Transcript Chapter 2 - Mesa Community College

Chapter 2 - Fractions
Section 2.1
Factors and Prime Numbers
1-2
Chapter 2 – Slide 2
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What Factors Mean and Why They
are Important

In a multiplication problem, the whole numbers
that we are multiplying are factors.
23  6
The 2 and 3 are both factors of 6.

1-3
To identify the factors of a whole number, we
divide the whole number by the numbers 1, 2, 3, 4,
5, 6, and so on, looking for remainders of 0.
Chapter 2 – Slide 3
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Divisibility Tests
1-4
Chapter 2 – Slide 4
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Example What are the factors of
54?
Is 54 divisible by
Answer
1
2
3
4
5
6
7
8
9
10
1-5
Chapter 2 – Slide 5
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Identifying Prime and Composite
Numbers

A prime number is a whole number that has
exactly two different factors: Itself and 1.

A composite number is a whole number that has
more than two factors. That is, whole numbers
that are not prime.
1-6
Chapter 2 – Slide 6
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Example
Indicate whether each number is prime or
composite.
a. 3
b. 96
c. 47
d. 39
1-7
Chapter 2 – Slide 7
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Finding the Prime Factorization of
a Number
The prime factorization of a whole number
is the number written as the product of its
prime factors.
A good way to find the prime factorization of
a number is by making a factor tree.
1-8
Chapter 2 – Slide 8
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Examples
What is the prime factorization of the following.
Create a factorization tree for each.
(a) 90
(b) 720
1-9
Chapter 2 – Slide 9
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Finding the Least Common Multiple
The least common multiple (LCM) of two or more whole
numbers is the smallest nonzero whole number that is a
multiple of each number.
To Computer The Least Common Multiple (LCM)
1. Find the prime factorization of each number.
2. Identify the prime factors that appear in each factorization.
3. Multiply these prime factors, using each factor the greatest
number of times that it occurs in any factorization.
1-10
Chapter 2 – Slide 10
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Examples
1.
2.
3.
1-11
Find the LCM of 9 and 15.
Find the LCM of 4, 6 and 18.
The prime factors are 3 and 5, 3 occurs twice and
5 occurs once. What is the LCM.
Chapter 2 – Slide 11
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Example
You walk dogs to make some extra money. You walk
a poodle every 4th day and a retriever every 6th day.
You walked both dogs today. In how many days will
you walk both dogs again?
1-12
Chapter 2 – Slide 12
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Section 2.2
Introduction to Fractions
1-13
Chapter 2 – Slide 13
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What Fractions Are and Why They
are Important





1-14
A fraction can mean part of a whole.
A fraction is any number that can be written in the form a /
b, where a and b are whole numbers and b is not zero.
The denominator (on the bottom) stands for the number of
parts into which the whole is divided.
The numerator (on top) tell us how many parts of the
whole the fraction contains.
The fraction line separates the numerator from the
denominator and stands for “out of” or “divided by”
Chapter 2 – Slide 14
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Fraction Diagrams and Proper
Fractions

3
4
The number
is an example of a proper fraction
because its numerator is smaller than its
denominator.
Example: In the diagram what does the
shaded portion represent?
The whole is divided into ____ parts.
 There are ___ shaded parts.
 The shaded portion represents _____.

1-15
Chapter 2 – Slide 15
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Example
The college of education accepted 115 out of 175
applicants for admission. What fraction of the
applicants were accepted?
1-16
Chapter 2 – Slide 16
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Mixed Numbers and Improper
Fractions
A number with a whole number part and a proper
fraction, is called a mixed number.
A mixed number can also be expressed as an
improper fraction. An improper fraction has a
numerator greater than or equal to its denominator.
1-17
Chapter 2 – Slide 17
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Example
Draw a diagram to show that
1 13
3  .
4 4
1
3 
4
13

4
1-18
Chapter 2 – Slide 18
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Changing Mixed Numbers to
Improper Fractions
1.
2.
3.
1-19
Multiply the denominator of the fraction by the
whole-number part of the mixed number.
Add the numerator of the fraction to this product.
Write this sum over the denominator to form the
improper fraction.
Chapter 2 – Slide 19
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Example
Write the following mixed number as an improper
fraction.
1. 4 2
2.
7
3.
1-20
2
2
9
4.
3
5
4
3
7
5
Chapter 2 – Slide 20
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Changing Improper Fractions to
Mixed Numbers
1.
2.
Divide the numerator by the denominator to get
the whole part.
If there is a remainder, write that remainder over
the denominator
Example Write each improper fraction as a mixed
number.
29
95
a.
b.
4
9
1-21
Chapter 2 – Slide 21
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Finding Equivalent Fractions
Equivalent fractions are fractions that have the same value as
the original fraction. For example, the whole number 2 can be
written as
4
6
40
or or
2
3
20
To find an equivalent fraction, multiply the numerator and
a
denominator of b by the same number n,
a an

