Fractions

1
Number Theory Fractions
Primes
Composites
Multiples
numerator
Denominator
Improper
2
Table of Contents - Fractions

• Vocabulary

• Comparing

• Order of Operations

• Fractions to Decimals

• Divisibility Rules

• Decimals to Fractions

• Adding Subtracting

• Equivalent Fractions

• Simplifying Fractions

• Multiplying

• Mixed Improper

• Dividing

• Fraction/Decimal Matches

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3
Vocabulary - for theory

• A prime number has exactly and only two factors -
  itself and one
• examples 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
  31, 37, 41, 43, 47, ...
• Remember the Barney song.
• A composite number has a set number of factors
  but more than two
• examples 4, 6, 8, 9, 10, 12, 14, etc.
• One and zero are neither prime nor composite

4

• Factors are numbers that divide evenly into your
  number
• Example for 24 1, 2, 3, 4, 6, 8, 12,
  24
• Multiples are your number multiplied by another
  number
• Example for 24 24, 48, 72, 96, 120,

5
Example for 24
Factors

• 1 24
• 2 12
• 3 8
• 4 6

Multiples 1x24 24, 2x24 48, 3x24 72,
4x24 96, 5x24 120, so on forever
6
Order of Operations
7
Order of Operations Please Excuse My Dear Aunt
Sally 1st Work within the parentheses (PEMDAS
too). 2nd Exponents 2 2x2x2x2x2 32 3rd
Multiplication division from left to right 4th
Addition subtraction from left to right
5
8
Priority of Operations

• ( ) Please
• x2 Excuse
• x, ? My Dear
• , - Aunt Sally

9
Priority of Operations

• Parenthesis
• Exponents
• Multiplication/Division
• Addition/Subtraction

10
Priority of Operations

• 16 20 ? 22
• What operation is performed first?

11
Priority of Operations

• Solve exponents
• 16 20 ? 22
• 16 20 ? 4
• What operation is performed next?

12
Priority of Operations

• Divide then subtract
• 16 20 ? 4
• 16 5
• 11

13
Priority of Operations

• 33 4 6
• What operation is performed first?

14
Priority of Operations

• Solve exponents
• 33 4 6
• 27 4 6
• What operation is performed next?

15
Priority of Operations

• Multiply then add
• 27 4 6
• 27 24
• 51

16
Priority of Operations

• 64 ? (8 4)
• What operation is performed first?

17
Priority of Operations

• Solve parenthesis then divide
• 64 ? (8 4)
• 64 ? 32
• 2

18
Priority of Operations

• 3 (18 7) 2
• What operation is performed first?

19
Priority of Operations

• Parenthesis within brackets
• 3 (18 7) 2
• 3 11 2
• What operation is performed next?

20
Priority of Operations

• Brackets then add
• 3 11 2
• 3 22
• 25

21
Left-to-Right Rule

• Operations with equal priority are performed left
  to right.
• Multiplication/Division
• Addition/Subtraction

22
Left-to-Right Rule

• 6 ? 3 5 ?
• Operations have equal priority, so perform them
  left to right.
• 6 ? 3 5
• 2 5
• 10

23
Left-to-Right Rule

• 38 70 ? 7 2 ?
• In what order are the operations performed?

24
Left-to-Right Rule

• Perform the operations with equal priority left
  to right, then perform lower priority operations.

25
Left-to-Right Rule

• Division and multiplication are equal priority so
  divide.
• 38 70 ? 7 2
• 38 - 10 2

26
Left-to-Right Rule

• Now multiply then subtract
• 38 - 10 2
• 38 20 18

27
Left-to-Right Rule

• 6 ? (17 11) 14 ?
• In what order are the operations performed?

28
Left-to-Right Rule

• Parenthesis
• 6 ? (17 11) 14
• 6 ? 6 14
• What operation is performed next?

29
Left-to-Right Rule

• Multiplication and division are equal priority so
  perform them left to right.
• 6 ? 6 14
• 1 14
• 14

30
Figure It Out

• 24 (2 8)2 ? 42 - 6 ?
• In what order are the operations performed?

31
Figure It Out

• Parenthesis
• Exponents
• Divide
• Add (L to R)
• Subtract

• 24 (2 8)2 ? 42 - 6
• 24 (16)2 ? 42 - 6
• 24 256 ? 16 - 6
• 24 16 - 6
• 40 - 6
• 34

32
Last One

• 18 23 - 5 6 ? 2 ?
• In what order are the operations performed?

