UNIT 2
Number
Operations
MODULE
MODULE
4
Operations with
Fractions
COMMON
CORE
6.NS.1, 6.NS.4
5
Operations with
MODULE
MODULE
Decimals
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COMMON
CORE
6.NS.2, 6.NS.3
CAREERS IN MATH
Chef The role of a chef is diverse and can
include planning menus, overseeing food
preparation, training staff, and ordering and
purchasing food items. Chefs use mathematics
when scaling recipes as well as in budgeting
and financial planning.
If you are interested in a career as a chef, you
should study these mathematical subjects:
• Basic Math
• Business Math
Research other careers that require the use of
scaling quantities, and financial planning.
Unit 2 Performance Task
At the end of the unit, check
out how chefs use math.
Unit 2
73
UNIT 2
Vocabulary
Preview
Use the puzzle to preview key vocabulary from this unit. Unscramble
the circled letters to answer the riddle at the bottom of the page.
1.
ISROPCALCER
2.
INERADRME
3.
RENNOOAIDTM
4.
DXMIE RUBMEN
5.
NEIDIDDV
6.
DRVIOSI
1. Two numbers whose product is one. (Lesson 4.2)
2. The whole number left over when you divide and the divisor doesn’t divide the dividend
evenly. (Lesson 5.1)
3. The part of a fraction that represents how many parts the whole is divided into.
(Lesson 4.1)
4. A number that is a combination of a whole number and a fraction. (Lesson 4.3)
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5. The amount that you want to divide in a division problem. (Lesson 5.4)
6. The number you divide by in a division problem. (Lesson 5.4)
Q:
Decimals always win in arguments with fractions.
What do decimals have that fractions don’t?
A:
74
Vocabulary Preview
!
Operations with
Fractions
?
4
MODULE
LESSON 4.1
ESSENTIAL QUESTION
Applying GCF and
LCM to Fraction
Operations
How can you use operations
with fractions to solve
real-world problems?
COMMON
CORE
6.NS.4
LESSON 4.2
Dividing Fractions
COMMON
CORE
6.NS.1
LESSON 4.3
Dividing Mixed
Numbers
COMMON
CORE
6.NS.1
LESSON 4.4
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Solving Multistep
Problems with
Fractions and Mixed
Numbers
COMMON
CORE
6.NS.1
Real-World Video
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To find your average rate of speed, divide the distance
you traveled by the time you traveled. If you ride in a
taxi and drive _12 mile in _14 hour, your rate was 2 mi/h
which may mean you were in heavy traffic.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
75
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this module.
Write an Improper Fraction
as a Mixed Number
EXAMPLE
13 _
__
= 55 + _55 + _35
5
= 1 + 1 + _35
= 2 + _35
= 2_35
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Online
Assessment and
Intervention
Write as a sum using names for one plus a
proper fraction.
Write each name for one as one.
Add the ones.
Write the mixed number.
Write each improper fraction as a mixed number.
1. _94
2. _83
23
3. __
6
11
4. __
2
17
5. __
5
15
6. __
8
33
7. __
10
29
8. __
12
Multiplication Facts
EXAMPLE
7×6=
7 × 6 = 42
Use a related fact you know.
6 × 6 = 36
Think: 7 × 6 = (6 × 6) + 6
= 36 + 6
= 42
9. 6 × 5
10. 8 × 9
11. 10 × 11
12. 7 × 8
13. 9 × 7
14. 8 × 6
15. 9 × 11
16. 11 × 12
Division Facts
EXAMPLE
63 ÷ 7 =
Think:
63 ÷ 7 = 9
So, 63 ÷ 7 = 9.
7 times what number equals 63?
7 × 9 = 63
Divide.
76
Unit 2
17. 35 ÷ 7
18. 56 ÷ 8
19. 28 ÷ 7
20. 48 ÷ 8
21. 36 ÷ 4
22. 45 ÷ 9
23. 72 ÷ 8
24. 40 ÷ 5
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Multiply.
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the triangle. Write the review word
that fits the description in each section of the triangle.
part
of a whole
top number
of a fraction
Vocabulary
Review Words
area (área)
✔ denominator
(denominador)
✔ fraction (fracción)
greatest common factor
(GCF) (máximo común
divisor (MCD))
least common multiple
(LCM) (mínimo común
múltiplo (m.c.m.))
length (longitud)
✔ numerator (numerador)
product (producto)
width (ancho)
Preview Words
bottom number of a fraction
mixed number (número
mixto)
order of operations (orden
de las operaciones)
reciprocals (recíprocos)
Understand Vocabulary
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In each grouping, select the choice that is described by the given
vocabulary word.
1. reciprocals
A 1:15
3 _
1
_
B 4÷6
C
3
_
and _53
5
2. mixed number
1 1
A _-_
3 5
1
B 3_
2
C -5
3. order of operations
A 5-3+2=0
B 5-3+2=4
C 5-3+2=6
Active Reading
Layered Book Before beginning the module,
create a layered book to help you learn the
concepts in this module. Label each flap with
lesson titles. As you study each lesson, write
important ideas, such as vocabulary and
processes, under the appropriate flap. Refer
to your finished layered book as you work on
exercises from this module.
Module 4
77
MODULE 4
Unpacking the Standards
Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to
learn in this module.
COMMON
CORE
6.NS.1
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions by
fractions, e.g., by using visual
fraction models and equations
to represent the problem.
Key Vocabulary
What It Means to You
You will learn how to divide two fractions. You will also understand
the relationship between multiplication and division.
UNPACKING EXAMPLE 6.NS.1
Zachary is making vegetable soup. The recipe makes 6_34 cups of
soup. How many 1_12 -cup servings will the recipe make?
6_34 ÷ 1_12
quotient (cociente)
The result when one number is
divided by another.
27 _
= __
÷ 32
4
fraction (fracción)
A number in the form _ba , where
b ≠ 0.
= _92
27 _
= __
× 23
4
= 4_12
The recipe will make 4_12 servings.
