Chapter 2 - Fractions I
Math Skills – Week 2
Today’s Schedule

Turn in Homework Assignment #1
 Quiz #1
 Lecture on first half of Chapter 2 (Fractions)
 Stuff:
1.
2.
Website
 Class listing and MyInfo
 Change your password
 Post Practice Final exam within the next week
Office hour location and time
1.
Virtual. Thursdays 6 – 6:45pm
Week 2 - Fractions
 Least Common Multiple (LCM) and Greatest Common
Factor (GCF)

Section 2.1
 Introduction to Fractions
 Section 2.2
 Writing Equivalent Fractions
 Section 2.3
 Arithmetic with Fractions (Pt. 1)
 Addition
 Section 2.4
 Subtraction
 Section 2.5
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 The multiples of a number are the products of that
number and the whole numbers 1, 2, 3, 4, 5, 6,…


Example: The multiples of 2 are:
2x1=2
2x2=4
2x3=6
2x4=8
…
Thus the multiples of 2 are
 2, 4, 6, 8, …
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 A number that is a multiple of two or more
numbers is called a common multiple of those
numbers

For Example…8 is a common multiple of 2 and 4.
 To find the Lowest Common Multiple (LCM) of a
set of numbers use one of the following two
methods
Least Common Multiple
(LCM) and Greatest Common
Factor (GCF) – Section 2.1
 Method 1 (Listing multiples)

Steps
1.
2.
3.
List the multiples of each number
Identify the common multiples
Identify which of those is the smallest number.
1.
This is the LCM
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Example: Find the LCM of 4 and 6
 Using Method 1
 Step 1:



Step 2:


The multiples of 4 are:
 4, 8, 12, 16, 20, 24, 28, 32, 36…
The multiples of 6 are:
 6, 12, 18, 24, 30, 36, 42,…
The common multiples of 4 and 6 are:
 12, 24, 36,…
Step 3:

By inspection, the LCM of 4 and 6 is:
 12
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Method 2: Using Prime Factorizations

Steps:
1.
2.
3.
4.
Write the prime factorization of each number
Organize these prime factors into a “table of prime
factors” (see pg. 65)
Circle the greatest product in each column
Multiply each of the circled quantities
1.
This product is the LCM
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1

Example: Find the LCM of 4 and 6

Use Method 2:
 Step 1:





The prime factorization of 4 is:
 2x2
The prime factorization of 6 is:
 2x3
Organize
Step 2:
Step 3:
Step 4:

Circle greatest
products
LCM = 2 x 2 x 3 = 12
2
4=
2x2
6=
2
3
3
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Which method is better? Which is easier?
 Tougher example: Find the LCM of 24, 36, and 50
 Method 1
 Step 1




Multiples of 24 are:
 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288,
312, 336, 360, 384, 408, 432, 456,…, 1800
Multiples of 36 are:
 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432,
468, 504, 540,…, 1800
Multiples of 50 are:
 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550,
600, 650, 700, 750, 800, 850,….., 1800
Step 2/3: LCM 1800
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Same example: Find the LCM of 24, 36 and 50


Method 2
Step 1

Prime factorization of 24:


Prime factorization of 36:




2x2x3x3
Prime factorization of 50:


2x2x2x3
Step 2: Prime Factors Table
Step 3: Circle largest products
Step 4: LCM is the product of
circled quantities

2
3
24 =
2x2x2
3
36 =
2x2
3x3
50 =
2
2x5x5
5
2 x 2 x 2 x 3 x 3 x 5 x 5 = 1800
5x5
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Group Examples: Find the LCM of the following sets of
numbers


14, 21
 Ans = 42
12, 27, 50
 Ans = 2700
 Class Examples:
 2, 7, 14
 Ans = 14
 5, 12, 15
 Ans = 60
Steps for finding LCM
Step 1: Find prime factorization of
each number
Step 2: Prime Factors Table
Step 3: Circle largest products
Step 4: LCM product of
circled quantities
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Recall that the factors of a number are the
numbers (1, 2, 3, 4, 5, …) that divide the
number evenly
 Common factors of a set of numbers are the
factors that those numbers have in common.
 The Greatest Common Factor of a set of
numbers is the largest number in the set of
common factors.
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 To find the GCF of a set of numbers use one of the
following two methods

Method 1 (Listing factors)
 Steps
1.
2.
3.
List the factors of each number
Identify the common factors .
Identify which of those is the largest number. That number is
the GCF
 Example: Find the GCF of 30 and 105
 The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
 The factors of 105 are: 1, 3, 5, 7, 15, 21, 35, 105
 The Common Factors are 1, 3, 5, 15
 The GCF is:

15
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Method 2 (Using Prime Factorization)

Steps
Find the Prime Factorization of each number
2. Write out Prime Factorization table.
3. Circle the smallest product in each column that is
not blank
1.

