Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.3 Simplest Form of a Fraction.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

2.3

Simplest Form of a Fraction

Martin-Gay, Basic Mathematics, 4e 22


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




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



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



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



Examples


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




Practice Problems


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




Practice Problems


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



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



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



Equality of Fractions

p 133




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




Practice Problem 8Practice Problem 8

27

21 and

9


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




Do Practice Problem 8 AgainDo Practice Problem 8 Again

27

21 and

9


SOLUTIONSOLUTION

219277

9

7

27

21?

189 189

Since the cross products is equal, are equivalent.27

21 and

9

7

P 132




Practice Problem 9Practice Problem 9

18

5 and

13


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




Do Practice Problem 9 AgainDo Practice Problem 9 Again

18

5 and

13


SOLUTIONSOLUTION

513184

13

4

18

5?

72 65

Since the cross products are not equal, they are not equivalent.

P 133



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



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



DONE



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

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.3 Simplest Form of a Fraction.

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