CHAPTE R
8
Decimals and Fractions
connectED.mcgraw-hill.com
The
BIG Idea
Investigate
How are decimals
and fractions
connected?
Animations
Vocabulary
Math Songs
Multilingual
eGlossary
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Virtual
Manipulatives
Audio
Foldables
Practice
Self-Check Practice
eGames
Worksheets
Assessment
Make this Foldable
to help you organize
information about
decimals and
fractions. Begin
with one sheet of
1
8 _ ” × 11” paper.
2
a ls
cim
De
alent
Equiv tions
rac imals
F
ions and Dec
ract
F
New Vocabulary
decimal decimal
a number that uses
place value, numbers,
and a decimal point
to show part of a whole
0.3
6 004A 112056
Key Vocabulary
English
Español
fracciones equivalentes
equivalent fractions
maximo común divisor
greatest common factor
forma reducida
simplest form
número mixto
mixed number
380
When Will I Use This?
Nicholas in Flying Feat
Boy, this paper
airplane contest is
going to be fun,
Mr. Grey!
Right, Nicholas!
Each student will use their tape
measure and I’ll write down
the distance their plane flies.
Your turn,
Olivia.
OK!
Wow!
How far
did it go?
Julian and
Teresa, how
about you?
My
airplane went
7 –34 feet.
Sweet!
5 –12 feet.
My
turn!
Good
throw,
Ethan!
6 –13 feet!
Mine
went
6 –14 feet.
Your
turn,
Nicholas.
Hope my
airplane goes
the farthest.
Your Tthuiisrn!
You will sollve
err.
teer
problem in the chap
Decimals and Fractions 381
Are You Ready
You have two options for checking
Prerequisite Skills for this chapter.
for the Chapter?
Text Option
Take the Quick Check below.
Write a fraction to describe the part that is green.
1.
2.
3.
5. eight-tenths
6. twenty-hundredths
Write each as a fraction.
4. four-tenths
7. Measurement On Tuesday, seven tenths of an inch of rain fell.
Write the amount of rain that fell as a fraction.
Algebra Copy and complete.
8.
9.
4
__
= ___
5
10
1
__
= ___
5
10
6
4
10. In Salvador’s aquarium, ___
of the fish are yellow, and ___
are blue.
10
10
Are there more blue or yellow fish in Salvador’s aquarium?
Explain how you know.
Online Option
382
Decimals and Fractions
Take the Online Readiness Quiz.
Multi-Part
Lesson
1
Decimals
A
PART
I will explore decimals.
Materials
IVOESFEUIT
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0OFT
UFOT
Workmat 6
POFT
C
D
E
Place Value Through
Tenths and Hundredths
Main Idea
IVOESFET
B
A decimal is a number that uses place value and a decimal
point to show part of a whole. The place-value chart shows
how decimals are part of the base-ten system. You can use
a place-value chart and models to represent decimals.
Hundreds Tens Ones Tenths Hundredths
hundredths grid
colored pencils
÷10
÷10
÷10
Use a place-value chart to show seven-tenths.
U
Step 1
Use a place-value chart.
Write 7 in the tenths column.
Hundreds Tens Ones Tenths Hundredths
Get ConnectED
GLE 0406.2.1
Understand place value of
numbers from hundredths
to the hundred-thousands
place.
÷10
0
Step 2
Use a model.
Step 3
Write the decimal.
0.7
decimal point
7
tenths place
So, seven-tenths can be written as 0.7.
Lesson 1A Decimals 383
Model forty-five hundredths. Write the decimal.
M
Step 1
Use a place-value chart.
Write 4 in the tenths column and 5 in the
hundredths column.
Hundreds Tens Ones Tenths Hundredths
0
Step 2
4
5
Use a hundredths grid.
Shade 45 of the 100 parts to show 45 hundredths.
Step 3
Write the decimal.
0.45
So, forty-five hundredths can be written 0.45.
About It
1. Describe the difference between the decimals two-tenths and
two-hundredths.
2. Use a hundredths grid to model the decimals in Exercise 1.
Identify the greater decimal.
and Apply It
Write a decimal for each shaded part.
4.
3.
5.
Write each decimal.
6. two-tenths
7. twenty-five
hundredths
8. forty-nine
hundredths
9. seventy-three
hundredths
Model each decimal.
10. 0.37
14.
E
11. 0.51
WRITE MATH Explain how to use a place-value chart to
write seventeen-hundredths.
384
3
12. 0.88
Decimals and Fractions
13. 0.9
Multi-Part
Lesson
1
Decimals
PART
A
Main Idea
I will model and
describe tenths as part
of the base-ten system.
Vocabulary
V
decimal
B
C
D
E
Tenths
In the previous lesson, you used parts of a whole to
show the place value. A decimal is a number that uses
place value, numbers, and a decimal point to show part
of a whole. A tenth is one of ten equal parts.
tenths
Write Tenths
W
Get ConnectED
GLE 0406.2.1
Understand place value of
numbers from hundredths
to the hundred-thousands
place. SPI 0406.2.1 Read
and write numbers from
hundredths to hundredthousands in numerals
and in words.
HORSES Lorena and Rebecca
went to 10 horseback riding
lessons. Three of the lessons
were outside the barn. What
part of horseback riding lessons
was spent outside?
Step 1 Find the parts.
There were 10 horseback riding lessons.
Three of the 10 lessons were spent outside.
Step 2
Model the parts.
Use a base-ten model to show three out of ten,
or three-tenths.
Read three-tenths
Write 0.3
So, 0.3 of the horseback riding lessons were outside.
Lesson 1B Decimals 385
Write Tenths
W
CAMPING A campground has 10 campsites. Eight of the
campsites have campers. The rest are empty. What part
of the campground is empty?
If eight out of ten campsites have campers, then two out
of ten campsites do not have campers.
You can use a place-value chart to write two out of ten,
or two-tenths.
Hundreds Tens Ones Tenths Hundredths Thousandths
0
2
So, 0.2 of the campground is empty.
Describe Tenths
Use words to describe 0.7.
Seven-tenths means seven out of ten equal parts.
So, 0.7 can be described as seven out of ten.
Model and write each decimal
decimal. SSee EExamples
l 1 andd 2
1. four out of ten
2. eight-tenths
3. two-tenths
Describe each decimal. See Example 3
4. 0.5
5. 0.1
7. A flower has ten petals. During a
storm, five of the petals fall off.
Write a decimal to show what
part of the petals is left.
9.
386
E
6. 0.3
8. A house has ten windows. During
spring cleaning, eight windows
were washed. Write a decimal to
show what part of the windows
still needs to be cleaned.
TALK MATH Explain why dimes could be used to model tenths.
Decimals and Fractions
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EXTRA
Begins on page EP2.
Model and write each decimal. See Examples 1 and 2
10. one out of ten
11. five out of ten
12. four-tenths
13. seven-tenths
14. nine-tenths
15. eight-tenths
Describe each decimal. See Example 3
16. 0.2
17. 0.6
19. Emilio had a math quiz today. He
answered nine of the ten questions
correctly. Write a decimal to show
what part of the quiz Emilio answered
correctly.
21. Jasmine had ten pennies in her
pocket. Some fell through a hole in
her pocket. Below are the pennies
she still has in her pocket. Write a
decimal to show how many pennies
she lost.
18. 0.9
20. Cody’s dog had ten chew toys. Cody
took six toys outside for his dog.
Write a decimal to show what part
of the toys is still inside.
22. There were ten goldfish in the
aquarium below. Some fish were
moved to a new aquarium. Use the
picture to write a decimal to show
how many fish are still in the original
aquarium.
Measurement
Meteorologists reported the amount
of rainfall for 4 weeks in April. Use the table to answer
the following questions.
23. Write each amount of rainfall in decimal form.
Rainfall
(in inches)
April
Week
Week
Week
Week
1
2
3
4
eight-tenths
four-tenths
nine-tenths
seven-tenths
24. Model the decimal described in week 1.
25. OPEN ENDED Write a decimal whose value is greater than five-tenths.
26.
E
WRITE MATH Write a real-world problem with 0.8 as the answer.
Lesson 1B Decimals 387
Multi-Part
Lesson
1
Decimals
PART
A
Main Idea
I will describe
hundredths as part of
the base-ten system.
B
C
D
E
Hundredths
A hundredth is one of one hundred equal parts.
Vocabulary
V
hundredths
thousandths
Get ConnectED
GLE 0406.2.1
Understand place value of
numbers from hundredths
to the hundred-thousands
place. SPI 0406.2.1 Read
and write numbers from
hundredths to hundredthousands in numerals
and in words.
1.0
one
0.1
one-tenth
0.01
one-hundredth
Write
Hundredths
Drew has a 100-piece puzzle.
There are four corner pieces
in the puzzle. What part of the
puzzle is corner pieces?
Four of the 100 puzzle pieces
are corner pieces.
4 of 100 parts
are shaded.
Read
four-hundredths
Write
0.04
So, 0.04 of the puzzle pieces are corner pieces.
388
Decimals and Fractions
Write Hundredths
Tiffany has the coins shown.
What part of a dollar are the
W
coins worth?
Step 1
Count the parts.
$1 = 100¢
10¢
Step 2
11¢
11
12¢
12
13¢
13
14¢
14
Compare the parts to the whole.
Each cent is one hundredth, or 0.01, of a dollar.