4.
Importante! If column has a blank for one of the
numbers, don’t circle anything for that column
The product of the circled quantities is the GCF
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Example using Method 2:
 Find the GCF of 90, 168, 420 (Using method 2)


Step 1
The Prime factorization of 90 is:


The Prime Factorization of 168 is:




2x2x2x3x7
The Prime Factorization
of 420 is:


2x3x3x5
2
3
2
3x3
158 =
2x2x2
3
420 =
2x2
3
5
7
2x2x3x5x7
Step 2: Prime Factors table
Step 3: Circle Smallest
Product (No Blanks)
Step 4: Product of circled
numbers is
GCF = 2 x 3 = 6
90 =
7
5
Least Common Multiple (LCM)
and Greatest Common Factor
(GCF) – Section 2.1
 Group Examples: Find the GCF of the following sets of
numbers


12, 18
 Ans = 6
21, 27, 33
 Ans = 3
 Class Examples:
 24, 64
 Ans = 8
 41, 67
 Ans = 1
Steps to find GCF
Step 1: Find Prime factorization of each
number
Step 2: Prime Factors table
Step 3: Circle Smallest Product in
each column (ignore columns with
blanks)
Step 4: Product of circled numbers is GCF
Introduction to Fractions – Section
2.2
 A fraction is the representation of a specified
portion of a whole number.
Numerator
Fraction Bar
Denominator
4
4
3
4
2
4
1
4
Introduction to Fractions – Section
2.2
 Definitions

Proper Fraction is a fraction that is less
than 1
 Numerator


is smaller than the denominator
Mixed number is a number greater than 1
 Whole
number part and a fractional part.
Improper Fraction is a fraction greater than
or equal to 1
 Numerator
3
4
is greater than the denominator
3
14
7
4
Introduction to Fractions – Section
2.2

Convert Improper fractions  Mixed numbers

Steps
1.
2.


Divide the Numerator into the Denominator
Fractional Part: Write any remainder as a fraction by placing it
over the original denominator
Example: Write 13/5 as a mixed number
Convert Mixed numbers  Improper Fractions

Steps
1.
2.
3.

Multiply the denominator of the fractional part by the whole
number part
Add this product to the numerator
Write the sum from step 2 over the denominator of the fractional
part
Example: Write 7 3/8 as an improper fraction
Introduction to Fractions – Section
2.2
 Class Examples:

Write 22/5 as a mixed number


Write 28/7 as a whole number


4
Write 14 5/8 as an improper fraction


4 2/5
117/8
Write 10/3 as a mixed number

3 1/3
Writing Equivalent Fractions –
Section 2.3

Equivalent fractions are equal fractions that look
different


Remember the ones property in multiplication?


Example 4/6 is equivalent to 2/3
1 x Number = Number
Agree? 2/3 x 1 = 2/3



2/3 x 1/1 = 2/3
2/3 x 4/4 = 2/3 = 8/12
2/3 x 5000/5000 = 2/3 = 10000/15000
Writing Equivalent Fractions –
Section 2.3

Example: (Finding equivalent fractions) What is an equivalent
fraction to 5/8 that has a denominator of 32?



Ask yourself…self…what do I have to multiply the denominator
of 5/8 by to get 32?
Or you could just divide 32 by 8

4
5/8 x 1 = 5/8 x 4/4 = 20/32


20/32 is a fraction with 32 in the denominator that is equivalent
to 5/8
Another example

Write 2/3 as an equivalent fraction that has a denominator of 42

Divide 42 by 3 = 14


2/3 x 14/14 = 28/42 is equivalent to 2/3
Example write 4 as a fraction with 12 in denominator
Writing Equivalent Fractions –
Section 2.3
 Class Examples:


