Transcript Write fractions in lowest terms.

Chapter 1
Section 1
1.1 Fractions
Objectives
1
Learn the definition of factor.
2
Write fractions in lowest terms.
3
Multiply and divide fractions.
4
Add and subtract fractions.
5
Solve applied problems that involve fractions.
6
Interpret data in a circle graph.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Definitions
Natural numbers: 1, 2, 3, 4,…,
Whole numbers: 0, 1, 2, 3, 4,…,
Fractions: 1 2 15
Numerator
, ,
Fraction Bar
2 3 7
Denominator
Proper fraction: Numerator is less than denominator and the value is
less than 1.
Improper fraction: Numerator is greater than or equal to denominator
and the value is greater than or equal to 1.
Mixed number: A combination of a natural number and a proper
fraction.
Example:
The improper fraction
23
4
can be written
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3
5
4
, a mixed number.
Slide 1.1-3
Objective 1
Learn the definition of factor.
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Slide 1.1-4
Learn the definition of factor.
In the statement 3 × 6 = 18, the numbers 3 and 6 are called factors of
18. Other factors of 18 include 1, 2, 9, and 18. The number 18 in this
statement is called the product.
The number 18 is factored by writing it as a product of two or more
numbers.
Examples:
6 · 3,
18 × 1, (2)(9), 2(3)(3)
A raised dot • is often used instead of the × symbol to indicate multiplication
because × may be confused with the letter x.
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Slide 1.1-5
Learn the definition of factor. (cont’d)
A natural number greater than 1 is prime if its factors include only 1
and itself.
Examples:
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.
Examples:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21…
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Slide 1.1-6
EXAMPLE 1 Factoring Numbers
Write 90 as the product of prime factors.
Solution:
 2  45
 2  3 15
 2  3 3 5
Starting with the least prime factor is not necessary. No matter which prime
factor we start with, the same prime factorization will always be obtained.
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Slide 1.1-7
Objective 2
Write fractions in lowest terms.
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Slide 1.1-8
Write fractions in lowest terms.
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 of a fraction are multiplied or
divided by the same nonzero number, the value of the fraction
remains unchanged.
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Slide 1.1-9
Write fractions in lowest terms. (cont’d)
Writing a Fraction in Lowest Terms
Step 1: Write the numerator and the denominator as the
product of prime factors.
Step 2: Divide the numerator and the denominator by the
greatest common factor, the product of all factors
common to both.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.1-10
EXAMPLE 2 Writing Fractions in Lowest Terms
12
Write
in lowest terms.
20
Solution:
3
3 4
=
5
54
When writing fractions in lowest terms, be sure to include the factor 1 in the
numerator or an error may result.
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Slide 1.1-11
Objective 3
Multiply and divide fractions.
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Slide 1.1-12
Multiply and divide fractions.
Multiplying Fractions
c
a c a c
a
If
and
are fractions, then
·
=
.
b
b d bd
d
That is, to multiply two fractions, multiply their numerators and then
multiply their denominators.
Some prefer to factor and divide out any common factors before multiplying.
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Slide 1.1-13
EXAMPLE 3 Multiplying Fractions
Find each product, and write it in lowest terms.
Solution:
7 12

9 14
2
7 3 2  2


3
33 2  7
1 3
3 1
3 4
10 7
 
3 4
257

3 2  2
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35

6
or
5
5
6
Slide 1.1-14
Multiply and divide fractions. (cont’d)
Dividing Fractions
If
a
and
b
c
are fractions, then
d
a
÷
b
c
=
d
ad
.
bc
That is, to divide by a fraction, multiply by its reciprocal; the fraction
flipped upside down.
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Slide 1.1-15
EXAMPLE 4 Dividing Fractions
Find each quotient, and write it in lowest terms.
Solution:
9 3

10 5
95

10  3
11 10
3
1
 
2 3
4 3
4
3
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335

253
3

2
11  3

4 10
33

40
or
1
1
2
Slide 1.1-16
Objective 4
Add and subtract fractions.
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Slide 1.1-17
Add and subtract fractions.
Adding Fractions
If
a
b
and
c
b
are fractions, then
a
b
+
c
b
=
ac
.
b
To find the sum of two fractions having the same denominator, add
the numerators and keep the same denominator.
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Slide 1.1-18
EXAMPLE 5 Adding Fractions with the Same Denominator
Find the sum
1 5

9 9
, and write it in lowest terms.
Solution:
1 5
9
6

9
23

33
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2

3
Slide 1.1-19
Add and subtract fractions. (cont’d)
Finding the Least Common Denominator
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.
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Slide 1.1-20
EXAMPLE 6 Adding Fractions with Different Denominators
Find each sum, and write it in lowest terms.
Solution:
7
2

30 45
21  4
7 3
22



90
30  3 45  2
25

90
55

2 335
5

18
5
1
4 2
6
3
29 7  2


6 3 2
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29  14

6
43

6
or
1
7
6
Slide 1.1-21
Add and subtract fractions. (cont’d)
Subtracting Fractions
If
a
b
and
c
b
are fractions, then
a c ac
 
b b
b
.
To find the difference between two fractions having the same
denominator, subtract the numerators and keep the same
denominator.
If fractions have different denominators, find the LCD using the same
method as with adding fractions.
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Slide 1.1-22
EXAMPLE 7 Subtracting Fractions
Find each difference, and write it in lowest terms.
Solution:
3 1

10 4
3  2 1 5


10  2 4  5
3 1
3 1
8 2
27 3  4


8 24
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65

20
1

20
27  12

8
15

8
or
7
1
8
Slide 1.1-23
Objective 5
Solve applied problems that involve
fractions.
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Slide 1.1-24
EXAMPLE 8 Adding Fractions to Solve an Applied Problem
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:
2
500
ft
4200 ft 2 
1gal
1gal
 4200 ft 
500 ft 2
2
42 100 gal

5 100
4200 gal

500
42

gal
5
2
8
5
Tran needs to buy 9 gallons of paint.
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Slide 1.1-25
Objective 6
Interpret data in a circle graph.
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Slide 1.1-26
EXAMPLE 9 Using a Circle Graph to Interpret Information
Recently there were about 970 million Internet users world wide. The
circle graph below shows the fractions of these users living in various
regions of the world.
Which region had the second-largest number of Internet Users?
Estimate the number of Internet users in Europe.
How many actual Internet users were there in Europe?
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Slide 1.1-27
EXAMPLE 9 Using a Circle Graph to Interpret Information (cont’d)
Solution:
a)
Europe
b)
3
1000million   300million
10
c)
3
970million   291million
10
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Slide 1.1-28