6.NS.4
Find the greatest common
factor of two whole numbers
less than or equal to 100 and
the least common multiple of
two whole numbers less than or
equal to 12. Use the distributive
property to express a sum of
two whole numbers 1–100 with
a common factor as a multiple
of a sum of two whole numbers
with no common factor.
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to see all the
Common Core
Standards
unpacked.
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78
Unit 2
What It Means to You
You can use greatest common factors and least common multiples
to simplify answers when you calculate with fractions.
UNPACKING EXAMPLE 6.NS.4
Add. Write the answer in simplest form.
1 _
Use the LCM of 3 and 6 as
_
+ 16 = _26 + _16
3
a common denominator.
2+1
= _____
6
Add the numerators.
= _36
3÷3
= ____
6÷3
Simplify by dividing by the GCF.
The GCF of 3 and 6 is 3.
= _12
Write the answer in simplest form.
© Houghton Mifflin Harcourt Publishing Company
COMMON
CORE
LESSON
4.1
?
Applying GCF and
LCM to Fraction
Operations
ESSENTIAL QUESTION
COMMON
CORE
6.NS.4
Find the greatest common
factor…and the least
common multiple of two
whole numbers…
How do you use the GCF and LCM when adding, subtracting,
and multiplying fractions?
Multiplying Fractions
To multiply two fractions you first multiply the numerators and then multiply
the denominators.
numerator × numerator
numerator
_____________________
= __________
denominator × denominator
denominator
Math On the Spot
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The resulting product may need to be written in simplest form. To write
a fraction in simplest form, you can divide both the numerator and the
denominator by their greatest common factor.
Example 1 shows two methods for making sure that the product of two
fractions is in simplest form.
EXAMPL 1
EXAMPLE
COMMON
CORE
6.NS.4
Multiply. Write the product in simplest form.
A _13 × _35
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1× 3
1 _
_
× 35 = _____
3
3× 5
Write the problem as a single fraction.
3
= __
15
Multiply numerators. Multiply denominators.
3÷ 3
= ______
15 ÷ 3
Simplify by dividing by the GCF.
The GCF of 3 and 15 is 3.
= _15
Write the answer in simplest form.
6 in the numerator and 3 in the
denominator have a common factor
other than one. Divide by the GCF 3.
B _67 × _23
6 _
6× 2
_
× 23 = _____
7
7× 3
Write the problem as a single fraction.
2
6× 2
= ______
7× 3
Simplify before multiplying using the GCF.
2× 2
= _____
7× 1
Multiply numerators. Multiply denominators.
1
= _47
Math Talk
Mathematical Practices
Compare the methods in
the Example. How do you
know if you can use the
method in B ?
Lesson 4.1
79
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
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Multiply. Write each product in simplest form.
1. _16 × _35
2. _34 × _79
3. _37 × _23
4. _45 × _27
8
7
× __
5. __
10
21
6. _67 × _16
Multiplying Fractions and
Whole Numbers
Math On the Spot
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To multiply a fraction by a whole number, you rewrite the whole number as a
fraction and multiply the two fractions. Remember to use the GCF to write the
product in simplest form.
EXAMPLE 2
COMMON
CORE
6.NS.4
A class has 18 students. The teacher asks how many students in the class have
pets and finds _59 of the students have pets. How many students have pets?
Estimate the product. Multiply the whole number by the nearest
benchmark fraction.
5
_
is close to _12 , so multiply _12 times 18.
9
1
_
× 18 = 9
2
STEP 2
Multiply. Write the product
in simplest form.
5
_
× 18
9
Math Talk
Mathematical Practices
How can you check
to see if the answer is
correct?
5
18
_
× 18 = _59 × __
1
9
2
Unit 2
5
__
×18
9
5
__
· 18
9
5
__
(18)
9
Rewrite 18 as a fraction.
× 18
= 5______
9×1
Simplify before multiplying using the GCF.
5×2
= ____
1×1
Multiply numerators. Multiply denominators.
10
= 10
= __
1
Simplify by writing as a whole number.
1
10 students have pets.
80
5
times 18
You can write __
9
three ways.
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white/Alamy
STEP 1
Reflect
7.
Analyze Relationships Is the product of a fraction less than 1 and a
whole number greater than or less than the whole number? Explain.
YOUR TURN
Multiply. Write each product in simplest form.
8.
5
_
× 24
8
9.
3
_
× 20
5
10.
1
_
× 8
3
11.
1
_
× 14
4
12.
7
× 7
3 __
10
13.
3
× 10
2 __
10
Personal
Math Trainer
Online Assessment
and Intervention
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Adding and Subtracting Fractions
You have learned that to add or subtract two fractions, you can rewrite the
fractions so they have the same denominator. You can use the least common
multiple of the denominators of the fractions to rewrite the fractions.
EXAMPL 3
EXAMPLE
COMMON
CORE
6.NS.4
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8
1
_
Add __
15 + 6 . Write the sum in simplest form.
STEP 1
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My Notes
Rewrite the fractions as equivalent fractions. Use the
LCM of the denominators as the new denominator.
8
8×2
16
__
→ _____
→ __
15
15 × 2
30
1
×
5
5
1
_ → ____ → __
6
6×5
30
STEP 2
Math On the Spot
The LCM of 15 and 6 is 30.
Add the numerators of the equivalent fractions. Then simplify.
16 __
5
21
__
+ 30
= __
30
30
÷3
_____
= 21
30 ÷ 3
7
= __
10
Simplify by dividing by the GCF.
The GCF of 21 and 30 is 3.
Reflect
14.
Can you also use the LCM of the denominators of the fractions to
8
_1
rewrite the difference __
15 - 6 ? What is the difference?
Lesson 4.1
81
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
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Add or subtract. Write each sum or difference in simplest form.
