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Chapter 4Fractions and Decimals

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Chapter 4Fractions and Decimals

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Lesson 4-1 Greatest Common Factor

Lesson 4-2 Problem-Solving Strategy: Make an Organized List

Lesson 4-3 Simplifying Fractions

Lesson 4-4 Mixed Numbers and Improper Fractions

Lesson 4-5 Least Common Multiple

Lesson 4-6 Problem-Solving Investigation: Choose the Best Strategy

Lesson 4-7 Comparing Fractions

Lesson 4-8 Writing Decimals as Fractions

Lesson 4-9 Writing Fractions as Decimals

Lesson 4-10 Algebra: Ordered Pairs and Functions

44Fractions and Decimals



Five-Minute Check (over Chapter 3)

Main Idea and Vocabulary

California Standards

Example 1

Example 2

Example 3

Example 4

Example 5

4-14-1 Greatest Common Factor

Greatest Common Factor



• I will find the greatest common factor of two or more numbers.

• common factor

• greatest common factor (GCF)



Preparation for Standard 6NS2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form for a fraction).


First, list the factors by pairs for each number.

Identify the common factors of 20 and 36.


Answer: The common factors of 20 and 36 are 1, 2, and 4.



Identify the common factors of 12 and 18.

A. 1, 2, 3, 6

B. 1, 2, 6

C. 1, 2, 4, 6

D. 1, 2, 3, 4, 6


Find the GCF of 36 and 48.

Write the prime factorization.

36


12 × 3

4 × 3 3

32 × 2 × 3 ×

48

Answer: The GCF of 36 and 48 is 2 × 3 or 6.

2 × 24

2 × 22 2 3×××

3 × 42 2 ××

2 × 122 ×


Check


Use a Venn diagram to show the factors. Notice that the factors 1, 2, 3, 4, 6, and 12 are common factors of 36 and 48 and the GCF is 12.




A. 1

B. 2

C. 3

D. 7



Answer: The GCF of 21 and 28 is 7.


21 28

7 × 3 2 × 14

2 × 72


Check


Use a Venn diagram to show the factors. Notice that the factors 1 and 7 are common factors of 21 and 28 and the GCF is 7.




A. 1

B. 2

C. 5

D. 15


Ana sells bags of different kinds of cookies. She made $27 selling bags of peanut butter cookies, $18 from chocolate chip cookies, and $45 selling bags of oatmeal cookies. Each bag of cookies costs the same amount. What is the most that Ana could have charged for each bag of cookies?



Factors of 27: 1, 3, 9, 27

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 45: 1, 3, 5, 9, 15, 45

Answer: The GCF of 27, 18, and 45 is 9. So, the most Ana could have charged for each bag of cookies is $9.




Joy bought presents for her three friends. She spent $48 on Jonah, $36 on Louise, and $60 on Brenden. Each gift cost the same amount. What is the most each gift could have cost?

A. $1

B. $4

C. $18

D. $12


Refer to Example 4 of this lesson. How many bags could Ana have sold if each bag cost $9?


There is a total of $27 + $18 + $45 or $90.

Answer: So, the number of bags of cookies is $90 ÷ $9 or 10.



If Joy spent $48 on Jonah, $36 on Louise, and $60 on Brenden, and each gift cost $12, how many gifts did she buy?

A. 48

B. 36

C. 12

D. 60


Five-Minute Check (over Lesson 4-1)

Main Idea


Example 1: Problem-Solving Strategy

4-24-2 Problem-Solving Strategy: Make an Organized List



• I will solve problems by making an organized list.



Standard 5MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

Standard 5NS1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors.


Jessica is setting up four booths in a row for the school carnival. There will be a dart game booth, a ring toss booth, a face-painting booth, and a virtual football booth. In how many ways can the four booths be arranged for the school carnival?



Understand

What facts do you know?

• There are four different booths: dart game, ring toss, face-painting, and virtual football.

• The booths will be set up in a row.

What do you need to find?

• Find how many different ways the booth can be arranged.



