Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.3 The Rational Numbers

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 5.3

The Rational Numbers


What You Will Learn

Rational NumbersMultiplying and Dividing FractionsAdding and Subtracting Fractions

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The Rational NumbersThe set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q ≠ 0.The following are examples of rational numbers:

1

3,

3

4,

7

8, 1

2

3, 2, 0,

15

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Fractions

Fractions are numbers such as:

The numerator is the number above the fraction line.The denominator is the number below the fraction line.

1

3,

2

9, and

9

53.

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Reducing Fractions

To reduce a fraction to its lowest terms, divide both the numerator and denominator by the greatest common divisor.

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Example 1: Reducing a Fraction to Lowest Terms

Reduce to lowest terms.

54

90

SolutionGCD of 54 and 90 is 18

54

90

54 18

90 18

3

5

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Mixed Numbers

A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. 3 ½ is read “three and one half” and means “3 + ½”.

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Improper FractionsRational numbers greater than 1 or less than –1 that are not integers may be written as mixed numbers, or as improper fractions.An improper fraction is a fraction whose numerator is greater than its denominator.An example of an improper fraction is .

12

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Converting a Positive Mixed Number to an Improper Fraction1. Multiply the denominator of the fraction in the mixed number by the integer preceding it.

2. Add the product obtained in Step 1 to the numerator of the fraction in the mixed number. This sum is the numerator of the improper fraction we are seeking. The denominator of the improper fraction we are seeking is the same as the denominator of the fraction in the mixed number.

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Example 2: Converting Mixed Numbers to Improper FractionsConvert the following mixed numbers to improper fractions.

a) 1

3

4

4 1 3

4

4 3

4

7

4

b) 3

7

8

8 3 7

8

24 78

31

8

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Converting a Positive Improper Fraction to a Mixed Number1. Divide the numerator by the denominator. Identify the quotient and the remainder.

2. The quotient obtained in Step 1 is the integer part of the mixed number. The remainder is the numerator of the fraction in the mixed number. The denominator in the fraction of the mixed number will be the same as the denominator in the original fraction.

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Example 3: From Improper Fraction to Mixed Number

Convert the following improper fraction to a mixed number.

a)

8

5

Solution

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Solution

The mixed number is

13

5.

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Example 3: From Improper Fraction to Mixed NumberConvert the following improper fraction to a mixed number.

b)

225

8

Solution

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Solution

The mixed number is

281

8.

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Terminating or Repeating Decimal Numbers

Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number.

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Terminating or Repeating Decimal Numbers

Examples of terminating decimal numbers are 0.5, 0.75, 4.65Examples of repeating decimal numbers 0.333… which may be written 0.2323… or and 8.13456456… or

0.3,

8.13456. 0.23,

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Example 5: Terminating Decimal NumbersShow that the following rational numbers can be expressed as terminating decimal numbers.

a)

3

5

b)

13

20 c)

23

16

= 0.6

= –0.65

= 1.4375

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Example 6: Repeating Decimal NumbersShow that the following rational numbers can be expressed as repeating decimal numbers.

a)

2

3 0.6

c) 1

5

36

b)

14

99 0.14

1.138

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Converting Decimal Numbers to FractionsWe can convert a terminating or repeating decimal number into a quotient of integers.The explanation of the procedure will refer to the positional values to the right of the decimal point, as illustrated here:

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Converting Decimal Numbers to Fractions

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Example 10: Converting a Repeating Decimal Number to a FractionConvert to a quotient of integers.

12.142

n 12.142

100n 1214.2

10 100n 10 1214.2

1000n 12142.2

1000n 12142.2

100n 1214.2

900n 10928

n

10,928

900

2732

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Multiplication of Fractions

a

bc

d

acb d

ac

bd, b 0, d 0

The product of two fractions is found by multiplying the numerators together and multiplying the denominators together.

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Example 11: Multiplying FractionsEvaluate

a)

3

57

8

375 8

c) 1

7

8

2

1

4

b)

2

3

4

9

2 4 3 9

15

89

4

21

40

8

27

135

32 4

7

32

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Reciprocal

3

1

31

The reciprocal of any number is 1 divided by that number.

The product of a number and its reciprocal must equal 1.

3

55

31

6

1

61

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Division of Fractions

a

b

c

d

a

bd

c

ad

bc, b 0, d 0, c 0

To find the quotient of two fractions, multiply the first fraction by the reciprocal of the second fraction.

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Example 12: Dividing FractionsEvaluate

a)

5

7

3

4

5 47 3

b)

3

5

7

8

385 7

20

21

24

35

5

74

3

3

58

7

24

35

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Addition and Subtraction of Fractions

a

c

b

c

a b

c, c 0;

a

c

b

c

a b

c, c 0

To add or subtract two fractions with a common denominator, we add or subtract their numerators and retain the common denominator.

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Example 13: Adding and Subtracting FractionsEvaluate

a)

1

8

3

8

4

8

b)

19

24

5

24

19 5

24

1

2

14

24

1 3

8

7

12

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Fundamental Law of Rational NumbersIf a, b, and c are integers, with b ≠ 0, and c ≠ 0, then

a

b

a

bc

c

acb c

a

b

acb c

and are equivalent fractions.

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Adding or Subtracting Fractions with Unlike DenominatorsWhen adding or subtracting two fractions with unlike denominators, first rewrite each fraction with a common denominator. Then add or subtract the fractions.

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Example 14: Subtracting Fractions with Unlike DenominatorsEvaluate

13

15

5

6

5

65

5

26

30

25

30

1

30

13

152

2

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Homework

P. 239 # 15 – 90 (x3)

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Documents

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 5.3 The Rational Numbers