5
15. __
+ _16
14
5
3
16. __
- __
12 20
5
17. __
- _38
12
3
18. 1 __
+ _14
10
19. _23 + 6 _15
20. 3 _16 - _17
Guided Practice
Multiply. Write each product in simplest form. (Example 1)
1. _12 × _58
2. _35 × _59
3. _38 × _25
4. 2 _38 × 16
5
5. 1 _45 × __
12
2
× 5
6. 1 __
10
Find each amount. (Example 2)
7. _14 of 12 bottles of water =
bottles
9. _35 of $40 restaurant bill = $
8. _23 of 24 bananas =
10. _56 of 18 pencils =
bananas
pencils
5
11. _38 + __
24
5
1
+ __
12. __
20
12
9
- _1
13. __
20 4
9
3
- __
14. __
10 14
5
15. 3 _38 + __
12
7
5
- __
16. 5 __
10 18
?
?
ESSENTIAL QUESTION CHECK-IN
17. How can knowing the GCF and LCM help you when you add, subtract,
and multiply fractions?
82
Unit 2
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Add or subtract. Write each sum or difference in simplest form.
Name
Class
Date
4.1 Independent Practice
COMMON
CORE
Personal
Math Trainer
6.NS.4
Solve. Write each answer in simplest form.
18. Erin buys a bag of peanuts that weighs
_3 of a pound. Later that week, the bag is _2
4
3
full. How much does the bag of peanuts
weigh now? Show your work.
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Online
Assessment and
Intervention
21. Marcial found a recipe for fruit salad that
he wanted to try to make for his birthday
party. He decided to triple the recipe.
Fruit Salad
3_12 cups thinly sliced rhubarb
15 seedless grapes, halved
_1 orange, sectioned
19. Multistep Marianne buys 16 bags of
potting soil that comes in _58 -pound bags.
a. How many pounds of potting soil does
Marianne buy?
2
10 fresh strawberries, halved
_3 apple, cored and diced
5
_2 peach, sliced
3
1 plum, pitted and sliced
_1 cup fresh blueberries
4
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b. If Marianne’s father calls and says he
needs 13 pounds of potting soil, how
many additional bags should she buy?
a. What are the new amounts for the
oranges, apples, blueberries, and
peaches?
20. Music Two fifths of the instruments in the
marching band are brass, one third are
percussion, and the rest are woodwinds.
a. What fraction of the band is
woodwinds?
b. One half of the woodwinds are
clarinets. What fraction of the band is
clarinets?
b. Communicate Mathematical Ideas
The amount of rhubarb in the original
recipe is 3_12 cups. Using what you know
of whole numbers and what you know
of fractions, explain how you could
triple that mixed number.
c. One eighth of the brass instruments
are tubas. If there are 240 instruments
in the band, how many are tubas?
Lesson 4.1
83
22. One container holds 1 _78 quarts of water and a second container holds
5 _34 quarts of water. How many more quarts of water does the second
container hold than the first container?
23. Each of 15 students will give a 1_12 -minute speech in English class.
a. How long will it take to give the speeches?
b. If the teacher begins recording on a digital camera with an hour
available, is there enough time to record everyone if she gives a
15-minute introduction at the beginning of class and every student
takes a minute to get ready? Explain.
c. How much time is left on the digital camera?
FOCUS ON HIGHER ORDER THINKING
24. Represent Real-World Problems Kate wants to buy a new bicycle from
a sporting goods store. The bicycle she wants normally sells for $360. The
store has a sale where all bicycles cost _56 of the regular price. What is the
sale price of the bicycle?
26. Justify Reasoning To multiply a whole number by a fraction, you can
first write the whole number as a fraction by placing the whole number
in the numerator and 1 in the denominator. Does following this step
change the product? Explain.
84
Unit 2
Work Area
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25. Error Analysis To find the product _37 × _49 , Cameron simplified _37 to _17 and
4
then multiplied the fractions _17 and _49 to find the product __
63. What is
Cameron’s error?
LESSON
4.2 Dividing Fractions
?
COMMON
CORE
6.NS.1
Interpret and compute
quotients of fractions, …,
e.g., by using visual fraction
models… .
ESSENTIAL QUESTION
How do you divide fractions?
COMMON
CORE
EXPLORE ACTIVITY 1
6.NS.1
Modeling Fraction Division
In some division problems, you may know a number of groups and need to
find how many or how much are in each group. In other division problems,
you may know how many there are in each group, and need to find the
number of groups.
A You have _34 cup of salsa for making burritos. Each
burrito requires _18 cup of salsa. How many burritos
can you make?
To find the number of burritos that can
be made, you need to determine how
many _18 -cup servings are in _34 cups.
Use the diagram. How many eighths
3
4
are there in _34 ?
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You have enough salsa to make
burritos.
1
8
B Five people share _12 pound of cheese
equally. How much cheese does each
person receive?
To find how much cheese each person
receives, you need to determine how
much is in each of
parts.
How much is in each part?
Each person will receive
pound.
Reflect
1. Write the division shown by each model.
Lesson 4.2
85
Reciprocals
Another way to divide fractions is to use reciprocals. Two numbers whose
product is 1 are reciprocals.
Math On the Spot
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3 _
12
_
× 43 = __
=1
4
12
_3 and _4 are reciprocals.
4
3
To find the reciprocal of a fraction, switch the numerator and denominator.
numerator · ___________
denominator = 1
____________
denominator
numerator
EXAMPLE 1
COMMON
CORE
Prep for 6.NS.1
Find the reciprocal of each number.
A _29
Math Talk
Mathematical Practices
How can you check
that the reciprocal in A is
correct?
9
_
2
Switch the numerator and denominator.
The reciprocal of _29 is _92.
B _18
8
_
1
Switch the numerator and denominator.
The reciprocal of _18 is _81, or 8.
C 5
5 = _51
5
_
1
Rewrite as a fraction.
1
_
5
Switch the numerator and the denominator.
The reciprocal of 5 is _15.
Reflect
3. Communicate Mathematical Ideas Does every number have a
reciprocal? Explain.
4. The reciprocal of a whole number is a fraction with
numerator.
in the
YOUR TURN
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Math Trainer
Online Assessment
and Intervention
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86
Unit 2
Find the reciprocal of each number.
