Fractions and Rational Numbers

6.1 The Basic Concepts of Fractions and Rational Numbers

6.2 Addition and Subtraction of Fractions6.3 Multiplication and Division of Fractions6.4 The Rational Number System

Copyright © 2012, 2009, and 2006, Pearson Education, Inc.

6.1

Slide 6-2

The Basic Concepts of Fractions and Rational Numbers

Copyright © 2012, 2009, and 2006, Pearson Education, Inc.

THE MEANING OF A FRACTION

To interpret the meaning of any fraction we must:

• agree on the unit;

• understand that the unit is subdivided into b parts of equal size;

• understand that we are considering a of the parts of the unit.

ab

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DEFINITION:FRACTION

A fraction is an ordered pair of integers a and b, b ≠ 0, written or a/b.

• The integer a is called the numerator of the fraction.

• The integer b is called the denominator of the fraction.

ab

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MODELS FOR FRACTIONS

A physical or pictorial representation of a fraction must clearly answer the following questions:

• What is the unit? (the whole)

• Into how many equal parts has the unit been subdivided? (the denominator)

• How many of these parts are under consideration? (the numerator)

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MODELS FOR FRACTIONS:COLORED REGIONS

A shape is chosen to represent the unit and is then subdivided into subregions of equal size.

14

412

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MODELS FOR FRACTIONS:THE SET MODEL

Each subset A of U corresponds to the fraction

310

.( )( )n An U

of the apples have worms.

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MODELS FOR FRACTIONS:FRACTION STRIPS

The unit is defined by a rectangular strip of cardstock. A set of fraction strips typically contains strips for the denominators 1, 2, 3, 4, 6, 8, and 12.

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MODELS FOR FRACTIONS:THE NUMBER-LINE

Fractions can be modeled by subdividing the unit interval into equal parts determined by the denominator and then counting off the number of those parts determined by the numerator.

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THE FUNDAMENTAL LAW OF FRACTIONS

Let be a fraction. Then ab

, for any integer 0. nanbn

ab

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THE CROSS-PRODUCT PROPERTY OF EQUIVALENT FRACTIONS

The fractions are equivalent if,

and only if, ad = bc. That is,

and c

d

ab

, if, and only if, . c

ad bcd

ab

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FRACTIONS IN SIMPLIEST FORM

A fraction is in simplest form if a and b have no common divisor larger than 1 and b is positive.

ab

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28

48

7 4

12 4

7

12

Example 6.3 Finding Common Denominators

Find equivalent fractions to with a common denominator of 12.

1 and

4

56

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DEFINITION:RATIONAL NUMBERS

A rational number is a number that can be represented by a fraction , where a and b are integers, b ≠ 0.

Two rational numbers are equal if, and only if, they can be represented by equivalent fractions.

ab

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Example 6.4 Representing Rational Numbers

How many different rational numbers are given in this list of five fractions?

4 39 7, 3, , , and

10 13 4

25

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4 3 39Since and , there are

10 1 132 7

three different rational numbers; , 3, and .5 4

25

6.2

Slide 6-16

Addition and Subtraction of Fractions

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DEFINITION:ADDITION OF FRACTIONS

Let two fractions have a

common denominator. Then their sum is the fraction given by

and a c

b b

.

a c a c

b b b

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MODELING ADDITION OF FRACTIONSWITH COLORED REGIONS

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MODELING ADDITION OF FRACTIONSWITH THE NUMBER-LINE

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MODELING ADDITION OF FRACTIONSWITH UNLIKE DENOMINATORS

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MIXED NUMBERS

A mixed number can always be rewritten in the standard form

bAc

Ac b

c c

Ac b

c

23

53 5 2

5

17

5

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MIXED NUMBERS

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Example 6.7 Working with Mixed Numbers

a. Give an improper fraction for 3 .17120

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3 173

1 120

17120

3 120 1 17

120

360 17

120

377

120

Example 6.7 Working with Mixed Numbers

b. Give a mixed number for

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.355133

355133

DEFINITION:SUBTRACTION OF FRACTIONS

Let be fractions.

Then

if, and only if,

and a c

b d

a c e

b d f

. a c e

b d f

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MODELING SUBTRACTION OF FRACTIONS WITH FRACTION STRIPS

1

4

7

12

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5

6

6.3

Slide 6-27

Multiplication and Division of Fractions

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DEFINITION:MULTIPLICATION OF FRACTIONS

Let be fractions.

Then their product is given by

and a c

b d

. a c ac

b d bd

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Example 6.10 Calculating Products of Fractions

5 2

8 3

ab

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Example 6.12 Multiplying Fractions on the Number Line

Illustrate why with a number-line diagram.

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4 8

5 15

23

THE INVERT-AND-MULTIPLY ALGORITHM FOR DIVISION OF FRACTIONS

, a c a d ad

b d b c bc

Note that this is a process for dividing fractions, not a definition of division.

where 0.c

d

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Example 6.15 Dividing Fractions

Compute.

a.

b.

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1

8

34

1

68

68

1

or 8

34

8 24

61 4

34

14 2

3

16

7 25 3

3 6 7

256

25 3

6 7

25 1

2 7

25

14

6.4

Slide 6-33

The Rational Number System

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DEFINITION:NEGATIVE OR ADDITIVE INVERSE

Let be a rational number.

Its negative, or additive inverse,

written is the rational number

a

b

,a

b.

ab

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Example 6.18 Subtracting Rational Nmbers

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7

6

34

22 4

3

14

4 7

4 6

3 64 6

18 2824

10

24

1 22 4

4 3

3 86

12 12 11

612

Compute.

a.

b.