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Chapter 2
 prime
Words To Know

number
 composite number
 prime factorization
 factor tree
 common factor
 equivalent fractions
 simplest form
 multiple
 least
common multiple
 least common
denominator
 improper fraction
 mixed number
Overview

2.1 Prime Factorization
2.2 Greatest Common Factor
2.3 Fundamental Fraction Concepts
2.4 Fractions in Simplest Form
2.5 Least Common Multiple
2.6 Comparing and Ordering Fractions
2.7 Mixed Numbers and Improper Fractions
2.1
What is a factor?

A nonzero whole number that divides another
nonzero whole number evenly
Or
A number that divides another without
remainders
1) What would the factors of 14 be?
Define: Prime Number

A number whose only whole number factors are
1 and itself
Or
A number whose factors are only 1 and itself
1) What would the factors of 3 be?
2) What would the factors of 2 be?
Define: Composite
Number

A number that has factors other than 1 and itself
1) What would the factors of 10 be?
2) What would the factors of 20 be?
Something to Ponder

What is 1?
It is neither a prime nor a
composite number
Practice

What are the factors of?
1) 12
2) 8
3) 15
4) 21
5) 35
Prime Factorization

This means writing the number as a product
of prime numbers
Write the prime factorization of …
1) 6
2) 15
3) 12
Practice

Find the prime factorization of each
18
150
64
99
75
345
98
222
140
2.2
Greatest Common Factor

The largest number that is a factor of two or
more nonzero whole numbers. (GCF)[also
can be called greatest common
divisor(GCD)]
This will be helpful to know how to do
when simplifying fractions.
Practice

Find the Greatest Common Factor by listing
all the factors of the numbers.
1. 14, 21
2. 72, 84
3. 18, 27, 45
4. 12, 35
5. 66, 96
2.3
Fractions

= Part of a Whole
Top Number?
Bottom Number?
3
4
= Numerator
= Denominator
What do they mean?

Numerator = number of objects that are
being looked at
Denominator = number of total equal parts
that make up the Whole
Note: the fraction bar means to divide the
numerator by the denominator
Easy Way To Remember!

Numerator = North
Whole Amount
3
4
What you have
Divided by
Denominator = Down
What is a Unit Fraction?

= one part of the whole
Or a fraction where the
numerator is one
What is a Ratio?

= a comparison of two quantities
Examples:
Miles per gallon
Girls to Boys
Write it like a fraction
Know the denominator does not have to = the
Whole
2.4
What are Equivalent
Fractions?

Different fractions that name the same value
Examples:
1
3
4
2
5
6
=
=
=
=
=
2
6
8
4
10
12
The numbers are different but the value is
the same!
Creating Equivalent
Fractions

Multiply the numerator AND denominator
by the same non zero whole number
Example:
1
2
x
18
18
=
36
18
They look different but they have the same value
Are they Equivalent?

Three methods
 Simplify all fractions
 Cross Multiply
 Get a common denominator
Simplify all Fractions

6
2x3
2
=
=
15
5x3
5
10
2x5
=
=
50
10 x 5
Reduced to
different numbers
Not Equivalent
2x1
2x5
1
=
5
Cross Multiply

1050
15
36
=
1080
30
70
Not Equivalent
Common Denominators

6
x
25
4
4
24
=
100
Not Equivalent
15
50
x
2
2
30
=
100
How do you know if …

A fraction is in its simplest form?
The numerator and denominator have a
greatest common factor of 1.
Simplifying Fractions

 Can it be reduced by 2?
 Can it be reduced by 3?
 Can it be reduced by 5?
 Can it be reduced by 7, 11 etc.?
Reducing by 2

Are both numbers even?
46
98
85
175
Yes
No
Reducing by 2

Divide both top and bottom by 2
46
98
÷
23
2
=
49
2
If answer comes out even repeat this step
Reducing by 3

Do both the numbers add up to a number
divisible by 3?
2+3 = 5
23
=
4 + 9 = 13
49
Can’t be reduced by 3
No
No
Reducing by 3

Do both the numbers add up to a number
divisible by 3?
No
8 + 5 = 13
85
=
1 + 7 + 5 = 13
175
Can’t be reduced by 3
No
Reducing by 5

Do both numbers end in either 5 or 0?
23
49
85
175
No
Yes
Reducing by 5

Divide both top and bottom by 5
85
175
÷
17
5
=
35
5
If answer comes out with a 5 in the top and
bottom repeat this step
Reducing by 7… etc.

Divide the top and bottom by 7
23
49
÷
3.3
7
=
7
7
Can’t be reduced by 7
Reducing by 7… etc.

Divide the top and bottom by 7
17
35
÷
7
2.4
=
7
5
Can’t be reduced by 7
Reducing

Simplified
23
49
17
35
Simplifying Improper
Fractions

Divide the numerator by the denominator
79
=
19
x4
19 79
- 76
3
Simplifying Improper
Fractions

The remainder becomes the new numerator
79
=
19
x4
19 79
- 76
3
Simplifying Improper
Fractions

The mixed number is
4
3
19
Check your Answer!