7
_
5. 8
6. 9
1
__
7. 11
© Houghton Mifflin Harcourt Publishing Company
2. Is any number its own reciprocal? If so, what number(s)? Justify your answer.
EXPLORE ACTIVITY 2
COMMON
CORE
6.NS.1
Using Reciprocals to Find
Equivalent Values
A Complete the table below.
Division
Multiplication
6 _
_
÷2=3
7 7
6 _
_
×7=
7 2
5 _
_
÷ 3 = _5
8 8 3
5 _
_
×8=
8 3
1 _
_
÷ 5 = _1
6 6 5
6
1× _
_
=
6 5
1 _
_
÷ 1 = _3
4 3 4
1 _
_
×3=
4 1
B How does each multiplication problem compare to its corresponding
division problem?
C How does the answer to each multiplication problem compare to the
answer to its corresponding division problem?
© Houghton Mifflin Harcourt Publishing Company
Reflect
8. Make a Conjecture Use the pattern in the table to make a conjecture
about how you can use multiplication to divide one fraction by another.
9. Write a division problem and a corresponding multiplication problem like
those in the table. Assuming your conjecture in 8 is correct, what is the
answer to your division problem?
Lesson 4.2
87
Using Reciprocals to Divide Fractions
Dividing by a fraction is equivalent to
multiplying by its reciprocal.
Math On the Spot
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1 _
_
÷ 14 = _45
5
EXAMPLE 2
4 _
1×_
_
= 45
1
5
COMMON
CORE
6.NS.1
Divide _59 ÷ _23 . Write the quotient in simplest form.
STEP 1
Rewrite as multiplication, using the reciprocal of the divisor.
5 _
_
÷ 2 = _59 × _32
9 3
Animated
Math
STEP 2
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2 __
The reciprocal of __
is 32 .
3
Multiply and simplify.
5 _
15
_
× 32 = __
9
18
= _56
5 _
_
÷ 23 = _56
9
Multiply the numerators. Multiply the denominators
Write the answer in simplest form.
15 ÷ 3 __
______
=5
18 ÷ 3 6
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
Divide.
9
11. __
÷ _35 =
10
9
10. __
÷ _25 =
10
my.hrw.com
Find the reciprocal of each fraction. (Example 1)
1. _25
2. _19
10
3. __
3
Divide. (Explore 1, Explore 2, and Example 2)
4. _43 ÷ _53 =
?
?
ESSENTIAL QUESTION CHECK-IN
7. How do you divide fractions?
88
3
5. __
÷ _45 =
10
Unit 2
6. _12 ÷ _25 =
© Houghton Mifflin Harcourt Publishing Company
Guided Practice
Name
Class
Date
4.2 Independent Practice
COMMON
CORE
6.NS.1
8. Alison has _12 cup of yogurt for making fruit
parfaits. Each parfait requires _18 cup of
yogurt. How many parfaits can she make?
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Fotolia
9. A team of runners is needed to run a _14 -mile
1
relay race. If each runner must run __
mile,
16
how many runners will be needed?
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Assessment and
Intervention
13. Jackson wants to divide a _34 -pound box of
trail mix into small bags. Each of the bags
1
will hold __
pound of trail mix. How many
12
bags of trail mix can Jackson fill?
14. A pitcher of contains _32 quart of lemonade.
If an equal amount of lemonade is
poured into each of 6 glasses, how much
lemonade will each glass contain?
10. Trevor paints _16 of the fence surrounding his
farm each day. How many days will it take
him to paint _34 of the fence?
15. How many tenths are there in _45 ?
11. Six people share _35 pound of peanuts
equally. What fraction of a pound of
peanuts does each person receive?
16. You make a large bowl of salad to share
with your friends. Your brother eats _13 of it
before they come over.
1
12. Biology If one honeybee makes __
12
teaspoon of honey during its lifetime, how
many honeybees are needed to make _12
teaspoon of honey?
a. You want to divide the leftover salad
evenly among six friends. What
expression describes the situation?
Explain.
b. What fractional portion of the original
bowl of salad does each friend receive?
Lesson 4.2
89
FOCUS ON HIGHER ORDER THINKING
Work Area
17. Interpret the Answer The length of a ribbon is _43 meter. Sun Yi needs
pieces measuring _31 meter for an art project. What is the greatest number
of pieces measuring _13 meter that can be cut from the ribbon? How much
ribbon will be left after Sun Yi cuts the ribbon? Explain your reasoning.
9
18. Represent Real-World Problems Liam has __
gallon of paint for painting
10
1
the birdhouses he sells at the craft fair. Each birdhouse requires __
gallon
20
of paint. How many birdhouses can Liam paint? Show your work.
19. Justify Reasoning When Kaitlin divided a fraction by _12, the result was
a mixed number. Was the original fraction less than or greater than _12 ?
Explain your reasoning.
1
21. Make a Prediction Susan divides the fraction _58 by __
. Her friend Robyn
16
5
1
_
__
divides 8 by 32 . Predict which person will get the greater quotient. Explain
and check your prediction.
90
Unit 2
© Houghton Mifflin Harcourt Publishing Company
20. Communicate Mathematical Ideas The reciprocal of a fraction less than
1 is always a fraction greater than 1. Why is this?
Dividing Mixed
Numbers
LESSON
4.3
?
COMMON
CORE
6.NS.1
Interpret and compute
quotients of fractions,
and solve word problems
involving division of fractions
by fractions… .
ESSENTIAL QUESTION
How do you divide mixed numbers?
COMMON
CORE
EXPLORE ACTIVITY
6.NS.1
Modeling Mixed Number Division
Antoine is making sushi rolls. He has 2_12 cups of rice and will use _14 cup
of rice for each sushi roll. How many sushi rolls can he make?
A To find the number of sushi rolls that can be made, you
need to determine how many fourths are in 2 _12. Use fraction
pieces to represent 2_12 on the model below.
1
1
4
1
4
1
2
1
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
B How many fourths are in 2_12?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Stockbyte /Getty
Images
Antoine has enough rice to make
sushi rolls.
Reflect
1.
Communicate Mathematical Ideas Which mathematical operation
could you use to find the number of sushi rolls that Antoine can make?
