Comparing and Ordering Rational Numbers

5-1

Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersPRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1

Write the prime factorization of 364.

22 x 7 x 13

Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersPRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1

Write the prime factorization of each number.

1. 20 2. 125 3. 45

4. 186 5. 621 6. 1,575

(For help, go to Lesson 4-3.)

Check Skills You’ll Need

5-1

PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1

Solutions

1. 20 2. 125 3. 452 • 10 5 • 25 3 • 152 • 2 • 5 5 • 5 • 5 3 • 3 • 5 20 = 22 • 5 125 = 53 45 = 32 • 5

4. 186 5. 621 6. 1,5752 • 93 3 • 207 3 • 5252 • 3 • 31 3 • 3 • 69 3 • 3 • 175186 = 2 • 3 • 31 3 • 3 • 3 • 23 3 • 3 • 5 • 35

621 = 33 • 23 3 • 3 • 5 • 5 • 71,575 = 32 • 52 • 7

Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers

5-1


Today, the school’s baseball and soccer teams had

games. The baseball team plays every 7 days. The soccer team

plays every 3 days. When will the teams have games on the

same day again?

7, 14, 21, 28, 35, 42, . . . List the multiples of 7.

3, 6, 9, 12, 15, 18, 21, . . . List the multiples of 3.

The LCM is 21. In 21 days both teams will have games again.


Quick Check

5-1


Find the LCM of 16 and 36.

= 144 Multiply.

16 = 24

36 = 22 • 32Write the prime factorizations.

The LCM of 16 and 36 is 144.

LCM = 24 • 32 Use the greatest power of each factor.


Quick Check

5-1


Find the LCM of 5a4 and 15a.

5a4 = 5 • a4

15a = 3 • 5 • a Write the prime factorizations.

= 15a4 Multiply.

The LCM of 5a4 and 15a is 15a4.

LCM = 3 • 5 • a4 Use the greatest power of each factor.


Quick Check

5-1


Graph and compare the fractions in each pair.

is on the left, so < .38

38

78

b. – 13

, – 16

is on the right, so > .– 16

– 16

– 13

a. 78

38

,

38

78

– 13

– 16


Quick Check

5-1


The softball team won of its games and the hockey

team won of its games. Which team won the greater fraction

of its games?

677

9

Step 1 Find the LCM of 7 and 9.7 = 7 and 9 = 32

LCM = 7 • 32 = 63Step 2 Write equivalent fractions with a denominator of 63.

6 • 97 • 97 • 79 • 7

=54634963

=

Step 3 Compare the fractions.5463

4963

67

79

> >, so

The softball team won the greater fraction of its games.


Quick Check

5-1


Order , , and from least to greatest.37

14

23

37

14

23

3 • 127 • 12

1 • 214 • 21

2 • 283 • 28

3684

2184

5684

=

=

=

=

=

=

The LCM of 7, 4, and 3 is 84.Use 84 as the common denominator.

2184

3684

5684

14

37

23

< < , so < < .


Quick Check

5-1


Find the LCM of each pair of numbers.

1. 8, 6 2. 12, 16

3. Compare and order , – , and – from least to greatest. 316

810

316

24 48

810

– –< <316

316


5-1

5-2


Fractions and DecimalsFractions and Decimals

Without actually dividing, which expression is the largest?

6.1 ÷2.8 12.6 ÷ 1.1 9.305 ÷ 2.99

12.6 ÷ 1.1

Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2

(For help, go to Skills Handbook page 728.)

Write the decimals in order from least to greatest.

1. 2.41, 0.241, 24.1, 12.4

2. 1.030, 13.03, 1.300, 1.003

3. 0.1, 0.01, –0.1, –0.01


5-2


Solutions

1. 0.241, 2.41, 12.4, 24.1

2. 1.003, 1.030, 1.300, 13.03

3. –0.1, –0.01, 0.01, 0.1

5-2


The fuel tank of Scott’s new lawn mower holds gal of

gasoline. Scott poured 0.4 gal into the tank. Did Scott fill the tank?

12

= 1 ÷ 2 = 0.5

Since = 0.5 and 0.5 > 0.4, Scott did not fill the tank.12

12

Quick Check

5-2


Write each fraction as a decimal. State the block of

digits that repeats.

a.

b.

56

5 ÷ 6 = 0.83333 … Divide.

Place a bar over the digit that repeats.= 0.8356

= 0.83; the digit that repeats is 3.

