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Multiplying and Multiplying and Dividing Rational Dividing Rational
NumbersNumbers
• 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