Explain.
2.
Multiple Representations Write the division shown by the model.
3.
What If? Suppose Antoine instead uses _18 cup of rice for each sushi roll.
How would his model change? How many rolls can he make? Explain.
Lesson 4.3
91
Using Reciprocals to Divide
Mixed Numbers
Math On the Spot
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My Notes
Dividing by a fraction is equivalent to multiplying by its reciprocal. You can
use this fact to divide mixed numbers. First rewrite the mixed numbers as
fractions greater than 1. Then multiply the dividend by the reciprocal of the
divisor.
EXAMPLE 1
COMMON
CORE
6.NS.1
One serving of Harold’s favorite cereal contains 1_25 ounces. How many
servings are in a 17_12 -ounce box?
STEP 1
Write a division statement to represent the situation.
17_12 ÷ 1_25
STEP 2
You need to find how many
2
1
17__
groups of 1__
2.
5 are in
Rewrite the mixed numbers as fractions greater than 1.
35 _
÷ 75
17_12 ÷ 1_25 = __
2
Rewrite the problem as multiplication using the reciprocal of
the divisor.
35 _
35 _
__
÷ 75 = __
× 57
2
2
STEP 4
7
5
__
The reciprocal of __
5 is 7 .
Multiply.
35 _
5 5__
__
×
= 35
× _57
7
2
2
1
5×5
_____
= 2× 1
25
, or 12_12
= __
2
Simplify first using the GCF.
Multiply numerators. Multiply denominators.
Write the result as a mixed number.
There are 12_12 servings of cereal in the box.
Reflect
92
Unit 2
4.
Analyze Relationships Explain how can you check the answer.
5.
What If? Harold serves himself 1_12 -ounces servings of cereal each
morning. How many servings does he get from a box of his favorite
cereal? Show your work.
© Houghton Mifflin Harcourt Publishing Company
STEP 3
YOUR TURN
6.
Sheila has 10 _12 pounds of potato salad. She wants to divide the potato
salad into containers, each of which holds 1_14 pounds. How many containers
does she need? Explain.
Personal
Math Trainer
Online Assessment
and Intervention
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Solving Problems Involving Area
Recall that to find the area of a rectangle, you multiply length × width. If you
know the area and only one dimension, you can divide the area by the known
dimension to find the other dimension.
Math On the Spot
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EXAMPL 2
EXAMPLE
COMMON
CORE
6.NS.1
The area of a rectangular sandbox is 56_23 square feet. The length of the
sandbox is 8_12 feet. What is the width?
STEP 1
Write the situation as a division problem.
56 _23 ÷ 8 _12
STEP 2
Math Talk
Rewrite the mixed numbers as fractions greater than 1.
170 __
÷ 17
56 _23 ÷ 8 _12 = ___
3
2
© Houghton Mifflin Harcourt Publishing Company
STEP 3
Rewrite the problem as multiplication using the reciprocal of
the divisor.
170 __
170 __
2
___
÷ 17
× 17
= ___
3
2
3
10
×2
______
= 170
3 × 17
20
, or 6 _23
= __
3
Mathematical Practices
Explain how to find
the length of a rectangle
when you know the area
and the width.
Multiply numerators. Multiply denominators.
1
Simplify and write as a mixed number.
The width of the sandbox is 6 _23 feet.
Reflect
7.
Check for Reasonableness How can you determine if your answer
is reasonable?
Lesson 4.3
93
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
8.
The area of a rectangular patio is 12_38 square meters.
The width of the patio is 2_34 meters. What is the length?
9.
1
The area of a rectangular rug is 14 __
12 square yards.
The length of the rug is 4 _13 yards. What is the width?
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Guided Practice
Divide. Write each answer in simplest form. (Explore Activity and Example 1)
1. 4_14 ÷ _34
2. 1_12 ÷ 2_14
3=
_____ ÷ __
_____ ÷ _____ =
_____ × _____ =
_____ × _____ =
4
4
4
2
4
2
3. 4 ÷ 1_18 =
4. 3_15 ÷ 1_17 =
5. 8_13 ÷ 2_12 =
6. 15_13 ÷ 3_56 =
7. A sandbox has an area of 26 square feet, and
the length is 5_21 feet. What is the width of the
sandbox?
8. Mr. Webster is buying carpet for an exercise
room in his basement. The room will have an
area of 230 square feet. The width of the room
is 12_12 feet. What is the length?
?
?
ESSENTIAL QUESTION CHECK-IN
9. How does dividing mixed numbers compare with dividing fractions?
94
Unit 2
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Write each situation as a division problem. Then solve. (Example 2)
Name
Class
Date
4.3 Independent Practice
COMMON
CORE
Personal
Math Trainer
6.NS.1
Online
Assessment and
Intervention
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10. Jeremy has 4 _12 cups of iced tea. He wants to divide the tea into _34 -cup
servings. Use the model to find the number of servings he can make.
1
1
4
1
4
1
1
4
1
4
1
4
1
4
1
1
4
1
4
1
4
1
4
1
1
4
1
4
1
4
1
4
1
1
4
1
4
1
4
1
4
1
4
1
4
11. A ribbon is 3 _23 yards long. Mae needs to cut the ribbon into pieces that are
_2 yard long. Use the model to find the number of pieces she can cut.
3
1
1
3
1
3
1
1
3
1
3
1
3
1
1
3
1
3
1
3
1
1
3
1
3
1
3
1
3
12. Dao has 2 _38 pounds of hamburger meat. He is making _14 -pound hamburgers.
Does Dao have enough meat to make 10 hamburgers? Explain.
13. Multistep Zoey made 5 _12 cups of trail mix for a camping trip. She wants
to divide the trail mix into _34 -cup servings.
© Houghton Mifflin Harcourt Publishing Company
a. Ten people are going on the camping trip. Can Zoey make enough
_3 -cup servings so that each person on the trip has one serving?
4
b. What size would the servings need to be for everyone to have a
serving? Explain.
c. If Zoey decides to use the _34 -cup servings, how much more trail mix
will she need? Explain.
