algebra demystified - rhonda huettenmueller

Because many find fraction arithmetic difficult, the first chapter is devoted almost exclusively to fractions.. Invert switchthe numerator and denominator the second fraction and the fractio

Calculus Demystified by Steven G Krantz

Physics Demystified by Stan Gibilisco

RHONDA HUETTENMUELLER

McGRAW-HILL

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spe-DOI: 10.1036/0071412107

CHAPTER 8 Linear Applications 197 CHAPTER 9 Linear Inequalities 285 CHAPTER 10 Quadratic Equations 319 CHAPTER 11 Quadratic Applications 353

This book is designed to take the mystery out of algebra Each section

con-tains exactly one new idea—unlike most math books, which cover several

ideas at once Clear, brief explanations are followed by detailed examples

Each section ends with a few Practice problems, most similar to the examples

Solutions to the Practice problems are also given in great detail The goal is

to help you understand the algebra concepts while building your skills and

confidence

Each chapter ends with a Chapter Review, a multiple-choice test designed

to measure your mastery of the material The Chapter Review could also be

used as a pretest If you think you understand the material in a chapter, take

the Chapter Review test If you answer all of the questions correctly, then

you can safely skip that chapter When taking any multiple-choice test, work

the problems before looking at the answers Sometimes incorrect answers

look reasonable and can throw you off Once you have finished the book,

take the Final Review, which is a multiple-choice test based on material from

each chapter

Spend as much time in each section as you need Try not to rush, but do

make a commitment to learning on a schedule If you find a concept difficult,

you might need to work the problems and examples several times Try not to

jump around from section to section as most sections extend topics from

previous sections

Not many shortcuts are used in this book Does that mean you shouldn’t

use them? No What you should do is try to find the shortcuts yourself Once

you have found a method that seems to be a shortcut, try to figure out why it

works If you understand how a shortcut works, you are less likely to use it

incorrectly (a common problem with algebra students)

Because many find fraction arithmetic difficult, the first chapter is devoted

almost exclusively to fractions Make sure you understand the steps in this

chapter because they are the same steps used in much of the rest of the book

For example, the steps used to compute 7

Even those who find algebra easy are stumped by word problems (alsocalled ‘‘applications’’) In this book, word problems are treated very care-fully Two important skills needed to solve word problems are discussedearlier than the word problems themselves First, you will learn how tofind quantitative relationships in word problems and how to representthem using variables Second, you will learn how to represent multiple quan-tities using only one variable.

Most application problems come in ‘‘families’’—distance problems, workproblems, mixture problems, coin problems, and geometry problems, toname a few As in the rest of the book, exactly one topic is covered ineach section If you take one section at a time and really make sure youunderstand why the steps work, you will find yourself able to solve a greatmany applied problems—even those not covered in this book

Good luck

RHONDAHUETTENMUELLER

I want to thank my husband and family for their patience during the manymonths I worked on this project I am also grateful to my students throughthe years for their thoughtful questions Finally, I want to express my appre-ciation to Stan Gibilisco for his welcome advice.

Fraction Multiplication

Multiplication of fractions is the easiest of all fraction operations All you

have to do is multiply straight across—multiply the numerators (the top

numbers) and the denominators (the bottom numbers)

Multiplying Fractions and Whole Numbers

You can multiply fractions by whole numbers in one of two ways:

1 The numerator of the product will be the whole number times thefraction’s numerator, and the denominator will be the fraction’sdenominator

2 Treat the whole number as a fraction—the whole number overone—then multiply as you would any two fractions

Fraction Division

Fraction division is almost as easy as fraction multiplication Invert (switchthe numerator and denominator) the second fraction and the fraction divi-sion problem becomes a fraction multiplication problem