Fourteen-hundredths is 0.14 of a dollar.
So, the coins are 0.14 of a dollar.
Write each decimal.
decimal SSee EExample
l 1
1.
2.
3.
Write a decimal for each part of a dollar shown.
See Example 2
4.
5.
6. A movie theater has 100 seats.
Seventy-eight of the seats are full.
What decimal shows the part of the
seats that is full?
7.
E
TALK MATH Explain why coins can
be used to model decimals.
Lesson 1C Decimals 389
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Begins on page EP2.
Write each decimal. See Example 1
8.
9.
10.
11.
12.
13.
Write a decimal for each part of a dollar shown. See Example 2
14.
15.
16.
17.
18. Emma earned eighty-nine out of
100 points on a math test. What
decimal shows the part of the points
that Emma earned?
19. A department store had one hundred
teapots. Six of the teapots were sold.
What decimal represents the teapots
that are left?
20. Darnell inserted the coins shown
into a vending machine. What part
of a dollar are the coins?
21. Bena received the coins shown as
change after buying pencils at the
school store. What part of a dollar
are the coins?
390
Decimals and Fractions
22. OPEN ENDED Write a real-world problem that results in an
answer of 0.72.
23. CHALLENGE What is the least number of coins needed to make
0.90 of one dollar? Which coins are used?
24.
E
WRITE MATH Write a decimal to represent the models
below. Explain.
Decimals
A whole number can be divided into 1,000 equal parts called
thousandths . Thousandths are smaller than hundredths.
Write Thousandths
W
CARNIVALS A school carnival sells 1,000 tickets.
On Saturday, 415 of the tickets are sold. The rest
are sold on Sunday. What part of the tickets is
sold on Sunday?
Subtract the number of tickets sold on Saturday.
1,000 – 415 = 585
On Sunday, 585 of the 1,000 tickets were sold.
Read
five hundred eighty-five thousandths
Write
0.585
So, 0.585 of the tickets were sold on Sunday.
Write each decimal.
25. three hundred forty-two
thousandths
26. twenty-six
thousandths
27. five-thousandths
28. An apple orchard can produce about 1,000 apples. If 35 of
the apples are rotten, what part of the apples is not rotten?
29.
E
WRITE MATH Explain why 0.040 is not the same as 0.004.
Lesson 1C Decimals 391
Multi-Part
Lesson
1
Decimals
PART
A
Main Idea
I will write decimals
greater than one.
Get ConnectED
GLE 0406.2.1
Understand place value of
numbers from hundredths
to the hundred-thousands
place. SPI 0406.2.1 Read
and write numbers from
hundredths to hundredthousands in numerals
and in words.
B
C
D
E
Decimals Greater
Than One
You have described decimals that have values that are less
than one whole. The value of decimals can also be greater
than one whole. These numbers contain a whole number and
a decimal part.
Model
Decimals
Model 5.02.
Since the least place value is the hundredths place,
use hundredths grids to model 5.02.
Step 1 Model the whole number.
Since the whole number is 5, shade 5
hundredths grids.
Step 2 Model the decimal.
Since there are two-hundredths, shade 2 squares of
a sixth hundredths grid.
two-hundredths = 2 out
of 100 squares shaded
So, 5.02 represents the whole number five and two
hundredths of one whole.
392
Decimals and Fractions
Representing Decimals
Words
Examples
Standard Form the usual way to write a
number using digits
Word Form a way to write a number using
words
Expanded Form a way to write a number
showing the value of each digit
6.25
six and twenty-five hundredths
6 + 0.2 + 0.05
Write Decimals
Hundredths
Expanded form: 8 + 0.7 + 0.05
Tenths
Standard form: 8.75
Ones
Write each digit of eight and
seventy-five hundredths in a
place-value chart.
8
7
5
Thousandths
A flamingo weighs about eight and seventy-five
hundredths pounds. Write eight and seventy-five
h
hundredths in standard and expanded form.
Read and Write Decimals
To write decimals in
word form, use the
word and for the
decimal point.
W
Write
sixteen and one hundred twenty-eight thousandths
in standard and expanded form.
Write each digit of sixteen and one hundred twenty-eight
thousandths.
Standard form: 16.128
Expanded form: 10 + 6 + 0.1 + 0.02 + 0.008
Lesson 1D Decimals 393
Model each decimal
l 1
decimal. SSee EExample
1. 2.4
2. 5.13
3. 9.07
4. Write seven and fifty-three hundredths
in standard and expanded form.
5.
E
TALK MATH Describe how to write
eighty-two and three-hundredths in
standard form.
See Examples 2 and 3
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Begins on page EP2.
Model each decimal. See Example 1
6. 1.6
7. 3.9
8. 4.27
9. 7.69
10. 11.37
11. 26.46
Write each decimal in standard and expanded form.
See Examples 2 and 3
12. five and thirty-seven hundredths
13. eight and ninety-three hundredths
14. ten and sixty-one hundredths
15. thirteen and forty-six hundredths
For Exercises 16–18, use the table at the right.
16. Write the length of a tibia in expanded
form.
17. Write the length of the shortest bone in
word form.
Longest Bones in the Human Body
Bone
Length (in.)
Femur (thigh bone)
Humerus (upper arm
bone)
Seventh Rib
14.35
Tibia (shin bone)
16.94
18. Write the length of the longest bone in word
and expanded form.
For Exercises 19–21, use the table at the right.
19. Write the length of a King Cobra in
expanded form.
Facts About Cobras
20. Write the length of King Cobra fangs
in word form.
• The fangs of a King Cobra
can be up to 1.25 cm long.
21. Write the distance a black-necked
cobra can spit venom in word form.
394
Decimals and Fractions
• King Cobras have been
found up to 5.5 m long.
• The black-necked cobra
can spit venom 2.4 m.
19.88
9.45
22. OPEN ENDED Write two decimals that are equivalent to 3.6.
Explain.
23.
E
WRITE MATH Explain how the word form of a decimal
can help you write the standard form of the decimal.
Test Practice
24. Which model represents 2.01?
(Lesson 1D)
A.
B.
C.
D.
25. Lydia has five pizzas. Each pizza
has ten slices. Four of the pizzas
are whole. One pizza has only
8 slices left. Which decimal
describes Lydia’s pizzas? (Lesson 1D)
F. 4.08
H. 4.8
G. 4.2
I. 5.4
26. Each box holds one hundred paper
clips. There are two full boxes. The
third box has only 24 paper clips.
Which decimal describes the boxes
of paper clips? (Lesson 1D)
A. 2.24
C. 3.24
B. 2.42
D. 3.42
Write each decimal. (Lesson 1C)
27. two-hundredths
28. nineteen-hundredths
29. forty-eight hundredths
30. two hundred sixty thousandths
31. A pronghorn antelope can run about 0.88 mile
per minute. Use a hundredths grid to model 0.88.
(Lesson 1C)
To assess partial mastery of SPI 0406.2.1, see your Tennessee Assessment Book.
395
Multi-Part
Lesson
1
Decimals
PART
A
Main Idea
I will compare and
order decimals.
Get ConnectED
B
C
D
E
Compare and
Order Decimals
To compare decimals, you can use a number line or
place value.
SPI 0406.1.2
Compare decimals using
concrete and pictorial
representations.
SPI 0406.2.6 Use the
symbols <, >, and = to
compare common
fractions and decimals in
both increasing and
decreasing order.
SCORES The table shows
the results from a
skateboarding
competition. Who has
the higher score,
Doria or Elise?
Doria has a score of 79.7,
while Elise has a score of 79.5.
One Way:
79.0
79.2
Number Line
Elise
Doria
79.5
79.7
79.4
79.6
79.8
80.0
79.7 is to the right of 79.5.
So, 79.7 > 79.5.
So, Doria has the higher score.
396
Decimals and Fractions
Tenths
Line up the decimal points.
Then compare the digits in
each place-value position.
In the tenths place, 7 > 5.
So, 79.7 is greater than 79.5.
Ones
Place Value
Tens
Another Way:
7
9
7
7
9
5
Order Decimals
Everyday Use
Order select items
you want
Order 9.86, 9.8, 9.92, and 9.09 from greatest to least.
Math Use
Order to arrange in
a logical pattern
Next, place zeros to the
right of the last digit so all
numbers have the same
number of decimal places.
First, line up the
decimal points.
Finally, compare
and order using
place value.
9.86
9.86
9.92
9.8
9.80
9.86
9.92
9.92
9.80
9.09
9.09
9.09
The order from greatest to least is 9.92, 9.86, 9.8, and 9.09.
Compare
> <
= See Example 1
Compare. Use >,
<, or =.
1. 0.2 0.6
2. 12.07 1.207
3. 5.60 5.6
Order from greatest to least. See Example 2
4. 3.2, 4.5, 3.9, 4.1
5. 0.12, 1.2, 1.21, 12.0
6. $6.52, $5.62, $6.50, $5.60
Use the number line to compare and order each set of
decimals from least to greatest.
4
7. 5.7, 5.2, 4.7, 4.2
4.5
5
5.5
6
8. 4.2, 4.8, 6.2, 5.8
9. Measurement Four friends are going to
different summer camps. The table shows the
distance between each camp and their hometowns.
Order the distances from least to greatest.
10.
E
TALK MATH Tell how to order 5.5, 5.3, 5.4, and 5.0
from greatest to least.