33
Last One

• Exponents
• Mult. (L to R)
• Divide
• Subtract

• 18 23 - 5 6 ? 2
• 18 8 - 5 6 ? 2
• 144 - 30 ? 2
• 144 - 15
• 129

Now you try.
34
8 9 3 5
17 3 5
14 5
19
35
7 5 2 3
35 6
41
36
18 5 x 2
18 10
8
37
(9 4 )(8 7)
13 x 1
13
38
(16 5) (13 2)
21 15
6
39
24 6 2
4 2
6
40
32 4 2
8 2
4
41
18 24 12 3
18 2 3
23
42
67 84 12 4 16
67 84 - 48 16
67 84 - 3
151 - 3
148
43
34 8 2 4 9
34 4 36
74
44
(15 21) 3
36 3
12
45
5 6 25 5 - 2
30 5 - 2
25 - 2
23
46
15 35 21 4
50 25
2
The division bar works like parenthesis around
the top and around the bottom so do the top first
and then the bottom second. Then divide.
47
Commutative Property (If all addition or all
multiplication, we can add or multiply in any
order.) 3 2 5 5 2 3 2 x 5 x 4 4 x 5
x 2 5 x 2 x 4, etc.
48
Associative Property We can remove or move
parentheses if the problem is all addition or all
multiplication. (2 3) 4 2 (3 4) 2
3 4 (2 X 3) X 4 2 X (3 X 4) 2 X 3 X 4
49
Divisibility Rules The easy way to simplify
fractions

• 2 - ends in a zero, 2, 4, 6, or 8 (even numbers)
• 3 - sum-of-the-digits - add all the digits and if
  three goes into the answer evenly, it will go
  into the number evenly. You can also use the
  pairing trick.
• 4 - check the last two digits OR does two go
  evenly evenly?

50

• 5 - ends in a zero or 5
• 6 - do two and three both go?
• 7 - hard luck
• 8 - check the last three digits OR does four go
  evenly evenly?
• 9 - sum-of-the-digits - add all the digits and if
  nine goes into the answer evenly it will go into
  the number evenly. You can also use the pairing
  trick.
• 10 - ends in a zero

51
Main ones 2 Even number 3 Sum-of-the-digits
divisible by 3 5 Ends in 5 or zero. 6
Divisible by both 2 3 9 Sum-of-the-digits
divisible by 9 10 Ends in zero
52
Equivalent Fractions
To make an equivalent fraction, either multiply
or divide both the top and the bottom numbers by
the same number.
53
Simplified Fractions
A fraction is simplified when the top and bottom
numbers have no common factors other than one.
If you can divide both by any number other than
one, it is not simplified. Use your divisibility
rules!
Simplified
Not Simplified
54
Prime Factorization
is breaking down a number until it is all prime
factors that can be multiplied back to equal it.
Use either a factor tree or ladder.
55
Factor tree
Step 1 Find two numbers that can be multiplied
to equal your number. Step 2 If either number
from the answer to step 1 is prime, leave.
Otherwise, break them down as in step 1. Repeat
until all are prime.
56
Factor tree example
24
4 x 6
2 x 2 x 2 x 3 Answer 2 to the 3rd
power x 3 or 23 x
3
57
Factor ladder
Use short division to divide but up-side-down by
primes and primes only starting with 2 if
possible and working up. Use 2 when possible
until you can not use 2 any more. Then move to
3, then 5, then 7, etc. (What pattern is this?
- Prime numbers)
58
Factor ladder example
24
2
12
2
6
2
3
The answer is still 23 x 3.
59
Self check 1 - Factors Multiples

• 1. List all factors for 40
• 2. List all factors for 25
• 3. List first five multiples for 8
• 4. List first five multiples for 12
• State the divisibility rule for three.

1, 2, 4, 5, 8, 10, 20, 40
1, 5, 25
8, 16, 24, 32, 40.
12, 24, 36, 48, 60.
Sum-of-the-digits divisible by 3
60
Mixed to Improper
Mixed is a mix of a whole number with a fraction
Improper is in only fraction form however, the
top is larger than the bottom. It equals more
than one.
61
Mixed to Improper
To change, multiply the bottom number of the
fraction by the whole number and then add the
top of the fraction. Place over the bottom of
the fraction.
wholes
pieces each
extra piece
3 x 4 1 over 4
13
4
62
When comparing fractions either
Cross multiply putting
the 3 5 product (answer) on
top and 4 6 compare products,
20
18
lt
change to decimals and compare, or
0.7500 0.8333
. 75
. 8333
6
3.00
4
5.00
find equivalent fractions with common
denominators.
3 4
5 6
lt
63
To change a fraction to a decimal
divide the top by the bottom Remember Top dog
in the dog house .
7 5 4
3 . 0 0 2 8
2 0
- 2 0
0
3
4
64
To change a decimal to a fraction
put it over its place value and simplify .72
.125
65
What is the very first thing you have to do
before adding or subtracting fractions?
Check to see if the denominators (bottoms) match.
66
If the denominators match we
add the tops only because the bottom is what we
are counting.
67
To add or subtract fractions with common
denominators

• just add (or subtract) the tops.
• Copy the bottom.
• If improper (top larger than the bottom), it is
  okay unless there is company - a mixed number
  (whole number with the improper fraction).
• Then the improper part must be changed to mixed
  and the whole numbers added.