711

7 ÷ 11 = 0.636363 … Divide.

= 0.63 Place a bar over the block of digits that repeats.

= 0.63; the block of digits that repeats is 63.7

11 Quick Check

5-2


Write the numbers in order, from least to greatest.

–0.8, , , 0.125312

54

–

–1.25 < –0.8 < 0.125 < 0.25 Compare the decimals.

3 ÷ 12 = 0.25Change the fractions to decimals.

–5 ÷ 4 = –1.25

From least to greatest, the numbers are , –0.8, 0.125, and .54

– 312

Quick Check

5-2


Write 1.72 as a mixed number in simplest form.

Keep the whole number 1.Write seventy-two hundredths as a fraction.

1.72 = 1 72100

Divide the numerator and denominator of the fraction by the GCF, 4.

72 ÷ 4100 ÷ 4= 1

Simplify.1825

1.72 = 1

Quick Check

5-2


Write 0.18 as a fraction in simplest form.

n Let the variable n equal the decimal.= 0.18

As a fraction in simplest form, 0.18 = .2

11

100n = 18.18 Because 2 digits repeat, multiply each side by 102, or 100.

1899

99n99

= Divide each side by 99.

n 18 ÷ 999 ÷ 9

= Divide the numerator and denominator by theGCF, 9.

211= Simplify.

100n = 18.18n = 0.18–

99n = 18

The Subtraction Property of Equality lets you subtract the same value from each side of the equation. So, subtract to eliminate 0.18.

Quick Check

5-2


1. Order , – , 0.625, –0.35 from least to greatest.

2. Write as a decimal.

Write each decimal as a fraction or mixed number in simplest form.

3. 6.04 4. 0.21

65

1720

13

65

13

–0.35, – , 0625,

0.85

61

257

33

5-2

5-3


Adding and Subtracting FractionsAdding and Subtracting Fractions

Five consecutive multiples of 6 have a sum of 270. What are they?

42, 48, 54, 60, 66

Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3

Find the LCM of each group of numbers or expressions.

1. 4, 8 2. 9, 18 3. 2n, 10

4. 3, 6, 9 5. 8, 5, 4 6. 10, n



5-3


Solutions

1. 4 = 22 2. 9 = 32 3. 2n = 2 • n

8 = 23 18 = 2 • 32 5 = 5

LCM = 23 LCM = 2 • 32 LCM = 2 • 5 • n

= 8 = 18 = 10n

4. 3 = 3 5. 8 = 23 6. 10 = 2 • 5

6 = 2 • 3 5 = 5 n = n

9 = 32 4 = 22 LCM = 2 • 5 • n

LCM = 2 • 32 LCM = 23 • 5 = 10n

= 18 = 40

5-3


Find each sum or difference. Simplify if possible.

a.

b.

49

29+

49

29+ =

4 + 29 Add the numerators.

69

= Simplify.

23

= Simplify.

5b

12b

–

– =5b

12b

12 – 5b Subtract the numerators.

7b= Simplify. Quick Check

5-3


Simplify each difference.

a.

b.

16

34

–

2y

– 516

4 – 1824

= Use the Order of Operations to simplify.

–1424

= Simplify.

= –724

Simplify.

2y

– =516

2 • 16 – 5 • yy • 16

Rewrite using a common denominator.

= 32 – 5y16y

Simplify.

16

34

– = 1 • 4 – 3 • 66 • 4


Quick Check

5-3


Suppose one day you rode a bicycle for 3 hours, and

jogged for 1 hours. How many hours did you exercise?14

12

You exercised for 4 hours.34

12

3 + = +1 14

72

54

Write mixed numbers as improper fractions.

= 7 • 4 + 5 • 22 • 4


= 28 + 108

Use the Order of Operations to simplify.

=

= 4 68

388

Write as a mixed number.

= 34

4 Simplify.

Quick Check

5-3


Find each sum or difference.

1. 2. 5 – 2

3. – – 4.

34

224

1024+

18

35

13

32c

47 +

12

583

1415

– 8c + 2114c

5-3

5-4


Multiplying and Dividing FractionsMultiplying and Dividing Fractions

Write each number as an improper fraction.

a. 4 b. 315

78

215

318

Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4

Write each mixed number as an improper fraction.

1. 2 2. 3 3. 1

4. 4 5. 7 6. 5

(For help, go to Skills Handbook page 738.)