Traveling to Camp
Name
Distance (mi)
Bill
64.25
Sami
42.5
Latesha
64.87
Irena
42.35
Lesson 1E Decimals 397
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Begins on page EP2.
Compare. Use >, <, or =. See Example 1
11. 0.74 7.4
12. 16.33 16.3
13. 0.56 0.58
14. 0.8 0.80
15. 1 0.09
16. 0.90 0.9
17. 82.6 82.60
18. 1.06 1.05
Order from greatest to least. See Example 2
19. $19.62, $19.56, $19.60
20. 0.5, 0.55, 0.6
21. 68.16, 81.6, 68.1
Use the number line to compare and order each set of decimals
from least to greatest.
6
7
8
9
22. 6.3, 8.1, 7.5, 7.7
23. 8.4, 7.9, 8.8, 8.6
24. 7.5, 6.2, 7.75, 6.25
25. 7.8, 8.25, 8.2, 8.50
26. Marlon averages 5.6 rebounds per
game. Tina averages 5.9 and Omar
averages 4.3 rebounds per game.
Who averages the most rebounds?
Explain.
27. Measurement Desiree ran the
100-meter dash four times. Her times
were 16.25, 15.36, 16.55, and 15.23
seconds. What was her slowest time?
28. NUMBER SENSE What number is halfway between 4.36 and
4.48 on a number line?
29. FIND THE ERROR Ashley wrote 3 decimals in order
from least to greatest. Find and correct her mistake.
6.27, 6.19,
6.29
30.
E
WRITE MATH Write a real-world problem about comparing and
ordering decimals.
398
To assess mastery of SPI 0406.1.2, see your Tennessee Assessment Book.
Multi-Part
Lesson
2
Fractions
A
PART
B
C
D
E
F
Equivalent Fractions
Main Idea
I will explore
equivalent fractions.
Like decimals, fractions are used to show part of a number.
Fractions that represent the same part of a number are
equivalent fractions.
Materials
Identify two fractions that are equivalent to
fraction tiles
1
4
1
1
3
Step 1
1
2
1
Model _.
1
3
1
_
fraction tile
3
1
to show _ .
Get ConnectED
3
1
3
Start with 1 whole.
Then, use the
_1 .
3
SPI 0406.2.5
Generate equivalent forms
of common fractions and
decimals and use them to
compare size.
Step 2
1
Find a fraction
equivalent to
_1 .
3
1
3
1
Using _ fraction
6
tiles, place
them below the
1
_
fraction tile.
1
6
1
6
3
How many
1
_
fraction tiles
6
are used?
Step 3
1
Find another
fraction equivalent
to
_1 .
1
3
3
1
Use _ fraction
12
tiles to equal the
length of the
1
_
fraction tile.
3
1
6
1
6
1 1 1 1
12 12 12 12
1
Count the number of _ fraction tiles.
12
1 2
4
So, _, _, and _ are equivalent fractions.
3 6
12
Lesson 2A Fractions 399
IIdentify equivalent fractions.
Step 1
Step 2
Draw three identical
number lines that show
zero and one.
0
1
0
1
0
1
2
4
Notice that _ = _ = _.
8
1
1
4
0
Divide the first number
line into fourths. Divide
the second number line
into eighths. Divide the
third number line into
sixteenths.
4
0
1
8
2
4
2
8
3
8
4
8
3
4
5
8
6
8
4
4
7
8
8
8
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16
16
About It
1
1. Algebra Copy and complete _ = _ = _.
3
9
15
3
2. Refer to Activity 2. Find two fractions equivalent to _.
4
and Apply It
Determine whether the fractions are equivalent.
Write yes or no. Use fraction tiles or number lines.
6
2
3. _ and _
4
12
6
5
4. _ and _
8
10
3
2
5. _ and _
3
5
9
3
6. _ and _
12
Find two equivalent fractions for each fraction. Use fraction
tiles or number lines.
1
7. _
5
11.
400
E
2
8. _
6
4
9. _
8
2
10. _
12
WRITE MATH Explain what it means for two fractions to be equivalent.
Decimals and Fractions
4
Multi-Part
Lesson
2
Fractions
PART
A
Main Idea
I will find equivalent
fractions.
B
C
D
E
F
Equivalent Fractions
4
1
The fraction models show that _ is the same as _. Fractions
2
8
Vocabulary
V
that represent the same amount are equivalent fractions .
equivalent fractions
1
Get ConnectED
1
2
SPI 0406.2.5
Generate equivalent forms
of common fractions and
decimals and use them to
compare size. Also
addresses GLE 0406.1.1,
GLE 0406.3.1.
Prefixes The prefix
equi- means equal.
1
8
1
8
1
8
1
8
Find Equivalent
Fractions
FISH Allie has 8 fish in an aquarium. Four fish are
_4 of the fish are green. Find three fractions
8
4
that are equivalent to _.
green. So,
8
You can use multiplication or division.
One Way:
Multiply
8
_
8 × 2 = 16
Multiply the
numerator
and the
denominator
by the same
number, 2.
4× 2
___
Another Way: Divide
Divide the
4÷ 2
2
_
_
numerator
=
8÷2
4
2÷ 2
_
_1
4÷2 = 2
and the
denominator
by the same
number, 2.
8 2
1
4
So, _, _, and _ are equivalent to _.
16 4
2
8
Lesson 2B Fractions 401
Write the fraction for the part that is shaded
shaded. Then find an
equivalent fraction. See Example 1
1.
2.
3.
Find an equivalent fraction for each fraction. See Example 1
1
4. _
4
4
5. _
1
6. _
6
10
10.
9. Javier has 4 juice boxes. Three are
grape flavored. Write two fractions
that describe the part of the juice
boxes that is grape.
1
8. _
3
8
7. _
5
E
TALK MATH Tell why
3 _
6
_
, and
4 8
9
_
are equivalent fractions. Give an
12
example of another set of three
equivalent fractions.
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Begins on page EP2.
Write the fraction for the part that is shaded. Then find an
equivalent fraction. See Example 1
11.
12.
13.
14.
15.
16.
Find an equivalent fraction for each fraction. See Example 1
2
17. _
2
18. _
6
19. _
2
20. _
2
21. _
6
22. _
1
23. _
9
24. _
1
25. _
12
26. _
7
8
5
2
10
27. A roller coaster has 16 cars. Six of the
cars are green. Write two fractions for
the part of the cars that is green.
402
Decimals and Fractions
18
12
6
3
20
1
28. Measurement Lucas ran _ mile.
2
4
Ayana ran _ mile. Did they run the
6
same distance? Explain.
Algebra Find the value of x.
3
12
29. _ = _
x
4
x
2
30. _ = _
5
8
x
31. _ = _
25
12
8
1
32. _ = _
x
64
3
Science
The state animal of Michigan is the
white-tailed deer. Female deer, called does, usually
have 1 to 3 young at a time. Young deer are called
fawns. The male deer, called bucks, grow antlers in
the spring.
33. Suppose 10 deer are traveling together. If 4 of
them are does and 3 are fawns, what fraction
represents the number of bucks? Write another
fraction that represents this number.
34. A doe has 3 fawns. If one of the fawns is male,
what fraction of the fawns is female? Write a fraction
equivalent to this fraction.
2
35. OPEN ENDED Write a fraction equivalent to _. Write a fraction
5
3
equivalent to _. Which fraction represents a greater number?
6
Explain.
36. FIND THE ERROR Julian is finding a fraction equivalent to _.
18
Find and correct his mistake.
6
6 ÷ 3 = __
3
_______
18 ÷ 3
6
25
37. CHALLENGE Identify a fraction that is equivalent to _.
100
38.
E
WRITE MATH Can you always find an equivalent fraction for a
fraction? Explain.
Lesson 2B Fractions 403
Multi-Part
Lesson
2
Fractions
PART
A
B
C
D
E
Problem-Solving Strategy:
F
Make a Model
Main Idea I will solve problems by making a model.
Brandi and her mom are at a pet store.
The pet store has 15 reptiles. One third
of the reptiles is turtles. Two are snakes,
and the rest are lizards. How many of each
reptile are there?
Understand
What facts do you know?
• There are 15 reptiles at the store.
• Two are snakes.
k
• One third is turtles.
• The rest are lizards.
What do you need to find?
• Find the number of each reptile.
Plan
Make a model to solve the problem.
Solve
• Draw 15 circles. Since the
1
fraction _ is used, place the
3
circles in 3 equal groups.
1
• To show the turtles, shade _
3
of the circles, that is, one of the
three equal groups. So, there are
5 turtles. There are 2 snakes, so
shade 2 circles to show the snakes.
turtles
• There are 8 circles not shaded.
This is the number of lizards.
So, there are 5 turtles, 2 snakes,
and 8 lizards at the pet store.
Check
Look back. 5 turtles + 2 snakes + 8 lizards = 15 reptiles.
The pet store has 15 reptiles. So, the answer is correct.
SPI 0406.2.10 Solve contextual problems using whole numbers, fractions, and decimals.
404
Decimals and Fractions
snakes
Refer to the problem on the previous page.
1. Explain why you used 15 circles.
1
2. You know that _ of the reptiles are
3. If the pet store had 24 reptiles, how
many of the reptiles would be lizards?
3
turtles. Explain why 5 circles were
shaded to show the number of turtles.
4. Check your answer to Exercise 3.
How do you know that it is correct?