Plan

Make a list of all the different possible arrangements. Use D for darts, R for ring toss, F for face-painting, and V for virtual football. Organize your list by listing each booth first as shown below.


D _ _ _ R _ _ _ F _ _ _ V _ _ _


Plan

Then fill in the remaining three positions with the other booths. Continue this process until all the different arrangements are listed in the second, third, and fourth positions.



Solve

Listing D first:


Listing R first:

D R F V

D R V F

D F R V

D F V R

D V R F

D V F R

R F V D

R F D V

R V D F

R V F D

R D F V

R D V F


Solve

Listing F first:

Answer: There are 24 different ways the booths can be arranged.


Listing V first:

F V D R

F V R D

F D R V

F D V R

F R V D

F R D V

V D R F

V D F R

V R F D

V R D F

V F D R

V F R D


Check

Look back. Is each booth accounted for six times in the first, second, third, and fourth positions?






Example 1

Example 2

Example 3

Example 4

4-34-3 Simplifying Fractions



• I will express fractions in simplest form.

• ratio

• equivalent fractions

• simplest form



Preparation for Standard 5NS2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less), and express answers in the simplest form.


Replace the x with a number so the fractions are equivalent.


613

x52

=

613

x52

=Since 13 × 4 = 52, multiply the numerator and denominator by 4.

Answer: So, x = 24.



Solve for x. Choose the correct answer.

A. 24

B. 28

C. 30

D. 7

712

x48

=


Replace the x with a number so the fractions are equivalent.


2440

3x

=

Since 24 ÷ 8 = 3, divide the numerator and denominator by 8.

Answer: So, x = 5.

2440

3x

=



Solve for x. Choose the correct answer.

A. 5

B. 10

C. 20

D. 15

525

1x

=



One Way: Divide by common factors.

1442

721

13

= = A common factor of 14 and 42 is 2.A common factor of 7 and 21 is 7.

Write in simplest form.1442



Another Way: Divide by the GCF.

factors of 14: 1, 2, 7, 14factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The GCF of 14 and 42 is 14.

1442

13

= Divide the numerator and denominator by the GCF, 14.



Write in simplest form.4850

A. 2425

B. 9

10

C. 1415

D. 4850


The GCF of 3 and 24 is 3.


324

18

=

1

8

Mentally divide both the numerator and denominator by 3.

Lin practices gymnastics 3 hours each day. There

are 24 hours in a day. Express the fraction in

simplest form.

324

Answer: So, Lin spends or 1 out of every 8

hours practicing gymnastics.

18



A. 5

10

Mark spends $10 of the $50 bill his mom gave him.

Express in simplest form.1050

D. 125

B. 12

C. 15





Example 1

Example 2

4-44-4 Mixed Numbers and Improper Fractions



• I will write mixed numbers as improper fractions and vice versa.

• mixed number

• proper fraction

• improper fraction



Standard 5NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.



If a spaceship lifts off the Moon, it must travel at

a speed of 2 kilometers per second in order to

escape the pull of the Moon’s gravity. Write this

speed as an improper fraction. Then graph the

improper fraction on a number line.

25



(2 × 5) + 2

5=2 2

5 =125

Answer: So, 2 = .125

25



Since is between 2 and 3, draw a number line

using increments of one fifth. Then, draw a dot at .

125 12

5



The average height of an adult man is 5 feet tall.

Choose the answer that shows 5 as an improper

fraction.

912

912

A. 5912

D. 6960

B. 9

108

C. 9

60


4


235

– 203

Write as a mixed number. Then graph the mixed

number on a number line.

234

34

Answer: So, = 5 . 234

34



Since 5 is between 5 and 6, draw a number line

from 5 to 6 using increments of one fourth. Then, draw

a dot at 5 .

34

34



Choose the answer that shows as a mixed

number.

476

A. 6 57

B. 7 56

C. 5 67

D. 7 78





Example 1

Example 2

Example 3

4-54-5 Least Common Multiple



• I will find the least common multiple of two or more numbers.

• multiple

• common multiples

• least common multiple (LCM)



Preparation for Standard 5SDAPS1.3 Use fractions and percentages to compare data sets of different sizes.