13

310

49

45

17

78


5-4


Solutions

1. 2. 3.

2 = 2 +13

13

= +63

13

=6 + 1

3

2 =13

73

=30 + 3

10

3 = 3 +3

103

10

= +3010

310

3 =3

103310

1 = 1 +49

49

= +99

49

=9 + 4

9

1 =49

139

5-4


Solutions (continued)

4. 5. 6.

4 = 4 +45

45

= +45

=20 + 4

5

4 =45

205

245

7 = 7 +78

78

= +78

=56 + 7

8

7 =78

568

638

5 = 5 +17

17

= +17

=35 + 1

7

5 =17

357

367

5-4


Find • .23

57

23

57

• = 2 • 5 Multiply the numerators.

= 1021 Simplify.

Multiply the denominators.3 • 7

Quick Check

5-4


34

23

a. Find • .

23

34

34

• = 23

23

•1 1

12Divide the common factors.

= 12

Multiply.

b. 5w

3w17

Find • .

5w

3w17

• = •5w

3w171

1Divide the common factors.

1517

= Multiply.

Quick Check

5-4


Keesha’s desktop is a rectangle 3 ft long and 1 ft

wide. What is the area of her desktop?

12

12

A Area of a rectangle = length • width.= 3 • 1 12

12

= 72

32

• Write 3 and 1 as improper fractions, and .12

12

72

32

Multiply.= 214

Write as a mixed number.14= 5

The area of Keesha’s desk is 5 ft2.14

Quick Check

5-4


a. Find ÷ .35

720

35

35

710÷ = •

107 Multiply by the reciprocal of the divisor.

= • 351

107

2

Divide the common factors.

= 67

Multiply.

5-4


(continued)

b. Find ÷ .9

4q

= 32

Multiply.

278q

94q÷ = •

4q9 Multiply by the reciprocal of the divisor.

278q

278q

= • 1

4q9

1

Divide the common factors.278q

3 1

2 1

= 1 12

Write as a mixed number.

Quick Check

5-4


Find 4 ÷ (–3 ).12

38

4 ÷ (–3 ) = ÷ (– ) Change to improper fractions.278

12

38

92

= • (– ) Multiply by – , the reciprocal of – . 827

278

827

92

1

1

= • – Divide the common factors. 827

92

4

3

= – , or –1 Simplify.43

13

Quick Check

5-4


Simplify each expression.

1. 2.

3. 4.12g25

52g

•

3z4

5z7÷

712•

38

–

1116÷ 15

38

– 732 , or 1

2120

120

13, or 3

103

65 , or 1

15

5-4

5-5


Using Customary Units of MeasurementUsing Customary Units of Measurement

12

In 2002, a disc jockey broadcast her show for 27 h each week.

How many hours did she broadcast that year?

1,430 h

Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5

Find each product.

1. 4 • 2. 3 •

3. 1 • 4. 4 •


14

112

81

14

161

12

13


5-5


Solutions

1. 2.

3. 4.

4 • = •41

112

112

=4

12

13=

3 • = •161

161

14

134

=208

4

= 52

=404

= 10

1 • = •54

14

81

81 4 • = •

12

13

92

13

=96

=32

= 112

5-5


Choose an appropriate unit of measure. Explain your

choice.

a. weight of a hummingbird

b. length of a soccer field

Measure its weight in ounces because a hummingbird is very light.

Measure its length in yards because it is too long to measure in feet or inches and too short to measure in miles.

Quick Check

5-5


Use dimensional analysis to convert 68 fluid ounces to cups.

68 fl oz = •Use a conversion factor that changes fluid ounces to cups.

68 fl oz 1

1 c8 fl oz

17

= Divide the common factors and units.68 fl oz • 1 c 8 fl oz2

17 2 = c Simplify.

= 8 c Write as a mixed number.12

There are 8 c in 68 fl oz.12 Quick Check

5-5


Fred’s fruit stand sells homemade lemonade in 6 -pint

bottles for $1.99. Jill’s fruit stand stand sells homemade

lemonade in 3 -qt containers for the same price. At which stand

do you get more lemonade for your money?

12

12

Since 7 pints > 6 pints, you get more lemonade for your money

at Jill’s stand.

12

3 qt = qt • Use a conversion factor that changes quarts to pints

12

72

2 pt1 qt

= • Divide the common factors and units.1 2 pt

1 qt7 qt 2 1

= 7 pt Multiply.

