copyright © 2011 pearson education, inc. publishing as prentice hall. 2.3 simplest form of a...
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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
2.3
Simplest Form of a Fraction
Martin-Gay, Basic Mathematics, 4e 22
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Writing Fractions in Simplest Form
Fractions that represent the same portion of a whole are called equivalent fractions.
There are many equivalent forms of a fraction. A special form of a fraction is called simplest form.
Simplest Form of a FractionA fraction is written in simplest form or lowest terms when the numerator and denominator have no common factors other than 1.
p 129
Martin-Gay, Basic Mathematics, 4e 33
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Writing Fractions in Simplest Form
Definition Example
Equivalent Fractions: Fractions that represent the same portion of a whole.
Simplest Form or Lowest Terms: When the numerator and the denominator of a fraction have no common factors other than 1.
3
2
6
4
12
8
42
4Both have a common factor
of 2.
21
2
242
24
There are no more common factors.
The simplest form of is 42
4.
21
2
P 129
732
22
42
4
21
2
73
2
Alternative method
Martin-Gay, Basic Mathematics, 4e 44
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Examples
Write in simplest form.
a.
b.
c.
9
273 3
3 3 3
5 2
21 2
3 3 1
3 3 3
1
3
10
42
5 51
21 21
8
36
4 2
4 9
2 2
19 9
p 129
Martin-Gay, Basic Mathematics, 4e 55
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Simplest Form
Writing a Fraction in Simplest FormTo write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors.
p 129
Martin-Gay, Basic Mathematics, 4e 66
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Examples
Write in simplest form.
a.
b.
c.
9
90
3 3
3 3 5 2
5 5 2
11 2
50
22
25
11
35
75
5 7
5 5 3
1
10
7
15
p 130
Martin-Gay, Basic Mathematics, 4e 77
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Writing Fractions in Simplest Form
Practice Problems
Write in simplest form.
1.
2.
3.
4.
P 130
factor 533
532
simply3
245
30
51
39
173
133
17
13
50
9
552
33
50
9
112
49
72222
77
2222
7
16
7
1 5
5
3
3equaland
No common Factors
Martin-Gay, Basic Mathematics, 4e 88
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Writing Fractions in Simplest Form
Practice Problems
Write in simplest form.
5.
6.
7.
P 131
522
222222
5
13or
5
16
20
64
56
8
7222
222
7
1 becomes
48
42
832
732
8
7
5
2222
7111
111
1or 1
1 becomes
2
2Each
Martin-Gay, Basic Mathematics, 4e 99
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Example
Determine whether are equivalent.6
2 and
9
3
Simplify each fraction.
3 3 1 = 9 3 3 3
2 2 1 =6 3 2
3
Since both of the simplified fractions are the same, they are equivalent.
p 133
Martin-Gay, Basic Mathematics, 4e 1010
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Determining Whether Two Fractions Are Equivalent
EXAMPLEEXAMPLE
6
2 and
9
3 Determine whether are equivalent.
SOLUTIONSOLUTION
3. offactor common a has 9
3 2. offactor common a has
6
2
33
3
32
2
3
1
3
1
Since both have a common simplified form of they are equivalent.6
2 and
9
3
3
1
P 132
Martin-Gay, Basic Mathematics, 4e 1111
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Equality of Fractions
p 133
Martin-Gay, Basic Mathematics, 4e 1212
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Determining Whether Two Fractions Are Equivalent
EXAMPLEEXAMPLE
11
4 and
5
2 Determine whether are equivalent by cross multiplying.
SOLUTIONSOLUTION
45112
5
2
11
4?
22 20
Since the cross products are not equal, are not equivalent.11
4 and
5
2
P 133
Martin-Gay, Basic Mathematics, 4e 1313
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Determining Whether Two Fractions Are Equivalent
Practice Problem 8Practice Problem 8
27
21 and
9
7 Determine whether are equivalent.
SOLUTIONSOLUTION
factors.common no has 9
7 3. offactor common a has
27
21
333
73
9
7
9
7
Since both have a common simplified form of they are equivalent.27
21 and
9
7
9
7
P 132
Martin-Gay, Basic Mathematics, 4e 1414
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Determining Whether Two Fractions Are Equivalent
Do Practice Problem 8 AgainDo Practice Problem 8 Again
27
21 and
9
7 Determine whether are equivalent by cross multiplying.
SOLUTIONSOLUTION
219277
9
7
27
21?
189 189
Since the cross products is equal, are equivalent.27
21 and
9
7
P 132
Martin-Gay, Basic Mathematics, 4e 1515
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Determining Whether Two Fractions Are Equivalent
Practice Problem 9Practice Problem 9
18
5 and
13
4 Determine whether are equivalent.
SOLUTIONSOLUTION
factors.common no has 13
4 factors.common no has
18
5
13
4
18
5
Since they do not have a common simplified form they are not equivalent.
P 133
Martin-Gay, Basic Mathematics, 4e 1616
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Determining Whether Two Fractions Are Equivalent
Do Practice Problem 9 AgainDo Practice Problem 9 Again
18
5 and
13
4 Determine whether are equivalent by cross multiplying.
SOLUTIONSOLUTION
513184
13
4
18
5?
72 65
Since the cross products are not equal, they are not equivalent.
P 133
Martin-Gay, Basic Mathematics, 4e 1717
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solving Problems by WritingFractions in Simplest Form
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
There are 5280 feet in a mile. What fraction of a mile is represented by 2640.
5280
2640 Both 2640 and 5280 have a common factor of 2640.
26405280
26402640
2
1
The fraction of a mile represented by 2640 is .2
1
P 133
Martin-Gay, Basic Mathematics, 4e 1818
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Solving Problems by WritingFractions in Simplest Form
Practice Problem 10 (alternate)Practice Problem 10 (alternate)
SOLUTIONSOLUTION
Eighty pigs were used in a recent study of olestra, a calorie-free fat substitute. A group of 12 of these pigs were fed a diet high in fat. What fraction of the pigs were fed the high-fat diet in this study? Write your answer in simplest form.
80
12
Both 12 and 80 have a common factor of 4.
204
34
20
3
The fraction for the group of 12 pigs is represented by .20
3
P 133
Martin-Gay, Basic Mathematics, 4e 1919
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
DONE
Martin-Gay, Basic Mathematics, 4e 2020
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Example
Determine whether are equivalent by cross multiplying.
2 4 and 5 11
Since 22 ≠20, then
5 42 11
5
2
11
4?22 20
2 4.
5 11
p 133