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Chapter Chapter 11Section Section 11
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Fractions
Learn the definition of factor.Write fractions in lowest terms.Multiply and divide fractions.Add and subtract fractions.Solve applied problems that involve fractions.Interpret data in a circle graph.
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44
33
22
66
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1.11.11.11.1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Definitions
Natural numbers: 1, 2, 3, 4,…,
, ,NumeratorFraction BarDenominator
Ex. The improper fraction can be written , a mixed number.
1
2
2
3
15
7
12
5
22
5
Whole numbers: 0, 1, 2, 3, 4,…,
Fractions:
Proper fraction: has a value of less then 1; the numerator is smaller than or equal to the denominator.
Improper fraction: has a value of greater then 1; the numerator is larger than the denominator.
Mixed number: is a combination of a whole number and a fraction.
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Objective 11
Learn the definition of factor.
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Learn the definition of factor.
In the statement 2 × 9 = 18, the numbers 2 and 9 are called factors. Other factors of 18 include 1, 3, 6, and 18. The number 18 in this statement is called a product.
The number 18 is factored by writing it as a product of two or more numbers.
Ex. 6 ·3, 18 × 1, (2)(9), or 2(3)(3)
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Learn the definition of factor. (cont’d)
A natural number greater than 1 is prime if its products include only 1 and itself.
Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,…
A natural number greater than 1 that is not prime is called a composite number.
Ex. 4, 6, 8, 9, 10, 12,…
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EXAMPLE 1
Starting with the smallest prime factor is not necessary. No matter which prime factor is started with the same prime factorization will always be found.
Write 90 as the product of prime factors.
Solution:
2 45 2 3 15 2 3 3 5
Factoring Numbers
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Objective 22
Write fractions in lowest terms.
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A fraction is in lowest terms, when the numerator and denominator have no common factors other than 1.
Basic Principle of Fractions:If the numerator and denominator are multiplied or divided by the same nonzero number, the fraction remains unchanged.
Writing fractions in lowest terms.
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Writing a Fraction in Lowest Terms:
Step 1: Write the numerator and the denominator as the product of prime factors.
Writing fractions in lowest terms. (cont’d)
Step 2: Divide the numerator and denominator by the greatest common factor, the product of all
factors common to both.
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EXAMPLE 2
When writing fractions in lowest terms, be sure to include the factor 1 in the numerator or an error may result.
Write in lowest terms.
Solution:
=
12
20
3 4
45
3
5
Writing Fractions in Lowest Terms
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Objective 33
Multiply and divide fractions.
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Multiply and divide fractions.
Multiplying Fractions:
If and are fractions, then · = .a
b
a
b
c
d
c
d
a c
b d
That is, to multiply two fractions, multiply their numerators and then multiply their denominators.
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Dividing Fractions:
If and are fractions, then ÷ = .a
b
a
bc
d
c
da d
b c
Multiply and divide fractions. (cont’d)
That is, to divide two fractions, is to multiply its reciprocal; the fraction flipped upside down.
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EXAMPLE 3
Find each product, and write it in lowest simple terms.
Solution:
2
3
7 3 2
3 2 7
2
3
1 33 1
3 4
10 7
3 4
5 7
3
2
22
35
6
55
6
7 12
9 14
or
Multiplying Fractions
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EXAMPLE 4
Find each quotient, and write it in lowest terms.
Solution:
or9 3
10 5
3 53
2 5 3
9 5
10 3
3
2
11
2
3 12 3
4 3
11 10
4 3 11 3
4 10
33
40
Dividing Fractions
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Objective 44
Add and subtract fractions.
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Add and subtract fractions.
Adding Fractions:
If and are fractions, then + = .a
b
c
b
a
b
c
b
a c
b
That is, to find the sum, the result of adding the numbers, having the same denominator, add the numerators and keep the same denominator.
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Add and subtract fractions. (cont’d)
If the fractions do not share a common denominator. The least common denominator (LCD) must first be found as follows:
Step 1: Factor each denominator.
Step 2: Use every factor that appears in any factored form. If a factor is repeated, use the largest number of repeats in the LCD.
Step 3: Find the number that can be multiplied by the denominator to get the LCD and multiply the
numerator and denominator by that number.
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Add and subtract fractions. (cont’d)
Subtracting Fractions:
If and are fractions, then .a
bc
b
a c a c
b b b
If fractions have different denominators, find the LCD using the same method as with adding fractions.
That is, to find the difference, the result of subtracting the numbers, between two fractions having the same denominator subtract the numerators and keep the same denominator.
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EXAMPLE 5
Find the sum , and write it in lowest terms.
Solution:
1 5
9 9
1 5
9
6
9
3
2
3
3
2
3
Adding Fractions with the Same Denominator
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Find each sum, and write it in lowest terms.
EXAMPLE 6
7 2
30 45
21 4
90
25
90
5
3 3
5
2 5
5
18
5 14 2
6 3 29 14
6
43
6
17
6or
Adding Fractions with DifferentDenominators
Solution:
3 2
3 2
7 2
30 45
29 7
6 23
2
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EXAMPLE 7 Subtracting Fractions
Find each difference, and write it in lowest terms.
EXAMPLE 7
3 1
10 4
3 13 1
8 2
6 5
20
1
20
27 12
8
15
8
71
8or
Solution:2 5
2
3 1
0 51 4
27 3
8 42
4
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Objective 55
Solve applied problems that involve fractions.
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22
14200
500
galft
ft
EXAMPLE 8
A gallon of paint covers 500 ft2. To paint his house, Tran needs enough paint to cover 4200 ft2. How many gallons of paint should he buy?
Solution:
Tran needs to buy 9 gallons of paint.
22 500
42001
ftft
gal 4200
500
gal
142
05
00
10
gal
42
.5
gal2
85
Adding Fractions to Solve an Applied Problem
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Objective 66
Interpret data in a circle graph.
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EXAMPLE 9
In November 2005, there were about 970 million Internet users world wide.
Using a Circle Graph to Interpret Information
How many actual Internet users were there in Europe?
Estimate the number of Internet users in Europe.
Which region had the second-largest number of Internet Users in November 2005?
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Solution:
a) Europe
b)
c)
EXAMPLE 9 Solutions
31000 300
10million million
3970 291
10million million
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