Write 3/5 as an equivalent fraction with a
denominator of 45
Fill in the blank
1.
2.
3.
½ = __ /32
2/3 = __ / 12
6 = __ / 11
Writing Equivalent Fractions –
Section 2.3
 A fraction is in simplest form when the
numerator and denominator have no common
factors (other than 1)

Example: 4/6 written in simplest form is 2/3
 To write a fraction in simplest form

Steps
1.
2.
Write prime factorization of the numerator and
denominator
Cancel (divide) out all common factors.
1.
Remaining products are the new Numerator and
Denominator
Writing Equivalent Fractions –
Section 2.3

Examples

Write 15/40 in simplest form


Write 6/42 in simplest form


= 3 x 5 / 2 x 2 x 2 x 5 = 3/8
= 2 x 3 / 2 x 3 x 7 = 1/7
Write 30/12 in simplest form

2 x 3 x 5 / 2 x 2 x 3 = 5/2 = 2 1/2
Writing Equivalent Fractions –
Section 2.3
 Class Examples:

Write the following in simplest form

16/24


8/56


2 x 2 x 2 / 2 x 2 x 2 x 7 = 1/7
15/32


= 2 x 2 x 2 x 2 / 2 x 2 x 2 x 3 = 2/3
= 3 x 5 / 2 x 2 x 2 x 2 x 2 = 15/32
48/36

= 2 x 2 x 2 x 2 x 3 / 2 x 2 x 3 x 3 = 4/3 = 1 1/3
Addition of Fractions and Mixed
Numbers 2.4

The key is the denominator. To add fractions together,
each fraction must have the same denominator.
 If the denominators are the same

Steps
1.
Add the Numerators
2.
Place the sum of the
Numerators over the
common denominator
1.
Write the sum in simplest form
5
12
+
11
12
=
16
12
= 4
3
Addition of Fractions and Mixed
Numbers 2.4

If denominators are not the same:

Steps
1.
Find the Lowest Common Denominator (LCD) of the two
fractions

Rewrite each fraction as an equivalent fraction with the
LCD as the denominator.
3.
Add the numerators
4.
Place this sum over the common denominator
Example: 1/2 + 1/3 = ?
 LCM = 6, then 3/6 + 2/6 = 5/6
2.

Note: this quantity is the LCM of the denominators
Addition of Fractions and Mixed
Numbers 2.4
 More Examples:

Find 7/12 more than 3/8

Lowest Common Denominator (LCD) = 24


Add 5/8 + 7/9

LCD = 72


14/24 + 9/24 = 23/24
45/72 + 56/72 = 101/72 = 1 29/72
Add 2/3 + 3/5 + 5/6

LCD = 30

20/30 + 18/30 + 25/30 = 63/30 = 2 3/30 = 2 1/10
Addition of Fractions and Mixed
Numbers 2.4
 Class Examples:

Find the sum of 5/12 and 9/16

LCM = 48


Add 7/8 + 11/15

LCM = 120


20/48 + 27/48 = 47/48
105/120 + 88/120 = 193/120 = 1 73/120
Add 3/4 + 4/5 + 5/8

LCM = 40

30/40 + 32/40 + 25/40 = 87/40 = 2 11/40
Addition of Fractions and Mixed
Numbers 2.4
 Addition of mixed numbers

Steps
1.
Find the Lowest Common Denominator (LCD) of the
two fractions

2.
3.
4.

Note: this quantity is exactly the (LCM) of the
denominators
Add the fractional parts
Add the whole number parts
Put fractional part in simplest form
Example: what is 6 14/15 added to 5 4/9 ?

LCM = 45, then5 20/45 + 6 42/45 = 11 62/45
= 11 + 1 17/45 = 12 17/45
Addition of Fractions and Mixed
Numbers 2.4
 More Examples:

Find 5 more than 3/8

LCD = Don’t need this


Add 17 + 3 3/8

LCD = Don’t need this


5 3/8
20 3/8
Add 5 2/3 + 11 5/6 + 12 7/9

LCD = 18

5 12/18 + 11 15/18 + 12 14/18 = 28 41/18 = 30 5/18
Addition of Fractions and Mixed
Numbers 2.4
 Class Examples:

Find the sum of 29 and 7 5/12

LCD = Don’t need this


Add 7 4/5 + 6 7/10 + 13 11/15

LCD = 30


46 5/12
7 24/30 + 6 21/30 + 13 22/30 = 26 67/30 = 28 7/30
Add 9 3/8 + 17 7/12 + 10 14/15