Examples2

34

5¼2

35

4¼10123

Reducing Fractions

When working with fractions, you are usually asked to ‘‘reduce the fraction

to lowest terms’’ or to ‘‘write the fraction in lowest terms’’ or to ‘‘reduce the

fraction.’’ These phrases mean that the numerator and denominator have no

common factors For example, 2

3 is reduced to lowest terms but 4

6 is not

Reducing fractions is like fraction multiplication in reverse We will first use

the most basic approach to reducing fractions In the next section, we will

learn a quicker method

First write the numerator and denominator as a product of prime

numbers Refer to the Appendix if you need to review how to find the

prime factorization of a number Next collect the primes common to both

the numerator and denominator (if any) at beginning of each fraction Split

each fraction into two fractions, the first with the common primes Now the

fraction is in the form of ‘‘1’’ times another fraction

3: 48

30¼2 2  2  2  3

2 3  5 ¼

ð2  3Þ  2  2  2ð2  3Þ  5 ¼

8: 240

165¼2 2  2  2  3  5

3 5  11 ¼

ð3  5Þ  2  2  2  2ð3  5Þ  11 ¼

3 5

3 5

2 2  2  211

¼15

1516

11¼1611

10: 150

30 ¼2 3  5  5

2 3  5 ¼

ð2  3  5Þ  5ð2  3  5Þ  1¼

Fortunately there is a less tedious method for reducing fractions to their

lowest terms Find the largest number that divides both the numerator

and the denominator This number is called the greatest common divisor

(GCD) Factor the GCD from the numerator and denominator and rewrite

the fraction In the previous examples and practice problems, the product of

the common primes was the GCD

might find it easier to reduce the fraction in several steps.

Adding and Subtracting Fractions

When adding (or subtracting) fractions with the same denominators, add (orsubtract) their numerators

2: 1

5þ3

5¼1þ 3

5 ¼45

When the denominators are not the same, you have to rewrite the fractions sothat they do have the same denominator There are two common methods ofdoing this The first is the easiest The second takes more effort but canresult in smaller quantities and less reducing (When the denominators have

no common divisors, these two methods are the same.)

The easiest way to get a common denominator is to multiply the firstfraction by the second denominator over itself and the second fraction bythe first denominator over itself

2þ3

7¼ 1

277



þ 3

722



¼ 7

14þ 6

14¼1314

8

151

2¼ 8

1522

 1

21515

¼16

3015

30¼ 130



 1

566



¼25

30 6

30¼1930

2: 1

3þ7

8¼ 1

388



þ 7

833



¼ 8

24þ21

24¼2924

3: 5

71

9¼ 5

799



 1

977



¼45

63 7

63¼3863

4: 3

14þ1

2¼ 3

1422

þ 1

21414

5: 3

4þ11

18¼ 3

41818

þ 11

1844

previous examples and practice problems, we found a common denominator

Now we will find the least common denominator (LCD) For example in



þ 1

633

1

3þ1

6¼ 1

322

common denominator When denominators get more complicated, either by

being large or having variables in them, you will find it easier to use the LCD

to add or subtract fractions The solution might require less reducing, too

In the following practice problems one of the denominators will be the

LCD; you only need to rewrite the other



¼1

8þ4

8¼58

2: 2

3 5

12¼ 2

344

3: 4

5þ 1

20¼ 4

544



¼ 5

24þ20

24¼2524There are a couple of ways of finding the LCD Take for example 1

12þ 9

14 Wecould list the multiples of 12 and 14—the first number that appears on eachlist will be the LCD:

12, 24, 36, 48, 60, 72, 84 and 14, 28, 42, 56, 70, 84

Because 84 is the first number on each list, 84 is the LCD for 1

12and 9

14 Thismethod works fine as long as your lists are not too long But what if yourdenominators are 6 and 291? The LCD for these denominators (which is582) occurs 97th on the list of multiples of 6

We can use the prime factors of the denominators to find the LCD moreefficiently The LCD will consist of every prime factor in each denominator(at its most frequent occurrence) To find the LCD for121 and149 factor 12 and

14 into their prime factorizations: 12¼ 2  2  3 and 14 = 2  7 There are two2s and one 3 in the prime factorization of 12, so the LCD will have two 2sand one 3 There is one 2 in the prime factorization of 14, but this 2 iscovered by the 2s from 12 There is one 7 in the prime factorization of 14, sothe LCD will also have a 7 as a factor Once you have computed the LCD,divide the LCD by each denominator Multiply each fraction by this numberover itself