14. The area of a rectangular picture frame is 30 _13 square inches. The length of
the frame is 6 _12 inches. Find the width of the frame.
Lesson 4.3
95
11
15. The area of a rectangular mirror is 11 __
16 square feet. The width of the
3
_
mirror is 2 4 feet. If there is a 5 foot tall space on the wall to hang the
mirror, will it fit? Explain.
16. Ramon has a rope that is 25 _12 feet long. He wants to cut it into 6 pieces
that are equal in length. How long will each piece be?
17. Eleanor and Max used two rectangular wooden boards to make a set for
the school play. One board was 6 feet long, and the other was 5 _12 feet
long. The two boards had equal widths. The total area of the set was
60 _38 square feet. What was the width?
FOCUS ON HIGHER ORDER THINKING
Work Area
19. Explain the Error To divide 14 _23 ÷ 2 _34 , Erik multiplied 14 _23 × _43 . Explain
Erik’s error.
20. Analyze Relationships Explain how you can find the missing number
= 2 _57 . Then find the missing number.
in 3 _45 ÷
96
Unit 2
© Houghton Mifflin Harcourt Publishing Company
2
18. Draw Conclusions Micah divided 11 _23 by 2 _56 and got 4 __
17 for an answer.
Does his answer seem reasonable? Explain your thinking. Then check
Micah’s answer.
LESSON
4.4
?
Solving Multistep Problems
with Fractions and Mixed
Numbers
ESSENTIAL QUESTION
COMMON
CORE
6.NS.1
...Solve word problems
involving division of fractions
by fractions...
How can you solve word problems involving more than one
fraction operation?
Solving Problems with
Rational Numbers
Sometimes more than one operation will be needed to solve a multistep
problem. You can use parentheses to group different operations. Recall
that according to the order of operations, you perform operations in
parentheses first.
EXAMPL 1
EXAMPLE
Problem
Solving
COMMON
CORE
Math On the Spot
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6.NS.1
Jon is cooking enough lentils for lentil barley soup and lentil salad. The
lentil barley soup recipe calls for _34 cup of dried lentils. The lentil salad
recipe calls for 1_12 cups of dried lentils. Jon has a _18 -cup scoop. How many
scoops of dried lentils will Jon need to have enough for the soup and
the salad?
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
Photodisc/Getty Images
Analyze Information
Identify the important information.
• Jon needs _34 cup of dried lentils for soup and 1_12 cups for salad.
• Jon has a _18 -cup scoop.
• You need to find the total number of many scoops of lentils he needs.
Formulate a Plan
(
)
You can use the expression _34 + 1_12 ÷ _18 to find the number of scoops of
dried lentils Jon will need for the soup and the salad.
Justify and Evaluate
Solve
Follow the order of operations. Perform the operations in parentheses first.
First add to find the total amount of dried lentils Jon will need.
3
_
+ 1_12 = _34 + _32
4
= _34 + _64
= _94
= 2_14
John needs
1
2 __
cups of
4
lentils.
Lesson 4.4
97
Jon needs 2 _14 cups of dried lentils for both the soup and the salad.
To find how many _18 -cup scoops he needs, divide the total amount of dried
lentils into groups of _18 .
2_14 ÷ _18 = _94 ÷ _18
= _94 × _81
×8
= 9____
4×1
2
Simplify before
multiplying using
the GCF.
1
18
= __
= 18
1
Jon will need 18 scoops of dried lentils to have enough for both the lentil
barley soup and the lentil salad.
Justify and Evaluate
You added _34 and 1_12 first to find the total number of cups of lentils. Then
you divided the sum by _18 to find the number of _18 -cup scoops.
YOUR TURN
Personal
Math Trainer
Online Assessment
and Intervention
1. Before conducting some experiments, a scientist mixes
_1 gram of Substance A with _3 gram of Substance B. If
4
2
the scientist uses _18 gram of the mixture for each
experiment, how many experiments can be conducted?
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1. An art student uses a roll of wallpaper to decorate two gift boxes. The
student will use 1 _13 yards of paper for one box and _56 yard of paper for the
other box. The paper must be cut into pieces that are _16 yard long. How
many pieces will the student cut to use for the gift boxes? (Example 1)
?
?
ESSENTIAL QUESTION CHECK-IN
2. How can you solve a multistep problem that involves fractions?
98
Unit 2
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Guided Practice
Name
Class
Date
4.4 Independent Practice
COMMON
CORE
6.NS.1
3. Naomi has earned $54 mowing lawns
the past two days. She worked 2 _12 hours
yesterday and 4 _14 hours today. If Naomi is
paid the same amount for every hour she
works, how much does she earn per hour
to mow lawns? (Example 2)
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Eveleigh/The Image Bank/Getty Images
4. An art teacher has 1_12 pounds of red clay
and _34 pound of yellow clay. The teacher
mixes the red clay and yellow clay
together. Each student in the class needs
_1 pound of the clay mixture to finish the
8
assigned art project for the class. How
many students can get enough clay to
finish the project?
5. A hairstylist schedules _14 hour to trim a
customer’s hair and _61 hour to style the
customer’s hair. The hairstylist plans to
work 3 _13 hours each day for 5 days each
week. How many appointments can the
hairstylist schedule each week if each
customer must be trimmed and styled?
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Math Trainer
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Online
Assessment and
Intervention
1
6. A picture framer has a thin board 10 __
12
3
_
feet long. The framer notices that 2 8 feet
of the board is scratched and cannot be
used. The rest of the board will be used to
make small picture frames. Each picture
frame needs 1 _23 feet of the board. At most,
how many complete picture frames can
be made?
7. Jim’s backyard is a rectangle that is 15 _56
yards long and 10 _52 yards wide. Jim buys sod
in pieces that are 1 _13 yards long and 1 _13 yards
wide. How many pieces of sod will Jim need
to buy to cover his backyard with sod?