Quick Check

5-5


Choose an appropriate unit of measure. Explain your choice.

1. weight of a dog

2. length of a football field

3. Use dimensional analysis to convert 1 mi to ft.

Pounds; dogs are too heavy to measure in ounces and too light to measure in tons.

Yards; it is too long to measure in feet or inches and too short to measure in miles.

7,920 ft

12

5-5

5-6


Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward

Use mental math to decide whether or not each pair of fractions is

yes, yes, no

equal: and , and , and3642

67

45

1620

615

1820

Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-ALGEBRA LESSON 5-6

Order from least to greatest.

1. 2.

3. 4.

12


110,

13

14

15, , ,

73

133,

53

33, ,

37

313,

35

33, , – 3

7 , – 35

313, – , – 3

5


5-6

3. 4.


Solutions

1. 2.12

110,

13

14

15, , ,

73

133,

53

33, ,

37

313,

35

33, , – 3

7 , – 35

313, – , – 3

5

0.50, 0.3, 0.25, 0.20, 0.10

12

110 ,

15

14

13, , ,

2.3, 1.6, 4.3, 1

33

133,

53

73, ,

0.43, 0.6, 0.23, 1

73

35,

35

33, ,

–0.43, –0.6, –0.23, 1

– 33 , – 3

53

13, – , –37

5-6


Your flight leaves the airport at 10:00 A.M. You must

arrive 2 hours early to check your luggage. The drive to the

airport takes about 90 minutes. A stop for breakfast takes about

30 minutes. It will take about 15 minutes to park and get to the

terminal. At what time should you leave home?

Move the hands of the clock to find the time you should leave home.

Write the starting time for each event.

5-6


(continued)

You should leave home at 5:45 A.M.

Quick Check

5-6


Solve by working backward.

1. Jason has a total of 212 sheep. Last spring, 52 lambs were born, but 9 died. How many sheep did he have before the lambs were born?

2. The bus picks up Tom every morning at 7:20 A.M. Tom walks his dog every morning before school for 36 minutes. It takes him 25 minutes to get ready for school and 7 minutes to walk to the bus stop. At what time does Tom need to wake up in order to get to the bus stop on time?

169 sheep

6:12 A.M.

5-6

5-7


Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting Fractions

Find the sum or difference. If possible, write each answer in lowest

terms.

a. + = ? b. 7 – 5 = ? 45

25

56

26

1 15

2 12

Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7

Find each sum or difference.

1. 1 – 2 2. 3 + 2

3. 5 – 3 4. –4 – 3


34

712

78

58

34

18

16

29


5-7


Solutions

1. 1 – 2 2. 3 + 434

78

58

= – = –198

18

238

78 – =

7 • 8 – 4 • 234 • 8

56 – 9232=

–3632=

712

348 + 44096=

78896=

5512+ =

29 • 12 + 8 • 558 • 12

298

= 8 = 82096

524

5-7



3. 5 – 3 4. –4 – 334

18

16

–225 – 17454=

–39954=

299+ =

–25 • 9 – 6 • 296 • 9

256

= –7 = –72154

718

184 – 10032=

=

258– =

23 • 8 – 4 • 254 • 8

234

= 2 = 22032

8432

58

29

–

5-7


One school recycles about of its waste paper. The

student council set a goal of recycling of the school’s waste

paper by the end of the year. By how much does the school need

to increase its paper recycling to reach the goal?

13

34

Words plus is

Let n = the increase.

Equation + n =

fraction school recycles

theincrease

student goal

34

13

5-7


(continued)

+ n =13

34

13

– + n = – Subtract from each side.13

34

13

13

n = Use 3 • 4 as the common denominator.3 • 3 – 1 • 4 3 • 4

n = Use the Order of Operations.9 – 4 12

n = Simplify. 512

To meet the student council goal, the school needs to recycle more of its waste paper.

512

5-7


(continued)

Check Is the answer reasonable? The present fraction of paper

waste that is recycled plus the increase must equal the

goal. Since + = + = = , the answer

is reasonable.

912

13

34

512

412

512

Quick Check

5-7


Solve x – = .23

19

x – + = + Add to each side.23

19

23

23

23

x – = 23

19

x = Use 9 • 3 as the common denominator. 1 • 3 + 9 • 2 9 • 3

x = Use the Order of Operations.3 + 18 27

x = Divide the common factors.2127

7

9

x = Simplify.79

Quick Check

5-7


Solve q – 6 = –1 .12

35

q – 6 = 1 12

35

q = Use the Order of Operations.–16 + 65 10

q = Simplify.4910

q =

–8 • 2 + 5 • 13 5 • 2

Use 5 • 2 as the common denominator.

q = – + Write mixed numbers as improper fractions.