LCD¼ 2  2  3  7 ¼ 84

84 12 ¼ 7: multiply 1

12by 77 84 14 ¼ 6: multiply 9

14by 66.1

12þ 9

14¼ 1

1277

þ 9

1466

¼ 7

84þ54

84¼6184

and 4

15by 2

2.5

6þ 4

15¼ 5

655



þ 4

1522

24þ 5

36¼ 17

2433

þ 5

3622

¼51

72þ10

72¼6172

 5

1822

¼33

3610

36¼2336

2: 7

15þ 9

20 ¼ 7

1544

þ 9

2033

3: 23

24þ 7

16 ¼ 23

2422

þ 7

1633

¼46

48þ21

48¼6748

4: 3

8þ 7

20 ¼ 3

855



þ 7

2022

¼15

40þ14

40¼2940

5: 1

6þ 4

15 ¼ 1

655



þ 4

1522

¼ 5

30þ 8

30¼1330

6: 8

75þ 3

10 ¼ 8

7522

þ 3

101515

¼ 16

150þ 45

150¼ 61150

7: 35

54 7

48 ¼ 35

5488

 7

4899

¼280

432 63

432¼217432

8: 15

88þ 3

28 ¼ 15

8877

þ 3

282222

¼105

616þ 66

616¼171616

9: 119

180þ 17

210 ¼ 119

18077

þ 17

21066

Adding More than Two Fractions

Finding the LCD for three or more fractions is pretty much the same as

finding the LCD for two fractions Factor each denominator into its

prime factorization and list the primes that appear in each Divide the

LCD by each denominator Multiply each fraction by this number over

15¼ 3  5

20¼ 2  2  5The LCD¼ 2  2  3  5 ¼ 60

þ 7

1544

þ 9

2033

3

10þ 5

12þ 1

18Prime factorization of the denominators: 10¼ 2  5

12¼ 2  2  3

18¼ 2  3  3LCD¼ 2  2  3  3  5 ¼ 180

þ 5

121515

þ 1

181010

¼ 54

180þ 75180

þ 10

180¼139

180



þ 7

1233

þ 3

101212

þ 1

81515

¼ 55

120þ 36

120þ 15120

¼106

120¼5360

þ 5

61010

þ 9

2033

¼2315

þ 9

1455

þ 7

1077

¼ 6

70þ45

70þ4970

¼100

70 ¼107

þ 3

1699

þ 1

62424

þ 7

91616

Whole Number-Fraction Arithmetic

A whole number can be written as a fraction whose denominator is 1 With

this in mind, we can see that addition and subtraction of whole numbers and

fractions are nothing new To add a whole number to a fraction, multiply

the whole number by the fraction’s denominator Add this product to the

fraction’s numerator The sum will be the new numerator

4: 2 4

5¼ð2  5Þ  4

5 ¼10 4

5 ¼65

To subtract a whole number from the fraction, again multiply the whole

number by the fraction’s denominator Subtract this product from the

frac-tion’s numerator This difference will be the new numerator

2 3 1 6

2 3

¼ 1 2

3¼ 1 3

2¼32

8 9

5¼8

9 5 ¼8

91

5¼ 845

¼

Mixed Numbers and Improper Fractions

An improper fraction is a fraction whose numerator is larger than its

denominator For example, 6

5 is an improper fraction A mixed numberconsists of the sum of a whole number and a fraction For example 11

5

(which is really 1þ1

5) is a mixed number We will practice going back andforth between the two forms

To convert a mixed number into an improper fraction, first multiply the

whole number by the fraction’s denominator Next add this to the

numerator The sum is the new numerator

In an improper fraction, the numerator is the dividend and the divisor is thedenominator In a mixed number, the quotient is the whole number, theremainder is the new numerator, and the divisor is the denominator.

  remainder

211

To convert an improper fraction to a mixed number, divide the numerator

into the denominator The remainder will be the new numerator and the

quotient will be the whole number

4 ¼ 31 4

3 ¼ 61 3

14¼ 211 14

5 ¼ 44 5

7 ¼ 35 7

Mixed Number Arithmetic

You can add (or subtract) two mixed numbers in one of two ways One way

is to add the whole numbers then add the fractions

 

¼ 7 þ 4

6þ3 6

 

¼ 7 þ7

6¼ 7 þ 1 þ1

6¼ 81 6

The other way is to convert the mixed numbers into improper fractions then

When multiplying mixed numbers first convert them to improper fractions,and then multiply Multiplying the whole numbers and the fractions isincorrect because there are really two operations involved—addition andmultiplication:

:Convert the mixed numbers to improper fractions before multiplying