Identify the first three common multiples of 3 and 9.

First, list the multiples of each number.


Multiples of 3: 3, 6, 9, 12, 15, 18, …

Multiples of 9: 9, 18, 27, 36, 45, 49, …

1 × 3, 2 × 3, 3 × 3, …

1 × 9, 2 × 9, 3 × 9, …

Notice that 9, 18, and 27 are multiples common to both 3 and 9.

Answer: So, the first three common multiples of 3 and 9 are 9, 18, and 27.



Identify the first three multiples of 6 and 12.

A. 6, 12, 18

B. 6, 12, 24

C. 12, 24, 36

D. 12, 24, 48


Find the LCM of 8 and 18.

Write the prime factorization of each number.


8

2 × 4

2 × 22

18

3 × 6

2 × 33


Identify all common prime factors.

8 = 2 × 2 × 2

Answer: The LCM is 2 × 2 × 2 × 3 × 3 or 72.


18 = 3 × 2 × 3

Find the product of the prime factors using each common prime factor only once and any remaining factors.



Find the LCM of 7 and 21.

A. 7

B. 42

C. 21

D. 14


Liam, Eva, and Bansi each have the same amount of money. Liam has only nickels, Eva has only dimes, and Bansi has only quarters. What is the least amount of money that each of them could have?

Find the LCM using prime factors.

Answer: The least amount of money they could all have is 5 × 5 × 2 or $0.50.


5

5

10

2 × 5

25

5 × 5



Samuel, John, and Uma were all paid the same amount in ones, fives, and tens, respectively. What is the least they could have been paid?

A. $10

B. $20

C. $25

D. $5



Main Idea


Example 1: Problem-Solving Investigation

4-64-6 Problem-Solving Investigation: Choose the Best Strategy



• I will choose the best strategy to solve a problem.



Standard 5MR2.6 Make precise calculations and check the validity of the results from the context of the problem.



Standard 5SDAP1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain which types of graphs are appropriate for various data sets.



TROY: This weekend, my family went to the zoo. We spent a total of $42 on admission tickets. We purchased at least 2 adult tickets for $9 each and no more than three children’s tickets for $5 each.

YOUR MISSION: Find how many adult and children’s tickets Troy’s family purchased.



Understand

What facts do you know?

• You know that the family spent a total of $42.

• At least 2 adult tickets were purchased for $9 each.

• No more than three children’s tickets were purchased for $5 each.



Understand

What do you need to find?

• You need to find how many of each ticket Troy’s family purchased.



Plan

Guess and check to find the number of adult and children’s tickets purchased.



Solve

Answer: So, Troy’s family bought 3 adult and 3 children’s tickets.



Check

Look back. Three adult tickets cost 3 × $9, or $27 and three children’s tickets cost 3 × $5 or $15. Since $27 + $15 = $42, the answer is correct.





Key Concept: Compare Two Fractions

Example 1

Example 2

Example 3

Example 4

4-74-7 Comparing Fractions



• I will compare fractions.

• least common denominator (LCD)



Standard 5SDAP1.3 Use fractions and percentages to compare data sets of different sizes.



Step 1 The LCM of the denominators is 21. So, the LCD is 21.

Step 2 Write an equivalent fraction with a denominator of 21 for each fraction.

821

821

=37

921

=

Replace the with <, >, or = in to make a true sentence.

821

37



Answer: So, < .821

37

Step 3 < , since 8 < 9.821

921



A. >

B. <

C. =

D. +

Which answer below makes a true sentence?

34

23



Step 1 The LCM of the denominators is 6. So, the LCD is 6.

Replace the with <, >, or = in 2 2 to make a true sentence.

13

26

Since the whole numbers are the same, compare

and .



Step 3 = , since 2 = 2.

Step 2 Write an equivalent fraction with a denominator of 6 for each fraction.

26

26

=13

26

=

Answer: So, 2 = 2 .13

26



A. >

B. <

C. =

D. +

Replace the with <, >, or = to make 4 4 a true sentence.