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Solve. Use the make a model strategy.
5. Measurement There are three trees
in a backyard. The second tree is half
as tall as the first. The third tree is
taller than the second tree and shorter
than the first tree. The total height of
the trees is 24 feet. What is the height
of each tree?
10 ft
x
y
8. There are 22 students
in Ms. Lane’s class.
Half of them packed
their lunches. Eight
students are buying
pizza. The rest are
buying salads. How
many students are
buying salads?
9. Measurement The table shows
how long Adam and Kenya rode their
bikes. Who biked more minutes? How
many more minutes?
Biking Schedule
6. Corey has 8 apples. Four are red, two
are green, and the rest are yellow.
Write two fractions that name the part
of the apples that are yellow.
7. Emil bought his mom a dozen roses.
Some of the roses are shown below.
The rest are white. Of which color
were there the most? What fraction of
the roses were that color? Write an
equivalent fraction.
Name
Time Spent Biking
Adam
1 of an hour
__
Kenya
15 minutes
3
10. There are 16 books on a shelf.
One-fourth of the books are about
animals. Two are adventure. The rest
are mystery. How many are mystery
books?
11.
E
WRITE MATH Look back at Exercise
10. Explain how you used the make a
model strategy to solve the problem.
Lesson 2C Fractions 405
Multi-Part
Lesson
2
PART
Fractions
A
Main Idea
I will compare and
order fractions.
Get ConnectED
B
D
C
E
F
Compare and Order
Fractions
To compare fractions, you can use models, number lines,
and equivalent fractions.
SPI 0406.2.6
Use the symbols <, >,
and = to compare
common fractions
and decimals in both
increasing and
decreasing order.
Compare Fractions
MEASUREMENT Ramon has an
insect collection. The table shows
the lengths of four insects in his
collection. Which is longer, a field
cricket or a whirligig beetle?
You can use fraction models to
5
3
compare _ and _.
8
8
5
_
Field cricket
3
_
Whirligig beetle
8
8
5
3
The models show that _ > _.
8
8
So, the field cricket is longer than the whirligig beetle.
Which is longer, a mosquito or lightning bug?
1
1
You need to compare _ and _.
4
0
0
1
4
2
4
1
2
3
4
1
1
2
1
_
Mosquito
1
_
Lightning bug
4
2
So, the lightning bug is longer than the mosquito.
406
Decimals and Fractions
Order Fractions
Order
_2 , _1 , and _7 from least to greatest.
3 2
12
Number
Lines
One Way:
Another Way:
Use a number line.
0
0
12
1
7
12
_1 , _7 , _2 .
2 12 3
3
12
12
12 12 12
_1 < _7 < _2
2
2×6
Compare the numerators.
Order from least to greatest.
_6 , _7 , _8
1
1
2
0
Find equivalent fractions
with the same denominator.
2×4
_
_8 _
_6
1×6
=
=
3×4
1
2
3
Equivalent
Fractions
So, the order from least to greatest is
_1 , _7 , _2 .
2 12 3
Compare. Use > , < , or =. SSee Examples
Compare
E
l 1 andd 2
1.
2.
0
5
1
_
_
8
1
4
0
8
1
6
2
4
2
6
1
3
4
3
6
4
6
5
6
1
1
1
_
_
4
3
1
3. _ _
4
6
3
3
4. _ _
2
6
4
Order from least to greatest. See Example 3
3 2 4
5. _, _, _
1 7 3
6. _, _, _
16 8 4
8 6 8
1
7. Measurement Griff worked for _ of
3
3
an hour. Sasha worked for _ of an
12
8.
E
TALK MATH Explain how to compare
7
2
the fractions _ and _.
6
12
hour. Who worked for a longer time?
Lesson 2D Fractions 407
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Compare. Use > , < , or =. See Examples 1 and 2
9.
10.
7
4
_
_
10
4
1
_
_
10
8
4
2
6
3
6
12.
11.
0
1
3
0
1
6
2
3
2
6
3
6
4
6
5
6
1
0
1
0 1
1
6
8
5
2
_
_
3
2
8
3
8
4
6
4
8
5
8
6
8
7
8
6
8
3
5
14. _ _
8
4
15. _ _
5
2
16. _ _
4
1
17. _ _
5
2
18. _ _
3
3
5
9
1
5
4
_
_
6
2
1
13. _ _
6
1
5
6
5
6
10
2
8
10
3
Order from least to greatest. See Example 3
4 1 3
19. _, _, _
3 2 7
20. _, _, _
6 3 3
3 3 3
21. _, _, _
4 3 8
22. Which meat makes up most of
Mr. Collin’s sandwich?
10 4 5
23. Allison took a survey. Find the favorite
weekend activity.
Favorite Weekend Activities
Activity
1
Decimals and Fractions
Basketball
25. Suzanne practiced volleyball for _
3
1
hour on Saturday and _ hour on
6
Sunday. Which day did she practice
longer?
26. Measurement The table shows how much time
each student needs to finish an art project. Does
Simón need more or less time than Phil? Explain.
408
Mall
ADMIT ONE
ADMIT ONE
24. Aisha ate _ of the carrots in the bag.
4 3
Enrique ate _ of the carrots in the
12
bag. Who ate more carrots?
Movie
Fraction of
Friends
2
6
1
4
5
12
2
Student Time
Simón
4
_
hour
Phil
3
_
hour
12
4
1
27. OPEN ENDED Write three fractions that are not greater than _.
2
28. WHICH ONE DOESN’T BELONG? Identify the set of fractions
that does not belong with the other three sets. Explain.
15
_1 , _5 , _
_2 , _1 , _1
7
_2 , _1 , _
9 3 2
4 8 16
5 2 10
_3 , _1 , _2
4 2 12
150
29. CHALLENGE Identify a fraction that is greater than _.
500
30.
E
WRITE MATH Explain how to decide if
3
3
_
is greater than or less than _.
4
5
Test Practice
31. What fraction is equivalent to
the fraction represented by
point M on the number line?
32. Which set of fractions is ordered
from greatest to least? (Lesson 2D)
3 6 2
F. _, _, _
(Lesson 2B)
5 15 10
M
1
0 8
1
4
3
8
1
2
5
8
7
8
1
A. _
1
C. _
3
B. _
8
3
D. _
4
2 3 6
G. _, _, _
1
10 5 15
2 6 3
H. _, _, _
10 15 5
2
6 3 2
I. _, _, _
15 5 10
4
33. Toya h
has 8 coins
her piggy bank. One fourth of
i iin h
the coins is quarters. Three of the coins are dimes.
The rest of the coins are pennies. How many pennies
does Toya have? (Lesson 2C)
Find an equivalent fraction for each fraction. (Lesson 2B)
1
34. _
2
35. _
3
36. _
7
37. _
1
38. _
4
39. _
2
10
3
8
To assess mastery of SPI 0406.2.6, see your Tennessee Assessment Book.
5
12
409
Multi-Part
Lesson
2
Fractions
PART
A
Main Idea
I will write a fraction
in simplest form.
B
C
D
E
F
Simplest Form
8
2 4
2
Among the equivalent fractions _, _, and _, the fraction _
3 6
Vocabulary
V
simplest form
greatest common
factor
Get ConnectED
Fractions in
Simplest Form
1
_
2
SPI 0406.2.4
Find factors, common
factors, multiples, and
common multiples of two
numbers. Also addresses
GLE 0406.3.1.
3
12
is in simplest form because its numerator and denominator
have no common factor other than 1.
2
_
3
_
3
4
3
_
7
Fractions NOT in
Simplest Form
5
_
2
_
4
8
4
_
8
_
6
8
_
10
16
_
24
12
To change a fraction into its simplest form, you can divide both
the numerator and the denominator by the greatest common
factor. A greatest common factor is the greatest of the
common factors of two or more numbers.
Simplest Form
QUIZ There are 12 questions on a quiz.
Esteban got 8 out of 12, or
_8 , of the
Name:
1.
12
questions correct. Two fractions
equivalent to
_8 are _2 and _4 .
8
Write _ in simplest form.
12
3
Step 1
3.
6
4.
5.
12
Find the common factors of
8 and 12.
7.
8.
2.
6.
Factors of 8:
1, 2, 4, 8
Factors of 12:
1, 2, 3, 4, 6, 12
9.
10.
11.
12.
Common factors: 1, 2 and 4
Step 2
Divide by the greatest common factor, 4.
8 ÷ 4
2
_
=_
3
12 ÷ 4
The numbers 2 and 3 have no
common factor other than 1.
8
2
_
So,
in simplest form is _.
3
12
410
Decimals and Fractions
To write a fraction in simplest form, you can also divide by
common factors.
Simplest Form
ANIMALS Koala bears sleep
up to 18 hours each day. What
part of a day do koalas sleep?
Write the fraction in simplest
form.
Step 1 Write a fraction.
18
_
24
Step 2
Divide until the
numerator and the
denominator have no
common factor other
than 1.
hours of sleep each day
total hours in a day
Divide by common factors.
18 ÷ 2
18
9
_
= _ =_
24
24 ÷ 2
12
A common factor of
18 and 24 is 2.
9÷ 3
9
3
_
= _=_
4
12
12 ÷ 3
A common factor of
9 and 12 is 3.
18
3
3
So, _ simplifies to _. Koalas sleep up to _ of a day.