85

13 2

q = 4 Write as a mixed number. 910

Quick Check

5-7

q – 6 + 6 = 1 + 6 Add 6 to each side.12

35

12

12

12


Solve the following equations.

1. + s = 2. k + = 3. u – =

4. – + g = 5. x – 2 = –4

37

1321

25

79

112

34

511

1922

13

415

421

1745

56

7221

1415–1

5-7

5-8


Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions

310Because of a mishap, of the oranges in a crate were ruined. If

the oranges originally weighed 50 lb, how many pounds of oranges

were ruined?

35 lb



Find each product.

1. 2.

3. 4.

•47

910

58 •

53–

–1 •51–

14 •

234

12


5-8



Solutions

1. 2.•47

58

910 •

53–

•47

58 =

4 • 57 • 8

= 2056

= 514

910 •

53– =

9 • –510 • 3

=–4530

= –112

–96=

5-8




3. 4.–1 •51–

14 •

234

12

=254

•54 =

–5 • –54 • 1–

–51

= 614

•92

23 =

9 • 22 • 3

= 186

= 3

5-8



Solve 7y = .13

• (7y) = • Multiply each side by , the reciprocal of 7.13

17

17

17

7y = 13

y = Simplify. 121

Quick Check

5-8



52

w = • Divide the common factors.1315

1

3

w = 2 Write as a mixed number.16

Solve w = .25

1315

• w = • Multiply each side by , the reciprocal of .25

52

52

52

25

1315

w = 25

1315

w = Simplify.13 6

Quick Check

5-8



Solve – c = .49

2027

– c = 49

2027

c = Divide common factors.27 • 420 • 9

13

15

= – Simplify.35

– – c = – Multiply each side by – , the

reciprocal of – .

49

2720

2027

2720

2720

2027

Quick Check

5-8



How many 2 -t trucks can you place on a rail car that

has a carrying capacity of 15 t?

12

Words times is

Let n = the number of trucks.

Equation 2 • n = 15

weight ofeach truck

the number of trucks

carrying capacity

12

5-8



(continued)

12 2 • n = 15

n = Divide common factors.2 • 155 • 1

3

1

n = 15 Write 2 as .52

52

12

• n = • 15 Multiply each side by , the reciprocal of .52

52

25

25

25

= 6 Simplify.

You can place 6 trucks on the rail car. Quick Check

5-8


– g =

Solve each equation.

1. 2. 3. 15 = –3 m= b


78

516

2532

58

613

514

45–

13–4

5-8


5-9


Powers of Products and QuotientsPowers of Products and Quotients

112 x 32 x 22 is the prime factorization for ?.

4,356

Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9


1. (22)3 2. (32)2 3. (15)4

4. (x3)6 5. (b2)5 6. (a7)4



5-9


Solutions

1. (22)3 2. (32)2 3. (15)4

= 22 • 3 = 32 • 2 = 15 • 4

= 26 = 34 = 120

= 64 = 81 = 1

4. (x3)6 5. (b2)5 6. (a7)4

= x3 • 6 = b2 • 5 = a7 • 4

= x18 = b10 = a28

5-9


Simplify (3z5)4.

(3z5)4 = 34 • (z5)4 Raise each factor to the fourth power.

= 34 • z5 • 4 Use the Rule for Raising a Power to a Power.

= 34 • z20 Multiply exponents.

= 81z20 Simplify.

Quick Check

5-9


a. Simplify (–3a)4.

b. Simplify –(3a)4.

(–3a)4 = (–3)4(a)4

= 81a4

–(3a)4 = (–1)(3a)4

= (–1)(3)4(a)4

= –81a4Quick Check

5-9


Find the area of a square with side length .x4

A = s2 s = length of a side

x2

42=

x2

16=

The area of the square is square units. x2

16

= x4

2

Quick Check

5-9



1. (12x2yz)2 2. (–4ab3)3 3. –(–2g2h)8

4. 5.u3

62

144x4y2z2 –64a3b9 –256g16h8

u6

36

3q8r s7

56

243q r s35

40 30

5-9

Documents

Comparing and Ordering Rational Numbers