68
1



is read as one-half plus one-half equals two
halves or two divided by two which equals 1.
69


70
-

71
1



72



Remember your divisibility rules!!!
73


74
-

75
Self check 2 Comparing adding like fractions
Compare lt, gt, or Add or subtract.
Simplify. 3) 4)
5)
1)
2)
lt

1


-
76
Example of improper with company
5
5

1
6
Divide the 4 by 3. You can make one whole group
of three with 1 out of 3 left over.
Sometimes, when subtracting and the second
fraction is larger than the top in mixed form,
you must go back to improper.
77
Adding or Subtracting unlike fractions
Only add or subtract like fractions so when they
do not match first make equivalent fractions with
like denominators.
78
If the denominators dont match we
find a common denominator and make new equivalent
fractions
79
Find the new numerators by multiplying the top by
what you multiplied
the bottoms by to get the new denominator.
80
Unlike example
11
15
81

x3


x5
3, 6, 9, 12, 15, 18, 21 5, 10, 15, 20, 25, 30
82



or
1
83



or
1
84

-

85



or
1
86

-

87
Self check 3 Adding and subtracting unlike
fractions
Add or subtract. Simplify.
1)
2)

-
3)
4)
5)


-
88
If mixed numbers have an improper fraction, it
must be changed to mixed and the whole numbers
added. Sometimes mixed numbers need to be
changed to make the fraction part improper in
order to be large enough to subtract the other
fraction.
89
Correct
Form
90
Correct
Form
91
Correct
Form
92
Correct
Form
93
Correct
Form
94
6

95
x3
x3
x7
x7
1
23 21

27
28
96
x3
x3
x2
x2
1
7 6

46
47
97
3 2
3 2

• 1
• 3
• 4
• 5

3 7 2 3


3 2
3 2
2 3 3 4
2 5 6 7


98
x
3
x
x
99
3
100
19
101
11
102
14
103
14
x3
x3
x7
x7
104
7
x3
x3
x4
x4
105
7
x5
x5
x4
x4
106
x5
x5
x4
x4
107
7
x5
x5
x7
x7
108
x2
x2
x7
x7
109
How are adding and subtracting mixed numbers
alike?
Common denominators are needed, new equivalent
fractions must be made, S.F.
110
How is adding mixed numbers different?
You have to change improper fractions to mixed
and add whole numbers.
111
How is subtracting mixed numbers different?
If the top fraction is too small, you have to
take from the whole number and rename to match
the denominator.
112
When multiplying or dividing

• 1. Change mixed numbers to improper and whole
  numbers to fraction by putting them over one.
• Examples 4 1/2 9/2 and 5 becomes 5/1
• 2. Division only - change to multiplication
  and flip the second number.
• 3. Cross simplify if possible.
• Multiply across the top write it on the top.
  Then same to bottom.
• Check simplification.

113
Examples for multiplication
2
x
x
2
1
1
3
x
x
6
or
1
1
114
x
1
x
4
5
2
x
14
15
3
115
x
2
1
x
5 x 2 4
7 x 1 3
5
7
2
2
14
x
5
1
1
116
x
4
1
x
2 x 4 1
6 x 1 5
2
6
3
9
x
33
11
2
6
4
2
117
x
4
x
2 x 4 1
1
2
2
x
9
1
9
2
2
4
118
x
4
4
x
2 x 4 1
4
2
1
2
x
9
18
4
2
1
1
119
x
2
x
2 x 2 1
3
2
10
1
x
5
3
3
2
10
4
2
120
x
2
1
x
2
3
3
1
10
5
5
121
x
2
x
2
1
1
2
1
x
2
1
1
1
2
1
122
What extra step do we have to do when dividing
fractions?
After changing mixed to improper and whole
numbers over one, change to multiplication and
flip the second number.
123
Examples for division

1
10
2
8
or
1
124

5
20
or
125

1
2
4
126

10
3
10

10
3
1
1
3
10
x
3
or
10
1
1
127

8
1
4
32
or
128

6
3
4
2
8
or
1
129
Self check 4 Multiplying dividing mixed
fractions
18
28
12
1)
2
1
2)
x
x
6
4
5
2
2


3)
4)
130
What is the last thing we do when working with
fractions?
Make sure it is simplified by dividing both the
top and the bottom by any common factors.
131
Fraction Matches
50
.5
25
.25
75
.75
60
.6
33 1/3
.333
87 1/2
.875
20
.2
66 2/3
.666
12 1/2
.125
80
.8
132
Fraction Matches
16 2/3
.166
62 1/2
.625
10
.1
83 1/3
.833...
40
.4
95
.95
5
.05
0
0
12 1/2
.125
100
1