24
4
4
Simplest Form
One Way
Divide the numerator and denominator
by the greatest common factor.
Another Way
Divide the numerator and denominator
by common factors until the only
common factor is 1.
Write each fraction in simplest form.
form If it is in simplest form,
form
write simplest form. See Example 1
2
1. _
4
6
2. _
10
5
3. _
6
18
4. _
30
5. Daniela read for 24 minutes. What
6. E TALK MATH How do you
4
fraction of an hour did she read? Write
know that _ is in simplest form?
5
as a fraction in simplest form. See Example 2
Lesson 2E Fractions 411
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Write each fraction in simplest form. If it is in simplest form,
write simplest form. See Example 1
2
7. _
3
8. _
7
9. _
6
10. _
4
11. _
10
12. _
18
13. _
24
14. _
32
15. _
42
16. _
10
12
27
15
20
9
40
16
56
42
Measurement What part of one day is each amount of time?
Write as a fraction in simplest form. See Example 2
17. 9 hours
18. 16 hours
19. 20 hours
Algebra Find the value of x to simplify each fraction.
15
x
20. _ = _
45
16
4
21. _ = _
x
28
3
20
x
22. _ = _
36
9
Write as a fraction in simplest form.
23. Six of the 16 marbles in a bag are
yellow. What fraction of the marbles
is yellow?
24. Adina read 12 out of 36 pages in a
comic book. What fraction of the pages
did she read?
Measurement The table lists the length of
some objects in a classroom. What part of
a foot is each object? Write as a fraction in
simplest form. See Example 2
25. pencil
26. crayon
27. scissors
28. pencil box
29. OPEN ENDED Write three fractions that are in simplest form.
30. WHICH ONE DOESN’T BELONG? Identify the fraction that does
not belong with the other three. Explain your reasoning.
31.
E
_2
_6
_1
_
11
3
15
2
12
WRITE MATH The simplest form of
6
2
_
is _. How can you
15
5
use multiplication to show that they name the same number?
412
Decimals and Fractions
Test Practice
32. In the circle below, _ of the
8
sections are shaded.
4
33. Mr. Henshaw used some eggs from
a full carton to make breakfast for
his family. The eggs that are left
are shown.
Which of the following is equal
4
to _? (Lesson 2B)
8
1
A. _
4
3
B. _
Identify the part of the eggs in
the carton Mr. Henshaw used.
6
6
C. _
(Lesson 2E)
5
D. _
4
2
G. _
4
10
3
F. _
12
H. _
1
3
2
I. _
3
34. Ashley has six coins. One third of the coins is quarters. She has
1 dime and the rest are nickels. What is the value of Ashley’s
coins in decimal form? (Lesson 2C)
Write the fraction for the part that is shaded. Then find an
equivalent fraction. (Lesson 2B)
35.
36.
37.
Determine whether the fractions are equivalent. Write yes or no.
(Lesson 2A)
5
6
38. _ and _
10
12
8
1
39. _ and _
4
12
6
3
40. _ and _
8
4
Lesson 2E Fractions 413
Multi-Part
Lesson
2
Fractions
PART
A
Main Idea
I will write mixed
numbers and improper
fractions.
Vocabulary
V
B
C
D
F
E
Mixed Numbers
A mixed number has a whole number part and a fraction part.
An improper fraction has a numerator that is greater than or
equal to its denominator.
mixed number
improper fraction
Mixed Numbers
1_
Get ConnectED
SPI 0406.2.2
Locate and place mixed
numbers on the number
line. SPI 0406.2.7 Convert
improper fractions into
mixed numbers and/or
decimals. Also addresses
GLE 0406.1.7.
2_
3_
3
4
1
2
Improper Fractions
3
_
5
6
11
_
FOOD Nyoko is selling pies
at a bake sale. Each pie has
5 slices. There are 7 slices
left. What fraction of the
pies is left?
Mixed Number
Count the wholes and the
parts.
1
5
Another Way:
1
5
1
5
1
5
1
5
1
5
1
5
_5 + _2 = 1_2
5
5
5
1
5
1
5
1
5
part
2
7
So, 1_, or _, of the pies are left.
5
5
1
5
1
5
_7
5
whole
Improper
Fraction
Count the parts.
1
5
1
5
Decimals and Fractions
6
Mixed Numbers and
Improper Fractions
One Way:
414
23
_
4
2
You can change from a mixed number to an improper
fraction. You can also change from an improper fraction to
a mixed number.
Mixed Number to Improper Fraction
_3 as an improper fraction.
Write 1
W
8
3
3
1_ = 1 + _
Write the mixed number as the sum
8 of a whole and part.
8
3
= _ + _ Write the whole number as a fraction.
8
8
8 + 3
= _
Add.
8
11
= _
8
8
Write
W
11
_
as a mixed number.
8
Divide the numerator by the denominator.
s
The fraction bar stand
11
_
for divided by. So, 8
1 R3
8 11
-8
numerator
−−
3
whole number
means 11 divided by 8.
denominator
3
11
So, _ = 1_ .
8
8
You can show improper fractions and mixed numbers on a
number line.
Use a Number Line
Identify point A as a mixed number and improper fraction.
A
4
5
6
Each interval on the number line is one third. So, point A
1
is 5 _.
3
A
4
5
51
6
3
3
3
3
3
3
1
1
5_ = _ + _ + _ + _ + _ + _
3
3
3
3
3
3
3
3+3+3+3+3+1
16
= __ = _
3
16
1
So, point A is 5 _, or _ .
3
3
3
Lesson 2F Fractions 415
Write a mixed number and an improper fraction for each model
model.
See Example 1
1.
2.
3.
Write each as an improper fraction or a mixed number.
Use models if needed. See Examples 2 and 3
4. 1_
5. 2_
9
6. _
3
4
2
5
13
7. _
4
3
Identify each point as a mixed number and an improper
fraction. See Example 4
B
8.
7
G
9.
8
4
9
10. Andrew’s family ate 1 _ pizzas and
3
8
4
Sheri’s family ate 1_ pizzas. Who ate
16
11.
5
6
E
TALK MATH Explain how to compare
3
17
2 _ and _.
5
5
more pizzas?
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Begins on page EP2.
Write a mixed number and an improper fraction for each model.
See Example 1
12.
13.
14.
Write each as an improper fraction or a mixed number.
Use models if needed. See Examples 2 and 3
15. 1_
16. 2_
17. 6_
7
8
18. 8_
7
19. _
17
20. _
45
21. _
50
22. _
3
4
3
416
7
10
5
Decimals and Fractions
8
5
8
6
Identify each point as a mixed number and an improper fraction.
See Example 4
23.
X
W
1
2
24.
3
M
4
N
5
Travel
A diagram of a horseback riding tour
is shown. There are resting stops along the trail.
25. Joaquin and his family started at the
stables on the left. They are at the
covered bridge. How many miles of
the trail have they traveled?
26. Joaquin reached the end of the trail
in 2 hours and 15 minutes. Write the amount of time
he spent on the trail as an improper fraction.
Use the information to solve the problem.
Flying
Flying
Feat Feat
Remember,
the class is having
a paper airplane
contest.
So far the
distances are:
Ethan—64_1 feet,
Julian—63_1 feet,
Olivia—74_3 feet, and
Teresa—52_1 feet.
My airplane
distance is
72_1 feet.
So, who
won the
contest?
27. Whose paper airplane went the farthest? Explain.
28. OPEN ENDED Name an improper fraction that can be
written as a whole number.
29.
E
WRITE MATH Compare fractions, mixed numbers, and
improper fractions.
Lesson 2F Fractions 417
Test Practice
30. Lionel used 2 _ cups of pancake mix.
2
Which improper fraction shows the
number of cups of pancake mix he
used? (Lesson 2F)
1
2
A. _
4
C. _
3
B. _
2
5
D. _
2
2
2
4
1
F. 3_
H. 3_
G. 3_
I. 3_
2
5
1
5
2
6
33. Caroline has a bag of marbles.
In simplest form, what fraction of
the marbles are blue? (Lesson 2E)
32. The table shows the number of
minutes Grant read each day.
Grant’s Reading Log
Minutes
Monday
30
Wednesday
55
Friday
45
What fraction of an hour did he read
on Friday? (Lesson 2E)
1
3
C. _
A. _
5
4
1
B. _
2
3
2
2
Day
31. What mixed number does point A
represent? (Lesson 2F)
A
3
D. _
3
F. _
H. _
G. _
4
I. _
4
6
10
2
5
5
4
Write each fraction in simplest form. If it is in simplest form,
write simplest form. (Lesson 2E)
4
34. _
8
35. _
6
5
36. _
10
8
Compare. Use >, <, or =. (Lesson 2D)
1
1
37. _ _
8
6
4
1
38. _ _
12
3
5
5
39. _ _
6
40. Zaria counted ten insects on her deck. Half of the insects were ants.
Three were ladybugs, and the rest were crickets. How many insects
were crickets? (Lesson 2C)
418
To assess mastery of SPI 0406.2.2 and partial mastery of SPI 0406.2.7,
see your Tennessee Assessment Book.
10
Mid-Chapter
Check
Write each decimal. (Lesson 1B)
1. three-tenths
3. one out of ten
2. eight-tenths
4. four out of ten
5. MULTIPLE CHOICE There were ten
peaches in a bag. Some were eaten.
The peaches below were not eaten.
