Pre-Algebra Lesson 5-1 Comparing and Ordering Rational Numbers.

Pre-AlgebraPre-AlgebraLesson 5-1

Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers


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.


Additional Examples


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.


Additional Examples


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.


Additional Examples


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


Additional Examples


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.


Additional Examples


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 < < .


Additional Examples

Pre-AlgebraPre-Algebra

Fractions and DecimalsFractions and Decimals

Lesson 5-2



Lesson 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

Additional Examples



Lesson 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

Additional Examples



Lesson 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

Additional Examples



Lesson 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

Additional Examples



Lesson 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.

Additional Examples


Adding and Subtracting FractionsAdding and Subtracting Fractions

Lesson 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.

Additional Examples



Lesson 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

Use a common denominator.

Additional Examples



Lesson 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

Rewrite using a common denominator.

= 28 + 108

Use the Order of Operations to simplify.

=

= 4 68

388

Write as a mixed number.

= 34

4 Simplify.

Additional Examples


Multiplying and Dividing FractionsMultiplying and Dividing Fractions

Lesson 5-4

Find • .23

57

23

57

• = 2 • 5 Multiply the numerators.

= 1021 Simplify.

Multiply the denominators.3 • 7

Additional Examples



Lesson 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.

Additional Examples



Lesson 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

Additional Examples



Lesson 5-4

a. Find ÷ .35

710

35

35

710÷ = •

107 Multiply by the reciprocal of the divisor.

= • 351

107

2

Divide the common factors.

= 67

Multiply.

Additional Examples



Lesson 5-4

(continued)

b. Find ÷ .9

4q

= 32

Simplify.

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.

Additional Examples



Lesson 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

Additional Examples


Using Customary Units of MeasurementUsing Customary Units of Measurement

Lesson 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.

Additional Examples



Lesson 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

Additional Examples



Lesson 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.

Additional Examples


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

Lesson 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.

Additional Examples


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

Lesson 5-6

(continued)

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

Additional Examples


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

Lesson 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

Additional Examples



Lesson 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

Additional Examples



Lesson 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

Additional Examples



Lesson 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

Additional Examples



Lesson 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

12

q – 6 + 6 = – 1 + 6 Add 6 to each side.35

12

12

12

Additional Examples


Solve 7y = .13

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

17

17

17

7y = 13

y = Simplify. 121

Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions

Additional Examples


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


Additional Examples


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


Additional Examples


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


Additional Examples


(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.


Additional Examples


Powers of Products and QuotientsPowers of Products and Quotients

Lesson 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.

Additional Examples



Lesson 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

= –81a4

Additional Examples



Lesson 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

Additional Examples

Pre-Algebra Lesson 5-1 Comparing and Ordering Rational Numbers.

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Transcript of Pre-Algebra Lesson 5-1 Comparing and Ordering Rational Numbers.