ =

175

⎛ ⎝

⎞ ⎠

54

⎛ ⎝

⎞ ⎠

=175

⎛ ⎝

⎞ ⎠

54

⎛ ⎝

⎞ ⎠

1

1

=17⋅11⋅4

=174

• Remember, when multiplying signed numbers...

Sign RulesSign Rules

1) 38

−25

⎛ ⎝

⎞ ⎠

Positive * Positive =

Negative * Negative =

Positive * Negative =

Positive.

Positive.

Negative.

=−640

÷2÷2

=−320

2) −3

10⎛ ⎝

⎞ ⎠ −

16

⎛ ⎝

⎞ ⎠ =

360

÷3÷3

=120

Multiply the following fractions and mixed numbers:

Try These: MultiplyTry These: Multiply

1) 65

−13

⎛ ⎝

⎞ ⎠ 2) 5

13

⋅65

3) −134

⎛ ⎝

⎞ ⎠ −3

12

⎛ ⎝

⎞ ⎠ 4)

49

⋅68

Solutions: MultiplySolutions: Multiply

1) 65

−13

⎛ ⎝

⎞ ⎠ =−

615

÷3÷3

=−25

2) 513

⋅65

=163

⋅65

=9615

÷3÷3

=325

3) −134

⎛ ⎝

⎞ ⎠ −3

12

⎛ ⎝

⎞ ⎠ = −

74

⎛ ⎝

⎞ ⎠ −

72

⎛ ⎝

⎞ ⎠ =

498

4) 49

⋅68

=2472

÷24÷24

=13

Solutions (alternative): MultiplySolutions (alternative): Multiply

1) 65

−13

⎛ ⎝

⎞ ⎠

2) 513

⋅65

=163

⋅65

4) 49

⋅68

Note: Problems 1, 2 and 4 could have been simplified before multiplying.

=−25

=325

1

2

2

1

=19

⋅62

1

2

=19

⋅31

1

3

=13

1

3

• When dividing fractions, they do NOT need to have a common denominator.

• To divide two fractions, change the operation to multiply and take the reciprocal of the second fraction (flip the second fraction). Keep-Change-Change.

Dividing FractionsDividing Fractions

25

÷92

=25

⋅29

Change Operation.

Flip 2nd Fraction.

• Finish the problem by following the rules for multiplying fractions.

Dividing FractionsDividing Fractions

25

÷92

=25

⋅29

=445

• Divide the following fractions & mixed numbers:

Try These: DivideTry These: Divide

1) 65

÷ −12

⎛ ⎝

⎞ ⎠ 2) −

32

÷ −12

⎛ ⎝

⎞ ⎠

3) 213

÷323

4) −73

÷123

Solutions: DivideSolutions: Divide

1) 65

÷ −12

⎛ ⎝

⎞ ⎠ =

65

⋅ −21

⎛ ⎝

⎞ ⎠ =−

125

2) −32

÷ −12

⎛ ⎝

⎞ ⎠ =−

32

⋅ −21

⎛ ⎝

⎞ ⎠ =

62

÷2÷2

=31

=3

3) 213

÷323

=73

÷113

=73

⋅311

=2133

÷3÷3

=711

4) −73

÷123

=−73

÷53

=−73

⋅35

=−2115

÷3÷3

=−75

2-1: Rational Numbers on the Number Line

(including absolute value) 

2-7: Square Roots and Real Numbers

(different types of numbers)

2-2: Adding and Subtracting Rational Numbers

2-3 & 4: Multiplying and Dividing Rational Numbers

Chapter 2 Test Chapter 2 Test