38

45



Step 1 Write each quantity as a fraction.

Ginny had 3 out of 4 hits in a baseball game. Belinda had 4 out of 6 hits in that game. Who has the greater fraction of hits?

Ginny: Belinda:



Step 2 The LCD of the fractions is 12. So, rewrite each fraction with a denominator of 12.

34

912

=46

812

=

Answer: Since > , the fraction of hits Ginny

made is greater.

912

812



Heidi got 10 out of 12 answers right on the math quiz. Tiffany got 5 out of 6 right on her math quiz. Who has the greater fraction of correct answers?

A. Tiffany

B. Heidi

C. They got the same fraction.

D. neither



You need to compare the fractions. The LCD of the fractions is 100.

Use the table to answer the following question. What did the least number of people say should be done with a penny?



Rewrite the fractions with the LCD, 100.

825

32100

=3

1003

100=

1320

65100

=

Answer: Since the least number is 3, the least number of people were undecided.



According to the data in the table, who walked the shortest distance?

A. Kayla

B. Nora

C. Mercedes

D. They all walked the same distance.





Example 1

Example 2

Example 3

Example 4

Example 5

4-84-8 Writing Decimals as Fractions



• I will write decimals as fractions or mixed numbers in simplest form.

• rational number



Standard 5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.


In the place-value chart, the last nonzero digit, 4, is in the tenths place. Say four tenths.

Write 0.4 as a fraction in simplest form.


0 0 0 0 4 0 0 0



0.4 =4

10Write as a fraction.

=4

10Simplify. Divide the numerator and denominator by the GCF, 2.=

25

2

5



Choose the answer below that shows 0.8 as a fraction in simplest form.

A. 45

C. 68

D. 23

B. 8

10



In the place-value chart, the last nonzero digit, 8, is in the hundredths place. Say thirty-eight hundredths.


0 0 0 0 3 8 0 0



0.38 =38


=Simplify. Divide the numerator and denominator by the GCF, 2.=

1950

19

50

38100




A. 57

C. 34

D. 75100

B. 75



In the place-value chart, the last nonzero digit, 7, is in the hundredths place. Say seven hundredths.


0 0 0 0 0 7 0 0



0.07 =7


=Simplify. Divide by the GCF, 1.

7100

=7

100




A. 4100

B. 2

50

C. 4

10

D. 1

25



In the place-value chart, the last nonzero digit, 4, is in the thousandths place. Say two hundred sixty-four thousandths.


0 0 0 0 2 6 4 0



0.264 =264

1,000Write as a fraction.

Simplify. Divide by the GCF, 8.

=264

1,000

=33

125

33

125




B. 123500

C. 1740

D. 4167

A. 246

1,000


In 1955, Hurricane Diane moved through New England and produced one of the region’s heaviest rainfalls in history. In a 24-hour period, 18.15 inches of rain were recorded in one area. Express this amount as a mixed number in simplest form.


18.15 =15100

Write as a fraction.

Simplify. =

=3

20

18

15100

18

18

3

20



Lee Redmond is the world record holder for the longest fingernails. Her thumbnail is 30.2 inches long. Choose the answer below that shows 30.2 inches written as a mixed number in simplest form.

A. 30 15

C. 30 34

D. 30 14

B. 30 12



Main Idea


Example 1

Example 2

Example 3

Example 4

4-94-9 Writing Fractions as Decimals



• I will write fractions as decimals.



Standard 5NS1.2 Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number.


= 0.7


Answer: 0.7

Write as a decimal.7

10

Since has a denominator of 10, it can be written as

a decimal using place value.

710



A. 0.029

B. 0.29

C. 2.9

D. 0.0029

Choose the answer below that shows written as a decimal.

29100


0.

Divide 1 by 4.


Write as a decimal.

4 1.002

– 820

– 200

5

Answer: Therefore, = 0.25.14



A. 0.58

B. 0.675

C. 0.625

D. 0.526


58



8 3.0003

– 2 460

– 564

70.38

0– 40

0


Answer: Therefore, = 0.375.38

5



A. 0.5

B. 0.2

C. 0.25

D. 0.4


25



Answer: The number of packs is 3.125.