LCD = 120

9 45/120 + 17 70/120 + 10 112/120 = 36 227/120
= 37 107/120
Addition of Fractions and Mixed
Numbers 2.4
 Word problems discussion

Pg 80, You Try It 9


Add all time spent together
Pg 80, You Try It 10


Add all time spent working overtime
Multiply total time spent working overtime by the
overtime hourly rate.
Subtraction of Fractions and
Mixed Numbers 2.5
Again…the key is the denominator. To subtract
fractions, each fraction must have the same
denominator.
 If the denominators are the same


Steps
1.
Subtract the Numerators
2.
Place the difference of
the new numerators
over the common denominator
1.
Write the difference in simplest form
11
12
-
5
12
=
7
12
Subtraction of Fractions and
Mixed Numbers 2.5

If denominators are not the same:

Steps
1.
Find the Lowest Common Denominator (LCD) of the two
fractions

Rewrite each fraction as an equivalent fraction with the
LCD as the denominator.
3.
Subtract the numerators
4.
Place this difference over the common denominator
Example: 5/6 – 1/4 = ?
 LCM = 12, thus 10/12 - 3/12 = 7/12
2.

Note: this quantity is exactly the Least Common Multiple
(LCM) of the denominators
Subtraction of Fractions and
Mixed Numbers 2.5
 More Examples:

Subtract 3/4 - 2/5

LCD = 20


Subtract 53/60 - 7/12

LCD = 60


15/20 – 8/20 = 7/20
53/60 – 35/60 = 18/60 = 3/10
11/16 – 5/12 = ?

LCD = 48

33/48 - 20/48 = 13/48
Subtraction of Fractions and
Mixed Numbers 2.5
 Class Examples:

Subtract 5/6 – 4/15

LCD = 30


25/30 – 8/30 = 17/30
Subtract 13/18 – 7/24

LCD = 72

52/72 – 21/72 = 31/72
Subtraction of Fractions and
Mixed Numbers 2.5

Subtraction of mixed numbers
 Steps
1. Find the Lowest Common Denominator (LCD) of
the two fractions

2.
Subtract the fractional parts
1.
3.
Note: this quantity is the LCM of the denominators
Borrow if necessary
 Borrow 1 from the whole number part and rewrite it
as an equivalent fraction to 1 using with the same
LCD
Subtract the whole numbers
Subtraction of Fractions and
Mixed Numbers 2.5

Subtraction of mixed numbers

Example: (No Borrowing) what is 5 5/6 subtracted from 2 3/4?

LCD = 12


Example: (With Borrowing): Subtract 5 – 2 5/8

LCD = Don’t need it


5 10/12 – 2 9/12 = 3 1/12
4 8/8 – 2 5/8 = 2 3/8
Example: (With Borrowing): Subtract 7 1/6 – 2 5/8

LCD = 24

7 4/24 - 2 15/24 = 6 28/24 – 2 15/24 = 4 13/24
Subtraction of Fractions and
Mixed Numbers 2.5
 More Examples:

Subtract 15 7/8 – 12 2/3

LCD = 24


Subtract 9 – 4 3/11

LCD = Don’t need this


15 21/24 – 12 16/24 = 3 5/24
8 11/11 – 4 3/11 = 4 8/11
Find 11 5/12 decreased by 2 11/16

LCD = 48

11 20/48 – 2 33/48 = 10 68/48 – 2 33/48 = 8 35/48
Subtraction of Fractions and
Mixed Numbers 2.5
 Class Examples:

Subtract 17 5/9 – 11 5/12

LCD = 36


Subtract 8 – 2 4/13

LCD = Don’t need this


17 20/36 – 11 15/36 = 6 5/36
7 13/13 – 2 4/13 = 5 9/13
Find 21 7/9 minus 7 11/12

LCD = 36

21 28/36 – 7 33/36 = 21 64/36 – 7 33/36 = 14 31/36
Subtraction of Fractions and
Mixed Numbers 2.5
 Word problems discussion

Pg 88,

6


Add all time spent together
You Try It 7

How would you approach this problem?
 Add all the weight lost over the first two months
 13 ¼ pounds lost in the first two months
 Subtract 13 ¼ from the total of 24. (10 ¾ pounds
left)