b bn
Where both b and n are nonzero.
1-22
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Chapter 2 – Slide 22
Example
Find two fractions equivalent to:
a. 3
5
b. 7
9
1-23
Chapter 2 – Slide 23
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Reducing Fractions
A fraction is said to be in simplest form (or
reduced to lowest terms) when the only common
factor of its numerator and denominator is 1.
1-24
Chapter 2 – Slide 24
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Example
Express each in simplest form (or reduce to lowest
terms).
a. 4
16
b. 42
35
1-25
Chapter 2 – Slide 25
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Like and Unlike Fractions
Definitions
Like fractions are fractions with the same
denominator. Unlike fractions are fractions with
different denominators.
To Compare Fractions
 If the fractions are like, compare their numerators.
 If the fractions are unlike, write them as equivalent
fractions with the same denominator and then
compare their numerators.
1-26
Chapter 2 – Slide 26
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Example
8
7
Compare
and . Which fraction is greater?
15
9
1-27
Chapter 2 – Slide 27
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Least Common Denominator (LCD)
For two or more fractions, their least common
denominator (LCD) is the least common multiple of
their denominators.
1-28
Chapter 2 – Slide 28
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Example
1. Write as equivalent fractions with a common
denominator of 18.
5
6
1
2
2. Order from smallest to largest:
1-29
13
18
5 1 13
, ,
6 2 18
Chapter 2 – Slide 29
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Section 2.3
Adding and Subtracting
Fractions
1-30
Chapter 2 – Slide 30
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Adding and Subtracting Like
Fractions
To add (or subtract) like fractions,
add (or subtract) the numerators.
Use the given denominator. Write
answer in simplest form.
1-31
Chapter 2 – Slide 31
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Examples
1.
4 8
Add:
 .
15 15
2.
5 12
7
Find the sum of
,
and
.
21 21
21
3.
Find the difference between
1-32
15
2
and .
9
9
Chapter 2 – Slide 32
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Adding and Subtracting Unlike
Fractions
To Add (or Subtract) Unlike Fractions



1-33
Write the fractions as equivalent fractions with the
same denominator, usually the LCD.
Add (or subtract) the numerators, keeping the same
denominator.
Writhe the answer in simplest form.
Chapter 2 – Slide 33
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Examples
1.
6
3
Add:

16 24
2.
Subtract: 7  3
12 8
3.
Combine:
1-34
2 5 1
 
3 6 10
Chapter 2 – Slide 34
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Adding Mixed Numbers
To Add Mixed Numbers




1-35
Write the fraction as equivalent fractions with the
same denominator, usually the LCD.
Add the fractions.
Add the whole numbers.
Write the answer in simplest form.
Chapter 2 – Slide 35
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Examples
1.
2
1
Add: 4  3
5
5
2.
Find the sum of 3 3 , 4 7 , and 7.
8 8
3.
Find the sum of
1-36
1
5
3
1  4 and 2 .
4
6
5
Chapter 2 – Slide 36
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Subtracting Mixed Numbers
To Subtract Mixed Numbers





1-37
Write the fractions as equivalent fractions with the
same denominator, usually the LCD.
Rename (or borrow from) the whole number on top if
the fraction on the bottom is larger than the fraction on
top.
Subtract the fraction.
Subtract the whole number.
Write the answer in simplest form.
Chapter 2 – Slide 37
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Examples
1.
2.
7 1
Subtract: 4  1
9 9
Subtract: 15 1  9 11
12
12
3.
Subtract:
4.
Subtract
1-38
2
8 1
9
3
1
10  4
8
16
Chapter 2 – Slide 38
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Example
5
5
1.
Add: 5  7
6
8
Sometimes it is easier to add or subtract mixed
numbers by first converting them to improper
fractions.
1
7
1.
Subtract: 13  11
4
8
1  4
3
2.
Combine and check. 6  1  2 
3  5
10 
1-39
Chapter 2 – Slide 39
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Section 2.4
Multiplying and Dividing
Fractions
2-40
Chapter 2 – Slide 40
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Multiplying Fractions
To Multiply Fractions
• Multiply the numerators.
• Multiply the denominators
• Write the answer in simplest form.
2-41
Chapter 2 – Slide 41
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Examples
2 8
1. 
3 9
4

2.  
9
2
3
3. What is of 18?
8
3 7 8
4.  
4 9 3
5. A recipe calls for ¼ cup of sugar. If Betty is
making ½ the recipe, how much sugar should she
use?
1-42
Chapter 2 – Slide 42
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Multiplying Mixed Numbers
To Multiply Mixed Numbers
• Convert the mixed numbers to improper
fractions.
• Multiply the fractions.
• Write the answer in simplest form.
2-43
Chapter 2 – Slide 43
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Examples
1.
1 3
2 1
2 4
 4  1
2.  5   3  2 
 9  7
8 ¼ ft
3. An area rug is being put down as depicted in the
following drawing. How many square feet of
flooring is not covered by the rug?
12 ½ ft
12 ft
15 ½ ft
1-44
Chapter 2 – Slide 44
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Dividing Fractions
To Divide Fractions and Mixed Numbers
• Convert mixed numbers to improper fractions.
• Change the divisor to it’s reciprocal (flip over)
and multiply.
• Write your answer in simplist form.
2-45
Chapter 2 – Slide 45
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Examples
3 7
1. 
4 8
3
2. 20  3
5
7
3. What is divided by 8?
15
7 2
4. 1  1
8 3
1-46
3
1
1
5. 4  2  3
8
7
4
Chapter 2 – Slide 46
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Example
1
Mrs. Henderson prepares a 5 pound roast for a
2
celebration. How many servings have been
prepared if each person receives 1/3 lb of roast?
1-47
Chapter 2 – Slide 47
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Checking Answers
As with adding and subtracting mixed numbers, it
is important to check our answers when
multiplying or dividing. We can check a product
or a quotient of mixed numbers by estimating the
answer and then confirming that our estimate
answers are reasonably close.
2-48
Chapter 2 – Slide 48
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Example
3
1
1
1. Simplify and check: 4  2  3
8
7
4
1-49
Chapter 2 – Slide 49
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