At a meeting, people drank 25 bottles of water. The

water came in packs of 8. This makes 3 packs.

Write the number as a decimal.

3 = 3 +

= 3

= 3 + 0.125 or 3.125

+125

1,000

Definition of a mixed number

Since 8 × 125 = 1,000, multiply the numerator and the denominator by 125.



A. 1.8

B. 1.10

C. 1.18

D. 1.08

At the party, kids drank 18 juice boxes. The juice

came in packs of 10. This makes 1 packs. Choose

the answer below that shows the number of packs

written as a decimal.

810





Example 1

Example 2

Example 3

Example 4

Example 5

4-104-10 Algebra: Ordered Pairs and Functions

Ordered Pairs and Functions



• I will use ordered pairs to locate points and organize data.

• coordinate plane

• origin

• x-axis

• y-axis

• ordered pair

• x-coordinate

• y-coordinate

• graph



Standard 5SDAP1.5 Know how to write ordered pairs correctly; for example, (x, y).


Step 1 Start at the origin. Move right along the x-axis until you are under point S. The x-coordinate of the ordered pair is 1.

Write the ordered pair that names the point S.


S


Step 2 Now move up until you reach point S. The y-coordinate is 2.

Answer: So, point S is named by the ordered pair (1, 2).


S



Write the ordered pair that names the point T.

A. (2, 1)

B. (1, 2)

C. (1, 1)

D. (2, 2)


• Start at the origin.

• Move 2 units to the right on the x-axis.

• Then move 2 units up to locate the point.

• Draw a dot and label the dot T.

Graph the point T(2, 2).


T



Which of the graphs show point N at (4, 3)?

A.




B.




C.




D.



B.




Graph the point U(1 , 0).

• Start at the origin.

• Then move 0 units up to locate the point.

• Draw a dot and label the dot U.

• Move 1 units to the

right on the x-axis.

U



Which of the graphs show point U at (2, 4 )?

A.




B.




C.




D.



C.



Amazi feeds her dog, Buster, 2 cups of food each day. Amazi made this table to show how much food Buster eats for 1, 2, 3, and 4 days. List the information as ordered pairs (days, food).


Answer: The ordered pairs are (1, 2), (2, 4), (3, 6), (4, 8).



Below is the continuation of the table in Example 4. Choose the answer that shows the information in ordered pairs.

A. (10, 5) (12, 6) (14, 7) (16, 8)

B. (5, 10) (6, 12) (7, 14) (8, 16)

C. (5, 6) (7, 8) (10, 12) (14, 16)

D. (5, 5) (6, 6) (7, 7) (8, 8)


Graph the ordered pairs from Example 4. Then describe the graph.


Answer: The points appear to lie on a line.

The ordered pairs (1, 2), (2, 4), (3, 6), and (4, 8) correspond to the points A, B, C, and D in the coordinate plane.

A

B

C

D



Choose the graph that has the ordered pairs (5, 3), (4, 2), (3, 1), and (2, 0) plotted correctly.

A.




B.




C.




D.



B.




Five-Minute Checks

Greatest Common Factor

Ordered Pairs and Functions


Lesson 4-1 (over Chapter 3)

Lesson 4-2 (over Lesson 4-1)










(over Chapter 3)

Find each sum or difference.

A. 4.1

B. 9.6

C. 0.4

D. 5.1

0.5 + 4.6


(over Chapter 3)

A. 8.76

B. 2.84

C. 8.66

D. 7.66


2.91 + 5.75


(over Chapter 3)

A. 2.7

B. 14.3

C. 3.7

D. 3.3


8.5 – 5.8


(over Chapter 3)

A. 9.46

B. 8.66

C. 9.44

D. 8.56


9.01 – 0.45


(over Chapter 3)

A. 8.56

B. 13.29

C. 9.42

D. 12.29


4.3 + 8.99


(over Chapter 3)

A. 9.32

B. 19.32

C. 8.22

D. 11.98


20 – 11.78


(over Lesson 4-1)

A. 3

B. 3 and 6

C. 1 and 6

D. 1 and 3

Identify the common factors of each set of numbers.