Multiply the Whole number by the denominator
Add the answer to the numerator
79
19
79 = 3
+
x 19 = 76
4
Practice

Give 2 equivalent fractions for each:
1
5
8
9
9
10
12
20
Practice

Is the fraction in it’s simplest form?
28
56
27
100
24
27
72
81
2.5
Least Common Multiple
Define:
 Multiple =

• the product of a number and any nonzero
whole number
Common Multiple =
• a multiple shared by two or more numbers
Least Common Multiple (LCM)=
• the smallest of all the common multiples of
two or more numbers
How to find LCM

Two ways to find them:
1. List the first several multiples of each
number and then compare the lists for the
common multiples and choose the lowest
one.
2. Compare their prime factorization
LCM Listing

Find the LCM of
1. 8, 10
8 = 16, 24, 32, 40, 48, 56, 64, 72, 80
10 = 20, 30, 40, 50, 60, 70, 80
Answer: 40
Practice Listing

Find the LCM of
1. 7, 11
2. 4, 6
3. 6, 8
4. 9, 11
5. 15, 25
LCM Prime Factorization

Find the LCM of
1. 12, 16
12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
• Circle the factors the two have in common
• Write out the factors of both, writing out
the ones they have in common only once
2 x 2 x 3 x 2 x 2 = 48
Answer: 48
Practice Prime
Factorization

Find the LCM of
1. 10, 14
2. 16, 20
3. 9, 33
4. 13, 39
5. 5, 9, 15
2.6
Sequence

Fractions that have the same denominator?
7
10
3
10
1
10
9
10
The numerator with the highest number is the
greatest fraction
So…
Sequence

1 3
10 10
7
10
Is the proper order
9
10
Sequence

Fractions with unlike denominators (and unlike
numerators)?
7
15
3
8
5
24
9
25
Convert them to equivalent fractions with common
denominators in order to compare them
Sequence

To find the least common denominator (LCD)
7
15
3
8
5
24
9
25
you have to find the least common
multiple of the denominators.
15
8
24
25
Denominators

x
x
x
x
40
75
25
24
=
=
=
=
600
600
600
600
7
3
5
9
Numerators

x
x
x
x
40
75
25
24
=
=
=
=
280
225
125
216
Compare:
Sequence

280 225
600 600
Set in order
125
600
216
600
Sequence

125 216
600 600
225 280
600 600
Sequence

Fractions with all the same numerator
1
6
1
9
1
8
1
5
As the denominator gets bigger the
fraction gets smaller.
Sequence

Fractions with all the same numerator
1
9
1
8
1
6
1
5
As the denominator gets bigger the
fraction gets smaller.
Cheat Sheet!

Practice

Compare Fractions:
1
3
5
12
4
9
5
6
2
3
5
18
3
8
7
30
2.7
What is a Proper
Fraction?

A fraction in which the numerator is less
than the denominator.
Example:
3
4
What is an Improper
Fraction?

A fraction in which the numerator is greater
than or equal to the denominator.
Examples:
4
4
7
4
What is a Mixed
Number?

A whole-number and a fraction
Examples:
16
3
4
Cheat Sheet!

Practice

Write as a proper fraction:
7
2
7
5
11
6
16
3
27
4
23
7
2.8
Convert Fractions to Decimals
3
= 0.3
10
323
= 0.323
1000

17
= 0.17
100
9
= 0.009
1000
Convert Fractions to Decimals

If you can turn the denominator into 10,
100, 1,000 (any power of 10) then it’s simple:
1
2
x
=
5
2
3
4
x
=
25
4
5
125
x
=
8
125
2
= 0.2
10
12
= 0.12
100
625
= 0.625
1000
Convert Fractions to Decimals

3
25
75
3
x
=
= .75
=
100
4 2 x 2 25
3
3
2
6
=
x =
= .06
2x5x5 2
50
100
Convert Fractions to Decimals

What about this?
7
75
=
7
3x5x5
That 3 makes it so you can’t use this method,
but there is another way…
Terminating Decimals

Decimals that stop!
3
= 0.03
100
1
= 0.125
8
1
= 0.04
25
Notice that these
denominators are
easily turned into
powers of 10
Repeating Decimals

Decimals that do not stop!
7.3333333333…
6.4545454545…
2.0188888888…
9.1234234234…
= 7.3
= 6.45
= 2.018
= 9.1234
The dot dot dot means it goes on forever
Converting Repeating
Decimals into Fractions

https://www.khanacademy.org/math/algebr
a/solving-linear-equations-andinequalities/conv_rep_decimals/v/coverting
-repeating-decimals-to-fractions-1
Writing a Fraction as
a Decimal

To write a fraction as a decimal, divide the
numerator by the denominator
Convert Fractions to
Decimals

7
75
=
7
3x5x5
So let’s address these kinds of fractions
7
75
=
0.0933… = 0.093
75 7.0
Convert Fractions to
Decimals

2
11
11 2.00
11 2
=
.1818…
=
.18
Convert Decimals to
Fractions

Remember to Simplify!
5
5
1
.5 =
÷
=
10
5
2
Convert Decimals to
Fractions

13
26
2
.26 =
÷
=
50
100
2
Convert Decimals to Fractions

13
325
25
.325 =
=
÷
40
1000
25
You can check your answer by…
13 ÷ 40 = .325
Write as a decimal
Practice

5
8
7
12
23
20
29
2
Write as a decimal
Practice
17
9
8
5

6
7
6
13
11
14
Write as an improper fraction
Practice

0.23
4.8
0.8
2.75
Write as an improper fraction
Practice

3.02
0.27
2.6
0.48