What decimal shows the part of the
peaches that was eaten? (Lesson 1B)
A. 0.4
C. 0.9
B. 0.6
D. 0.10
6. Write a decimal for the part of a dollar
shown. (Lesson 1C)
Find an equivalent fraction for each
fraction. (Lesson 2B)
1
14. _
3
15. _
5
16. _
4
17. _
6
9
5
10
18. Stewart has a collection of 12
footballs. Nine footballs have been
signed by professional football players.
The others have not. What part of the
footballs has not been signed? Write
an equivalent fraction. (Lesson 2C)
Write each fraction in simplest form.
(Lesson 2E)
8
_
3
19. _
20.
6
21. _
12
22. _
6
30
10
20
23. Write a mixed number and an
improper fraction for the model.
(Lesson 2F)
7. Mark has six apples. He cut each apple
into 10 slices to share with his family.
There are 13 slices left after his family
eats. Write a decimal that describes
the leftover apples. (Lesson 1D)
Compare. Use >, <, or =. (Lesson 1E)
8. 0.49 0.7
10. 82.6 82.60
9. 1 0.90
24. MULTIPLE CHOICE Which set of
fractions is in order from greatest
to least? (Lesson 2D)
3 6 2
F. _, _, _
11. 1.06 1.07
5 15 10
2 3 6
G. _, _, _
10 5 15
Order from least to greatest. (Lesson 1E)
12. 0.98, 0.79, 0.89, 1.87
13. 5.3, 5.23, 4.62, 5.20
25.
E
2 6 3
H. _, _, _
I.
10 15 5
6 _
3 2
_
, ,_
15 5 10
3
WRITE MATH Explain why ___
10
30
and ____
are equivalent. (Lesson 2B)
100
Mid-Chapter Check 419
Multi-Part
Lesson
3
MP-Title
-TitlFractions
e
Relate
and Decimals
A
PART
B
C
D
E
Fractions and Decimals
Main Idea
I will explore fractions
and decimals.
A fraction shows part of a whole. A decimal also shows part
of a whole. A decimal is a number that uses place value,
numbers, and a decimal point to show part of a whole.
Get ConnectED
GLE 0406.2.4
Understand and use the
connections between
fractions and decimals.
W
Write
a fraction and a decimal
ffor the shaded part.
Step 1
Find the shaded parts and
the total number of parts.
There are 34 shaded parts out
of a total of 100 parts.
Step 2
Write a fraction.
34
_
100
Step 3
Write a decimal.
Write thirty-four hundredths as a decimal.
0.34
hundredths place
34
So, the shaded part of the model is _, or 0.34.
100
and Apply It
Write a fraction and a decimal for each model.
1.
4.
420
2.
E
3.
WRITE MATH Explain how to find the decimal equivalent of
Decimals and Fractions
2
_
.
100
Multi-Part
Lesson
3
Relate Fractions and Decimals
PART
A
Main Idea
I will identify, read,
and write tenths,
hundredths, and
thousandths as
decimals and as
fractions.
B
C
D
E
Fractions and Decimals
A decimal is a number that uses place value and a
decimal point to show part of a whole.
Get ConnectED
MONEY It costs 85 cents
for a child to ride a
light rail system. Write
85 cents as a fraction
and as a decimal.
The amount 85 cents
means 85 pennies out
of 1 dollar.
Read
Write
eighty-five
hundredths
85
_
or 0.85
100
Read
Write
Hundredths
Place Value
Tenths
Draw a hundreds model.
Shade 85 out of 100 parts
to show 85 cents.
Another Way:
Ones
Use a Model
Tens
One Way:
Hundreds
GLE 0406.2.4
Understand and use the
connections between
fractions and decimals.
Also addresses GLE 0406.1.6.
0
8
5
eighty-five
hundredths
85
_
or 0.85
100
85
So, 85 cents is _ as a fraction and 0.85 as a decimal.
100
Lesson 3B Relate Fractions and Decimals
421
Some fractions can be written as tenths, hundredths, or
thousandths.
Write Fractions as Decimals
Write
W
_5 as two different decimals.
10
One Way:
Another Way:
Write Tenths
Write Hundredths
0.5
0.50
The decimals 0.5 and 0.50 are equivalent decimals.
Write
W
Read
Write
23
_
as a decimal.
1,000
twenty-three thousandths
0.023
23
So, _ can be written as 0.023.
1,000
Write a fraction and a decimal for each shaded part
part. SSee Example
E
l 1
1.
2.
3.
Write as a fraction and as a decimal. See Examples 1–3
4. one-tenth
7.
E
5. twenty-five hundredths
6. seven-thousandths
TALK MATH Shade all of the boxes along the outer edge of a
hundredths grid. Write a fraction and decimal for the shaded part.
Why is it not 0.40?
422
Decimals and Fractions
EXTRA
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Begins on page EP2.
Write a fraction and a decimal for each shaded part.
See Example 1
8.
9.
10.
11.
12.
13.
Write as a fraction and as a decimal. See Examples 1–3
14. sixty-two hundredths
15. two-tenths
16. thirty-five hundredths
17. eight-tenths
18. fourteen-thousandths
19. six-thousandths
Write each fraction as a decimal.
22
20. _
2
21. _
100
50
23. _
100
100
375
24. _
1,000
26. On Monday, it snowed _ of
10
an inch.
6
28. Each state has a representation
2
of _ in the United States Senate.
100
2
22. _
10
80
25. _
1,000
3
27. A car traveled _ of a mile in 18
10
seconds.
29. Wesley learned that _ of the
10
students in his class are left
handed.
2
2
30. OPEN ENDED Write a fraction whose decimal value is between _
10
25
and _. Write the fraction and its decimal equivalent.
100
31. CHALLENGE Decide whether the following sentence is true or
6
false. Explain. The fraction _ equals 0.0006.
10,000
32.
E
WRITE MATH Write a summary statement about decimals
equivalent to fractions that have denominators of 10 and 100.
Lesson 3B Relate Fractions and Decimals
423
in Music
Decimal
Musical notes are a universal
language. Musical notes are based
on fractions. The most common
musical notes include whole, half,
quarter, eighth, and sixteenth
notes. These values represent the
duration of the notes. The
durations of the notes are not
o
o
N te-ati n
specific; they are relative to the
other notes. For example, a
one-eighth note is twice as
long as a one-sixteenth note, a
one-fourth note is twice as long
as a one-eighth note, and so on.
Beethoven was
the first musician
to use the
one hundred
twenty-eighth note.
424 Decimals and Fractions
Note
Notation
Fractional
Equivalent
Whole
1
_
Half
1
_
Quarter
1
_
1
2
4
Use the table above to solve each problem.
m.
Write the value of each musical note as a decimal.
Refer to Exercises 1–3. Draw a number line that shows these values.
Write the value of each group of musical notes as a mixed number.
Draw three musical notes that
represent a value of 2.5.
Draw four musical notes that
1
represent a value of 2 _ .
4
Problem Solving in Music 425
Multi-Part
Lesson
3
Relate Fractions and Decimals
PART
A
B
C
D
E
Problem-Solving Investigation
Main Idea I will choose the best strategy to solve a problem.
Clarissa visited a fair on Saturday and
played 0.20 of the games. She visited
the fair again Sunday. By the end of
the day she still had not played 0.62
of the games. What part of the 100
fair games did she play on Sunday?
Understand
What facts do you know?
• There are 100 games at the fair.
• She played 0.20 of the games on Saturday.
• She did not play 0.62 of the games.
What do you need to find?
• The part of the 100 fair games she played Sunday.
Plan
Work backward to find the part of the games she played Sunday.
Solve
0.62 = 62 games out of 100 not played
0910
100
- 62
−−−−
38
total games
games not played
38 out of 100,
total games played
or 0.38
38
- 20
−−−−
18
total games played
games played Saturday
18 out of 100,
games played Sunday
or 0.18
So, she played 18 out of 100, or 0.18,
of the games Sunday.
Check
Add the total number of games she played to the number
of games she did not play.
62 + 38 = 100
So, the answer is correct. GLE 0406.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and
reasonableness of the solution. GLE 0406.2.6 Solve problems involving whole numbers, fractions, and/or decimals using all
four arithmetic operations.
426
Decimals and Fractions
EXTRA
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Begins on page EP2.
• Make a model.
• Make a table.
• Choose an operation.
6. Algebra The table shows the number
of shapes after a pattern repeats itself
five times. How many of each shape
are in the original pattern?
Use any strategy to solve each problem.
Pattern
1. There are 18 students in Mr. Case’s
class. Twelve of the students play an
instrument. What fraction of the
students play an instrument?
Shape
Number
Circles
Squares
Hearts
15
5
10
7. Mr. Lawrence used 100 tiles when
he put a new floor in the kitchen.
Thirty-four tiles are blue and 16 tiles
are gray. Write the total amount of
blue and gray tiles as a decimal and
a fraction.
2. Mrs. Keys has 36 pens. Yesterday she
had half that amount plus 2. How
many pens did she have?
3. There are three lines. The first line is
3 times as long as the second. The
second line is 4 inches longer than the
third. The third line is 2 inches long.
How long is the first line?
4. The table below shows Adelais’
schedule. She finished at 5:00 P.M.
What time did she start her lunch?
8. Cathy baked 36 muffins. She gave half
the muffins to her neighbor and ate
the muffins shown below. She brought
the rest to school. What fraction of the
muffins did Cathy bring to school?