8. Eva wants to make two pieces of pottery.
She needs _35 pound of clay for one piece and
7
__
10 pound of clay for the other piece. She
has three bags of clay that weigh _45 pound
each. How many bags of clay will Eva need
to make both pieces of pottery? How many
pounds of clay will she have left over?
9. Mark wants to paint a mural. He has 1 _13
gallons of yellow paint, 1 _14 gallons of green
paint, and _78 gallon of blue paint. Mark
plans to use _34 gallon of each paint color.
How many gallons of paint will he have left
after painting the mural?
Lesson 4.4
99
10. Trina works after school and on weekends. She always works three days
each week. This week she worked 2 _34 hours on Monday, 3 _35 hours on
Friday, and 5 _12 hours on Saturday. Next week she plans to work the same
number of hours as this week, but will work for the same number of
hours each day. How many hours will she work on each day?
FOCUS ON HIGHER ORDER THINKING
Work Area
11. Represent Real-World Problems Describe a real-world problem that
can be solved using the expression 29 ÷ ( _38 + _56 ). Find the answer in the
context of the situation.
13. Multiple Representations You are measuring walnuts for bananawalnut oatmeal and a spinach and walnut salad. You need _38 cup of
walnuts for the oatmeal and _34 cup of walnuts for the salad. You have a
_1 -cup scoop. Describe two different ways to find how many scoops of
4
walnuts you will need.
100
Unit 2
© Houghton Mifflin Harcourt Publishing Company
12. Justify Reasoning Indira and Jean begin their hike at 10 a.m. one
1
morning. They plan to hike from the 2 _25 -mile marker to the 8 __
-mile
10
marker along the trail. They plan to hike at an average speed of 3 miles
1
per hour. Will they reach the 8 __
-mile marker by noon? Explain your
10
reasoning.
MODULE QUIZ
Ready
Personal
Math Trainer
4.1 Applying GCF and LCM to Fraction Operations
Online Assessment
and Intervention
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Solve.
1. _45 × _34
9
2. _57 × __
10
3. _38 + 2 _12
4. 1 _35 − _56
4.2 Dividing Fractions
Divide.
5. _13 ÷ _79
6. _13 ÷ _58
7. Luci cuts a board that is _34 yard long into pieces that
are _38 yard long. How many pieces does she cut?
4.3 Dividing Mixed Numbers
Divide.
8. 3 _13 ÷ _23
10. 4 _14 ÷ 4 _12
9. 1 _78 ÷ 2 _25
11. 8 _13 ÷ 4 _27
© Houghton Mifflin Harcourt Publishing Company
4.4 Solving Multistep Problems with Fractions
and Mixed Numbers
12. Jamal hiked on two trails. The first trail was 5 _13 miles
long, and the second trail was 1 _34 times as long as
the first trail. How many miles did Jamal hike?
ESSENTIAL QUESTION
13. Describe a real-world situation that is modeled by dividing two
fractions or mixed numbers.
Module 4
101
MODULE 4 MIXED REVIEW
Personal
Math Trainer
Assessment
Readiness
6. What is the reciprocal of 3 _37 ?
Selected Response
1. Two sides of a rectangular fence are 5 _58 feet
long. The other two sides are 6 _14 feet long.
What is the perimeter?
B 13 feet
5
D 35 __ feet
32
18
2. Which shows the GCF of 18 and 24 with __
24
in simplest form?
3
A GCF: 3; _
4
6
B GCF: 3; _
8
3
C GCF: 6; _
4
6
D GCF: 6; _
8
B $1.52
C
D $1.71
4. Which of these is the same as _35 ÷ _47 ?
5
A -_
8
1
B -_
8
1
C _
4
D 0.5
Mini-Task
a. What is the area of each piece of paper
that Jodi is cutting out?
5. Andy has 6 _23 quarts of juice. How many
_2 -cup servings can he pour?
3
4
A 4_
9
B 6
C
D 10
Unit 2
1
A 4 _ feet
8
1
B 8 _ feet
4
1
C 16 _ feet
2
9. Jodi is cutting out pieces of paper that
measure 8 _12 inches by 11 inches from a
larger sheet of paper that has an area of
1,000 square inches
3
7
A _÷ _
5
4
4 3
B _÷_
7 5
3
4
C _× _
5
7
3 7
D _×_
5 4
102
7. A rectangular patio has a length of 12 _12 feet
and an area of 103 _18 square feet. What is
the width of the patio?
8. Which number is greater than the absolute
value of -_38 ?
A $1.49
7
3
B _
7
24
__
D
7
D 33 feet
3. A jar contains 133 pennies. A bigger jar
contains 1 _27 times as many pennies.
What is the value of the pennies in the
bigger jar?
$1.68
7
A __
24
7
C _
3
b. What is the greatest possible number of
pieces of paper that Jodi can cut out of
the larger sheet?
© Houghton Mifflin Harcourt Publishing Company
7
A 11 _ feet
8
3
C 23 _ feet
4
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Online
Assessment and
Intervention
Operations with
Decimals
?
5
MODULE
LESSON 5.1
ESSENTIAL QUESTION
Dividing Whole
Numbers
How can you use operations
with decimals to solve
real-world problems?
COMMON
CORE
6.NS.2
LESSON 5.2
Adding and
Subtracting Decimals
COMMON
CORE
6.NS.3
LESSON 5.3
Multiplying Decimals
COMMON
CORE
6.NS.3
LESSON 5.4
Dividing Decimals
COMMON
CORE
6.NS.3
LESSON 5.5
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Applying Operations
with Rational
Numbers
COMMON
CORE
6.NS.3
Real-World Video
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The gravitational force on Earth’s moon is less than
the gravitational force on Earth. You can calculate
your weight on the moon by multiplying your
weight on Earth by a decimal.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
103
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this module.
Represent Decimals
EXAMPLE
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Online
Assessment and
Intervention
Think: 1 square = 1 of 100 equal parts
1 , or 0.01
= ____
100
10 squares = 10 of 100 equal parts
1
= ___
, or 0.1
10
So, 20 squares represent 2 × 0.1, or 0.2.
Write the decimal represented by the shaded square.
1.