9, 15


A. 1 and 3

B. 1, 2, 3, and 6

C. 1, 2, and 3

D. 1, 3, and 6

(over Lesson 4-1)

Identify the common factors of each set of numbers.

6, 42


A. 3

B. 5

C. 1

D. 2

(over Lesson 4-1)

Find the GCF of each set of numbers.

13, 15


A. 2

B. 4

C. 1

D. 11

(over Lesson 4-1)


22, 104


A. 3

B. 2

C. 6

D. 12

(over Lesson 4-1)


24, 42, 72


(over Lesson 4-2)

A. 3

B. 6

C. 12

D. 9

Solve. Use the make an organized list strategy. Luis is displaying sports balls for sale. He has a soccer ball, a baseball, and a basketball. How many different ways can he arrange these balls on a table?


(over Lesson 4-3)

C.34

A.9

12

B.3

24

Write in simplest form. If the fraction is already

in simplest form, write simplest form.

1824

D. simplest form


B. 57

(over Lesson 4-3)

A. 1517

C. 1014



3549

D. simplest form


D. simplest form

(over Lesson 4-3)



411

A. 8

22

B. 25

C. 13


(over Lesson 4-3)

D. simplest form



19105

A. 8

13

C. 38210

B. 15


B. 517

(over Lesson 4-3)



30102

A. 1034

D. simplest form

C. 1551


C. 296

(over Lesson 4-4)

Write 4 as an improper fraction.56

A.96

B. 246

D. 256


D. 93

Write 3 as an improper fraction.

(over Lesson 4-4)

A.33

B. 39

C.13


(over Lesson 4-4)

Write as a mixed number in simplest form.195

A. 3 45

D. 1 145

B. 2 95

C. 2 45


C. 4

(over Lesson 4-4)

A. 123

B. 2126

D. 366



(over Lesson 4-4)


B. 1

D. 0

A. 11

17

C. 217


(over Lesson 4-5)

Find the LCM of 9, 12.

A. 3

B. 72

C. 1

D. 36



A. 3

B. 90

C. 45

D. 14

(over Lesson 4-5)


A. 33

B. 99

C. 3

D. 66


(over Lesson 4-5)


Find the LCM of 4, 6, 12.

A. 24

B. 12

C. 6

D. 36

(over Lesson 4-5)


(over Lesson 4-5)

Find the LCM of 2, 4, 7.

A. 14

B. 21

C. 56

D. 28


(over Lesson 4-6)

Solve this problem. A clothing store sells 4 different styles of shoes in 3 different colors. How many combinations of style and color are possible?

A. 24

B. 7

C. 12

D. 4


(over Lesson 4-7)

A. <

B. >

C. =

Replace each with <, >, or = to make a true sentence.

58

56


(over Lesson 4-7)

A. <

B. >

C. =


3 37

1223


(over Lesson 4-7)

A. <

B. >

C. =

Replace each with <, >, or = to make a true sentence.1316

45


(over Lesson 4-7)

A. <

B. >

C. =


13

732


D. 1120

(over Lesson 4-8)


C. 11100

A.55100

B. 510


B. 225

(over Lesson 4-8)


A.8

100

D. 45

C.8

10


C. 318

(over Lesson 4-8)

Write 3.125 as a mixed number in simplest form.

A. 35

40

B. 31251000

D. 31251000


B. 4 125

(over Lesson 4-8)

Write 4.04 as a mixed number in simplest form.

A. 404100

C. 4 4100

D. 4 250


(over Lesson 4-9)

A. 0.7

B. 0.07

C. 7.10

D. 0.71



A. 0.505

B. 5.50

C. 0.55

D. 0.055

(over Lesson 4-9)



B. 2.625

A. 0.625

(over Lesson 4-9)

Write 2 as a decimal.58

D. 8.25

C.318


D. 3.4545...

A. 3.5

(over Lesson 4-9)

Write 3 as a decimal.511

C. 0.4545

B. 3811

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