Write in simplest form.
Adelais’ Schedule
Activity
Time
Lunch
1 hour
Scrapbooking
3 hours
9. Giselle played with some friends on
Monday. She played with 2 times as
many friends on Wednesday. This was
4 more than on Friday. On Friday there
were 4. How many did she play with
on Monday?
5. Dirk ran eight-tenths of a mile at track
80
practice. Leslie ran _ of a mile. Who
100
ran farther? Write each distance as a
decimal.
10.
E
WRITE MATH Look back at
Exercise 9. Explain which strategy
you used to solve the problem.
Lesson 3C Relate Fractions and Decimals
427
Multi-Part
Lesson
3
Relate Fractions and Decimals
PART
A
Main Idea
I will explore fraction
and decimal
equivalents.
B
C
D
E
Model Fraction and
Decimal Equivalents
You can use a number line to find fraction and decimal
equivalents.
Get ConnectED
GLE 0406.2.4
Understand and use the
connections between
fractions and decimals.
SPI 0406.2.5 Generate
equivalent forms of
common fractions and
decimals and use them
to compare size.
_1 are equivalent.
Determine whether 0.2 and
D
Step 1
5
Model 0.2.
Use a number line to model 0.2.
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Step 2
Model
_1 .
5
1
Use a number line to model _ .
5
Step 3
0
5
1
5
2
5
3
5
4
5
5
5
0
0.2
0.4
0.6
0.8
1
Compare.
1
Compare the locations of 0.2 and _ on the
5
number lines.
0
10
1
0.2 and 5 are located in
the same position on the
number lines.
1
10
2
10
3
10
4
10
7
10
8
10
9
10
10
10
0
5
1
5
2
5
3
5
4
5
5
5
0
0.2
0.4
0.6
0.8
1
5
Decimals and Fractions
6
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
So, 0.2 and _ are equivalent.
428
4
5
10
Find fraction and decimal equivalents.
F
Write a fraction and a decimal to describe
the shaded part of the model.
Step 1
Identify the fraction.
Determine which fraction the
model represents.
One out of four equal parts of
the model is shaded.
1
So, the model represents _.
4
Step 2
Find a decimal equivalent.
Divide the model into 100
equal parts. Then count the
shaded squares.
25
25 out of 100, or _, or 0.25
100
of the squares are shaded.
1
So, 0.25 is a decimal equivalent to _ .
4
About It
3
1. Do 0.6 and _ represent equivalent amounts? Explain.
5
3
2. Is the fraction _ equivalent to 0.7? Explain.
4
and Apply It
Write a decimal equivalent for each fraction.
1
3. _
2
6.
E
4
4. _
5
3
5. _
4
WRITE MATH Explain how to find an equivalent decimal for
_2 .
4
Lesson 3D Relate Fractions and Decimals
429
Multi-Part
Lesson
3
PART
Relate Fractions and Decimals
A
Main Idea
I will find fraction and
decimal equivalents.
Get ConnectED
B
C
E
D
Fraction and Decimal
Equivalents
When a fraction and a decimal name the same amount,
they are fraction and decimal equivalents.
GLE 0406.2.4
Understand and use the
connections between
fractions and decimals.
SPI 0406.2.5 Generate
equivalent forms of
common fractions and
decimals and use them
to compare size.
TRAVEL Austin’s family is
taking a trip. The odometer
(mileage tracker) shows
they have driven 0.5 mile.
Austin wants to find this
in a fraction. Use a model
to find an equivalent
fraction.
Use tenths and hundredths
grids to model that 0.5 and
0.5 =
_5 = _1
2
0.50 =
2
10
_1 name the same amount.
50
1
_
=_
2
100
The number lines also show that they name the
same amount.
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
2
1
2
2
2
0
0.5
1.0
So, 0.5 and
430
Decimals and Fractions
_1 are equivalent.
2
To find a decimal that is equivalent to a fraction, it helps to write
the fraction with a denominator of 10 or 100.
Find Fraction and Decimal Equivalents
W
Write
a fraction and a decimal
to describe the shaded part
of the model.
3×2
25
75
_
=_
4 × 25
100
75
_
= 0.75
100
THINK What number can you multiply
the denominator by to get 100?
75
Write _ as a decimal.
100
3
So, _ and 0.75 describe the shaded part of the model.
4
Here are some common fraction and decimal equivalents.
Fraction-Decimal Equivalents
1
_
= 0.5
1
_
= 0.25
2
_
= 0.5
3
_
= 0.75
1
_
= 0.2
2
_
= 0.4
3
_
= 0.6
4
_
= 0.8
2
5
4
5
4
4
5
5
decimal. SSee EExample
Use a model to find an equivalent fraction or decimal
l 1
1. 0.6
2. 0.06
2
3. _
4
4
4. _
5
Write a fraction and a decimal to describe the shaded part of
each model. See Example 2
5.
6.
9. Lupe got 20 out of 25 questions
correct on a quiz. Write her score as
a decimal and a fraction.
7.
10.
8.
E
TALK MATH What do you notice
3 6
12
about _ , _ , and _ ?
4 8
16
Lesson 3E Relate Fractions and Decimals
431
EXTRA
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Begins on page EP2.
Use a model to find an equivalent fraction or decimal. See Example 1
11. 0.78
12. 0.4
13. 0.6
14. 0.35
1
15. _
4
16. _
7
17. _
1
18. _
4
5
25
10
Write a fraction and a decimal to describe the shaded part of
each model. See Example 2
19.
20.
21.
22.
23.
24.
25.
26.
Write each amount as a fraction and a decimal.
27. Measurement Adelina swam 3 out
of 4 laps.
28. Vince read 6 out of 10 pages in a
chapter of a book.
29. Jun ate 8 out of her 10 orange slices.
30. Cathy has completed 18 out of 20
math problems.
31. Maya has painted 1 out of 4 walls of
her bedroom.
32. Jack has opened 3 out of 5 of his
birthday presents.
33. OPEN ENDED Create a model and shade in a fraction of it.
Write two fractions and a decimal to describe the shaded
area of the model.
34. CHALLENGE Carra collects stuffed frogs. She has 25 frogs, and
2
_
of them are multicolored. The rest are green. How many green
25
frogs are in her collection? Explain how you found your answer.
35.
E
WRITE MATH Demetri is completing 0. =
Explain how he can find the correct answer.
432
5
_
.
50
To assess mastery of SPI 0406.2.5, see your Tennessee Assessment Book.
Fraction and Decimal Game
You will need: 10 index cards
Compare Decimals to Fractions
Get Ready!
Players: 2
Get Set!
On each index card, write a
statement using >, <, or =. Write
5 true statements and 5 false
statements. A few examples are
shown at the right.
0.25 < _31
10
0.5 > _
20
0.75 = _4
75
0.8 < _
3
100
GO!
Shuffle the cards.
Spread out the cards
facedown on a desk.
Player 1 turns over an index
card and must say whether
the statement is true or false.
Player 1 keeps the card if
the answer is correct and
draws again. If Player 1 is
wrong, the index card is put
back. Player 2 takes a turn.
The player who collects
more cards wins.
Game Time Fraction and Decimal Game
433
Chapter Study
Guide and Review
Be sure the following Key
Concepts are noted in your
Foldable.
Key Vocabulary
decimal
equivalent fractions
greatest common factor
Decim
als
Frac
t
ions
mixed number
Equ
F ival
and ractio ent
Dec ns
ima
ls
simplest form
Vocabulary Check
Complete each sentence with the
correct vocabulary word.
Key Concepts
Decimals (Lesson 1)
1. The
• Tenths , hundredths , and thousandths
are used to describe decimals as part of
the base-ten system.
Fractions (Lesson 2)
• Equivalent fractions represent the same
amount.
1
_
2
_
4
_
4
2
2
2 14
2.5
2 12
2.75
3
2 34
3
• When a fraction and decimal name the
same amount, they are equivalents.
434
Decimals and Fractions
2
2. The fraction _ is written in
3
?
.
?
has a whole
3. A(n)
number part and a fraction part.
?
is
10
• Mixed numbers can be written as
decimals.
2.25
of 8 and 12 is 4.
4. An example of a(n)
2
5_.
8
Fractions and Decimals (Lesson 3)
2
?
5. Fractions that represent the
?
same amount are
.
?
is a number that
6. A
uses place value, a number, and
a decimal point to show part of a
whole.
Multi-Part Lesson Review
Lesson 1
Tenths
Decimals
(Lessons 1A and 1B)
EXAMPLE 1
Model and write each decimal.
Sandra knocked down seven out of ten
pins with her first bowling ball. What
part of the set of pins is still standing?
7. one out of ten
8. five out of ten
9. nine out of ten
10. Caleb ate six out of ten pretzels.
Write a decimal to show what part
of the pretzels Caleb did not eat.
Hundredths
If seven out of ten pins were knocked
down, then three pins are still standing.
The amount of pins still standing is three
out of ten, three-tenths, or 0.3.
(Lesson 1C)
Write a decimal for each part of a
dollar shown.
11.
EXAMPLE 2
Write a decimal for the part of a dollar
shown.
12.
Each cent is one hundredth, or 0.01, of a
dollar.
So, thirty-seven cents is 0.37 of a dollar.
