Chapter 2 - Fractions I Math Skills – Week 2
Dec 25, 2015
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. Website Class listing and MyInfo Change your password Post Practice Final exam within the next week
2. 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:
2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 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. List the multiples of each number2. Identify the common multiples3. 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: 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,… Step 2:
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. Write the prime factorization of each number
2. Organize these prime factors into a “table of prime factors” (see pg. 65)
3. Circle the greatest product in each column
4. 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:
2 x 2 The prime factorization of 6 is:
2 x 3 Step 2: Step 3: Step 4:
LCM = 2 x 2 x 3 = 12
4 =
6 =
2 3
2 x 2
2 3
Organize
Circle greatest products
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: 2 x 2 x 2 x 3
Prime factorization of 36: 2 x 2 x 3 x 3
Prime factorization of 50: 2 x 5 x 5
Step 2: Prime Factors Table Step 3: Circle largest products Step 4: LCM is the product of circled quantities
2 x 2 x 2 x 3 x 3 x 5 x 5 = 1800
24 =
36 =
50 =
2 3 5
2 x 2 x 2
2 x 2
2
3
3 x 3
5 x 5
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
Step 1: Find prime factorization of each numberStep 2: Prime Factors TableStep 3: Circle largest productsStep 4: LCM product of circled quantities
Steps for finding LCM
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)
Steps1. List the factors of each number2. Identify the common factors .3. 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
1. 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
Importante! If column has a blank for one of the numbers, don’t circle anything for that column
4. 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:
2 x 3 x 3 x 5 The Prime Factorization of 168 is:
2 x 2 x 2 x 3 x 7 The Prime Factorization of 420 is:
2 x 2 x 3 x 5 x 7 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 =
158 =
420 =
2 3 5
2 x 2 x 2
2 x 2
3
3 x 3
5
7
2
3
7
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
Step 1: Find Prime factorization of each numberStep 2: Prime Factors tableStep 3: Circle Smallest Product in each column (ignore columns with blanks)Step 4: Product of circled numbers is GCF
Steps to find GCF
Introduction to Fractions – Section 2.2
A fraction is the representation of a specified portion of a whole number.
24
14
34
44
Numerator
Denominator
Fraction Bar
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 is greater than the denominator
34
341
74
Introduction to Fractions – Section 2.2
Convert Improper fractions Mixed numbers Steps
1. Divide the Numerator into the Denominator2. 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. Multiply the denominator of the fractional part by the whole number part
2. Add this product to the numerator3. 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
4 2/5 Write 28/7 as a whole number
4 Write 14 5/8 as an improper fraction
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
Example 4/6 is equivalent to 2/3
Remember the ones property in multiplication? 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. ½ = __ /32
2. 2/3 = __ / 12
3. 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. Write prime factorization of the numerator and denominator
2. 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
= 3 x 5 / 2 x 2 x 2 x 5 = 3/8 Write 6/42 in simplest form
= 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 = 2 x 2 x 2 x 2 / 2 x 2 x 2 x 3 = 2/3
8/56 2 x 2 x 2 / 2 x 2 x 2 x 7 = 1/7
15/32 = 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
512
+1112
1612
= 43
=
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 Note: this quantity is the LCM of the denominators
2. 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
Addition of Fractions and Mixed Numbers 2.4
More Examples: Find 7/12 more than 3/8
Lowest Common Denominator (LCD) = 24 14/24 + 9/24 = 23/24
Add 5/8 + 7/9 LCD = 72
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 20/48 + 27/48 = 47/48
Add 7/8 + 11/15 LCM = 120
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 Note: this quantity is exactly the (LCM) of the
denominators
2. Add the fractional parts3. Add the whole number parts4. 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 5 3/8
Add 17 + 3 3/8 LCD = Don’t need this
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 46 5/12
Add 7 4/5 + 6 7/10 + 13 11/15 LCD = 30
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
1112
-5
127
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 Note: this quantity is exactly the Least Common Multiple
(LCM) of the denominators
2. 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
Subtraction of Fractions and Mixed Numbers 2.5
More Examples: Subtract 3/4 - 2/5
LCD = 20 15/20 – 8/20 = 7/20
Subtract 53/60 - 7/12 LCD = 60
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 Note: this quantity is the LCM of the denominators
2. Subtract the fractional parts1. Borrow if necessary
Borrow 1 from the whole number part and rewrite it as an equivalent fraction to 1 using with the same LCD
3. 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 5 10/12 – 2 9/12 = 3 1/12
Example: (With Borrowing): Subtract 5 – 2 5/8 LCD = Don’t need it
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 15 21/24 – 12 16/24 = 3 5/24
Subtract 9 – 4 3/11 LCD = Don’t need this
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 17 20/36 – 11 15/36 = 6 5/36
Subtract 8 – 2 4/13 LCD = Don’t need this
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)