(Lesson 1D)
Hundredths
15. On average, a United States citizen
eats eleven and eighty-eight
hundredths pounds of breakfast
cereal each year. Write this amount
in standard and expanded form.
Write fourteen and thirty-seven
hundredths in standard
and expanded form.
Write each digit
in the decimal in a
place-value chart.
Tenths
14. eight and ninety-four hundredths
EXAMPLE 3
Ones
Write each decimal in standard and
expanded form.
13. two and eighteen hundredths
Tens
Decimals Greater Than One
1
4
3
7
Standard form:
14.37
Expanded form:
10 + 4 + 0.3 + 0.07
Chapter Study Guide and Review 435
Chapter Study Guide and Review
Compare and Order Decimals
(Lesson 1E)
1
sandwich and said, “Here is your _
2
of the sandwich.” Her brother said,
2
“Actually, you ate _ of it.” Who is
4
correct? Explain.
Lesson 2
Compare 5.34 and 5.32.
4>2
5.34 > 5.32
5
5
3
3
4
2
So, Stephanie has been jogging longer.
Fractions
Equivalent Fractions
(Lessons 2A and 2B)
Find an equivalent fraction for each
fraction.
EXAMPLE 5
1
20. _
1
21. _
1
22. _
4×2
8
_
=_
6×2
12
6
23. _
7
24. _
9
25. _
5
8
3
14
4
12
Find a fraction equivalent to
Solve. Use the make a model strategy.
1
26. Jeff had 28 grapes. He ate _ of them
2
for lunch. Then he ate 10 more as a
snack. How many are left?
27. Marisa has a marble collection.
One-fourth of her 16 marbles is blue.
She has 5 red marbles. The rest of
the marbles are green. How many
are green?
Decimals and Fractions
_4 .
6
Multiply the numerator and
the denominator by the same
number, 2.
8
4
_
So,
is equivalent to _.
6
12
Problem-Solving Strategy: Make a Model
436
Hundredths
19. Della gave her brother part of a
Stephanie has been
jogging for 5.34 minutes.
Todd has been jogging
for 5.32 minutes. Who
has been jogging longer?
Tenths
16. 4.35 3.45
17. 6.40 6.40
18. 2.27 2.72
Ones
EXAMPLE 4
Compare. Use >, <, or =.
(Lesson 2C)
EXAMPLE 6
_1
Marcela has 24 crayons. Of them, is
3
blue. Four are yellow, and the rest are
green. How many crayons are green?
Make a model.
There are 12
parts left.
So, 12 of the
crayons are green.
Compare and Order Fractions
(Lesson 2D)
EXAMPLE 7
Compare. Use >, <, or =.
4
8
28. _ _
5
6
5
29. _ _
6
10
Dakota has a red and a blue pencil. The
1
red pencil is of a foot long. The blue
3 2
pencil is of a foot long. Which pencil
8
is longer?
Find an equivalent fraction.
6
2 3 4
30. _, _, _
1 3 7
31. _, _, _
3 7 35
_
_
Order from least to greatest.
4 16 8
1
4
4
_
×_=_
3
32. Measurement Patrick took _ of
4
3
an hour to finish a test. Jose took _ of
8
an hour to finish. Who took more time
to finish the test?
Simplest Form
4
2
3
4
_
> _
8
8
8
So, the red pencil is longer than the blue
pencil.
(Lesson 2E)
Write each fraction in simplest form. If
it is in simplest form, write simplest
form.
4
33. _
2
34. _
6
35. _
3
36. _
10
13
3
18
EXAMPLE 8
Write
20
Factors of 15: 1, 3, 5, 15
Factors of 20: 1, 2, 4, 5, 10, 20
The greatest common factor is 5.
15
5
3
_
÷_
=_
20
Mixed Numbers
1
15
_
in simplest form.
5
4
(Lesson 2F)
Write a mixed number and an improper
fraction for each model.
EXAMPLE 9
37.
A fourth grade class had a pizza party.
The amount of pizza eaten can be
1
represented as 3_ . Write the amount
5
of pizza eaten as an improper fraction.
38.
5+5+5+1
5
5
5
1
1
3_ = _ + _ + _ + _ = __
5
5
16
=_
5
5
5
5
5
16
1
So, 3 _ = _.
5
5
Chapter Study Guide and Review 437
Chapter Study Guide and Review
LLesson
esson 3
Relate Fractions and Decimals
Fractions and Decimals
(Lessons 3A and 3B)
39. Write a fraction and a decimal for the
shaded part.
EXAMPLE 10
Write eight tenths as two different
decimals.
Write tenths. Write hundredths.
Write as a fraction and as a decimal.
40. three-tenths
41. twenty-two hundredths
0.8
0.80
Eight tenths can be written as 0.8.
Problem-Solving Investigation: Choose a Strategy
(Lesson 3C)
Use any strategy to solve each problem.
EXAMPLE 11
42. Jonas earned $5.25. Ian earned
$5.50. Who earned more money?
A one-year-old puppy has three new
toys. A day later, the puppy has
one-third of the toys. How many toys
does the puppy have left?
43. Zach has $30. Tickets to the school
baseball game cost $7. How much
money will Zach have left if he buys
3 tickets?
Fraction and Decimal Equivalents
Write a fraction and a decimal to
describe the shaded part of each model.
44.
45.
Find _ of 3. You do not need to know
3
the age of the puppy.
So, the puppy has one toy left.
1
(Lessons 3D and 3E)
EXAMPLE 12
Write a fraction and a decimal to
describe the shaded part of each model.
Thirty-two squares
are shaded. So,
32
that is _, or 0.32.
100
438
Decimals and Fractions
Practice
Chapter Test
Tell whether each statement is true or
false.
1
1. _ and 0.5 are equivalent.
5
2. To find an equivalent fraction,
multiply or divide the numerator
and denominator by the same
number.
Write as a fraction and as a decimal.
3. one-tenth
4. nine-tenths
Compare. Use > , < , or = .
9. 4.75 4.77
10. 3.21 3.20
Write each as a mixed number or an
improper fraction.
11. 2_
27
12. _
3
4
5
13. There are 12 fish in an aquarium.
One-half of the fish is goldfish. Four of
the fish are tetras. The rest of the fish
are rainbow fish. How many of
the fish are rainbow fish?
Identify each point as a mixed number.
5. eighteenhundredths
6. threehundredths
A
14.
5
7. Madison emptied her pockets. Her
coins are shown. What decimal
represents the part of a dollar
shown?
6
15.
B
2
3
16. Abby read _ of a book on Saturday.
10
Then she read 0.4 of the book on
Sunday. What fraction of the book
does Abby still have to read?
3
8. MULTIPLE CHOICE Which number is
NOT equivalent to the shaded area of
the circle?
1
A. _
2
2
B. _
4
C. 0.5
D. 0.4
17. MULTIPLE CHOICE Which set of
numbers is in order from least to
greatest?
18.
F. 0.6, 0.76, 0.67
H. 0.6, 0.67, 0.76
G. 0.76, 0.67, 0.6
I.
E
WRITE MATH
0.67, 0.6, 0.76
2
Explain how _
6
and _ are equivalent fractions.
7
21
Practice Chapter Test
439
Test Practice
On the number line below, what mixed number
does point K represent?
K
7
8
A. 7_
Count the intervals between
whole numbers on a
number line to find how the
number line is divided.
4
C. 7_
3
10
4
B. 7_
6
10
4
D. 7_
5
Read the Question
You need to find which number point K represents.
Solve the Question
Look at the number line. The 9 marks between the
1
7 and 8 let you know that each line is _. Fill in
10
the numbers up to point K.
K
7 7 1 7 2 7 3 7 4 7.5
8
10 10 10 10
4
So, point K represents 7_ on the number line.
10
The answer is C.
Read each question. Then fill in the correct answer on the answer
sheet provided by your teacher or on a separate sheet of paper.
1. On the number line below, what
number does point A represent in
decimal form?
A
2
440
1
22
3
A. 2.3
C. 2.75
B. 2.5
D. 2.8
Decimals and Fractions
2. Which symbol makes the following
sentence true?
13.08
1.308
F. >
H. =
G. <
I. -
4
3. In the rectangle below, _ of the
8
sections are shaded.
6.
GRIDDED RESPONSE The
function table shows some input and
output values.
Input (x)
4
Which of the following is equal to _ ?
A. _
B. _
1
4
4.
1
2
C. _
D. _
1
8
8
4
4
GRIDDED RESPONSE The
elephants below each weigh
16,500 pounds. How many pounds
do they weigh altogether?
1
1
2
4
3
7
4
10
5
What is the missing value?
7. The table shows how long Tara read
each night during a week. What
fraction of an hour did she read on
Wednesday?
Day
5. Which order shows the distances run
from greatest to least?
Track Practice
Runner
Miles
Darin
1.24
Kirk
1.5
Damon
1.31
Output (y)
Minutes Read
Monday
17
Tuesday
24
Wednesday
36
Thursday
30
Friday
32
A. _
3
C. _
1
2
2
B. _
3
5
36
D. _
100
4
8. Trina grew _ of an inch in a year.
F. Kirk, Damon, Darin
5
G. Kirk, Darin, Damon
4
Which decimal is equivalent to _ ?
H. Damon, Darin, Kirk
F. 0.7
H. 0.8
I. Damon, Kirk, Darin
G. 0.75
I. 0.85
5
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Test Practice 441