2.
3.
4.
Multiply Decimals by Powers of 10
EXAMPLE
6.574 × 100
Count the zeros in 100: 2 zeros.
6.574 × 100 = 657.4
Move the decimal point 2 places to the right.
5. 0.49 × 10
6. 25.34 × 1,000
7. 87 × 100
Words for Operations
EXAMPLE
Write a numerical expression for
the product of 5 and 9.
5×9
Think: Product means “to multiply.”
Write 5 times 9.
Write a numerical expression for the word expression.
8. 20 decreased by 8
10. the difference between 72 and 16
104
Unit 2
9. the quotient of 14 and 7
11. the sum of 19 and 3
© Houghton Mifflin Harcourt Publishing Company
Find the product.
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the chart. You may put more
than one word in each section.
450 ÷ 9 = 50
÷
Dividing Numbers
450
___
9
Not division
Vocabulary
Review Words
decimal (decimal)
✔ denominator
(denominador)
divide (dividir)
✔ dividend (dividendo)
✔ divisor (divisor)
✔ fraction bar (barra de
fracciones)
✔ multiply (multiplicar)
✔ numerator (numerador)
✔ operation (operación)
✔ product (producto)
✔ quotient (cociente)
✔ rational number (número
racional)
✔ symbol (símbolo)
whole number (número
entero)
Understand Vocabulary
© Houghton Mifflin Harcourt Publishing Company
Match the term on the left to the definition on the right.
1. divide
A. The bottom number in a fraction.
2. denominator
B. The top number in a fraction.
3. quotient
C. To split into equal groups.
4. numerator
D. The answer in a division problem.
Active Reading
Booklet Before beginning the module, create
a booklet to help you learn the concepts in this
module. Write the main idea of each lesson on
its own page of the booklet. As you study each
lesson, record examples that illustrate the main
idea and make note of important details. Refer to
your finished booklet as you work on assignments
and study for tests.
Module 5
105
MODULE 5
Unpacking the Standards
Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to
learn in this module.
COMMON
CORE
6.NS.2
Fluently divide multi-digit
numbers using the standard
algorithm.
What It Means to You
You will use your prior knowledge of division of whole numbers
to perform division with decimals.
Key Vocabulary
UNPACKING EXAMPLE 6.NS.2
quotient (cociente)
The result when one number is
divided by another.
Eugenia and her friends bought frozen
yogurt for 45 cents per ounce. Their
total was $11.25. How many ounces
did they buy?
Divide 11.25 by 0.45.
25
⎯
0.45⟌ 11.25
90
225
225
0
They bought 25 ounces of frozen yogurt.
6.NS.3
Fluently add, subtract, multiply,
and divide multi-digit decimals
using the standard algorithm
for each operation.
What It Means to You
You will use your prior knowledge of operations with whole
numbers to perform operations with decimals.
UNPACKING EXAMPLE 6.NS.3
Key Vocabulary
algorithm (algoritmo)
A set of rules or a procedure
for solving a mathematical
problem in a finite number
of steps.
Visit my.hrw.com
to see all the
Common Core
Standards
unpacked.
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106
Unit 2
Estimate and find the exact answer.
A. 3.25 × 4.8
B. 132.5 - 18.9
3 × 5 = 15
133 - 19 = 114
3.25
× 4.8
2600
13000
15.600
132.5
-18.9
113.6
© Houghton Mifflin Harcourt Publishing Company
COMMON
CORE
LESSON
5.1
?
Dividing Whole
Numbers
COMMON
CORE
6.NS.2
...Divide multi-digit
numbers using the standard
algorithm....
ESSENTIAL QUESTION
How do you divide multi-digit whole numbers?
EXPLORE ACTIVITY
COMMON
CORE
6.NS.2
Estimating Quotients
You can use estimation to predict the quotient of multi-digit whole numbers.
A local zoo had a total of 98,464 visitors last year. The zoo was open
every day except for three holidays. On average, about how many
visitors did the zoo have each day?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©John Coletti/
Getty Images
A To estimate the average number of visitors per day, you can divide
the total number of visitors by the number of days. To estimate the
quotient, first estimate the dividend by rounding the number of visitors
to the nearest ten thousand.
98,464 rounded to the nearest ten thousand is
quotient
⎯
divisor⟌ dividend
.
B There were 365 days last year. How many
days was the petting zoo open?
C Estimate the divisor by rounding the number of days that the zoo was
open to the nearest hundred.
rounded to the nearest hundred is
D Estimate the quotient.
÷
.
=
The average number of visitors per day last year was about
.
Reflect
1.
How can you check that your quotient is correct?
2.
Critical Thinking Do you think that your estimate is greater than or
less than the actual answer? Explain.
Lesson 5.1
107
Using Long Division
The exact average number of visitors per day at the zoo in the Explore Activity
is the quotient of 98,464 and 362. You can use long division to find this
quotient.
Math On the Spot
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EXAMPLE 1
COMMON
CORE
6.NS.2
A local zoo had a total of 98,464 visitors last year. The zoo was open every
day except three holidays? On average, how many visitors did the zoo
have each day?
STEP 1
Math Talk
Mathematical Practices
How does the estimate from
the Explore Activity compare
to the actual average
number of visitors
per day?
STEP 2
362 is greater than 9 and 98, so divide 984 by 362. Place the first
digit in the quotient in the hundreds place. Multiply 2 by 362 and
place the product under 984. Subtract.
2
⎯
362⟌ 98,464
- 72 4
26 0
Bring down the tens digit. Divide 2,606 by 362. Multiply 7 by 362
and place the product under 2,606. Subtract.
27
⎯
362⟌ 98,464
-72 4
26 06
-25 34
72
Bring down the ones digit. Divide the ones.
272
⎯
362⟌ 98,464
-72 4
26 06
-25 34
724
-724
0
The average number of visitors per day last year was 272.
YOUR TURN
Personal
Math Trainer
Find each quotient.
Online Assessment
and Intervention
3. 34,989 ÷ 321
my.hrw.com
108
Unit 2
4. 73,375 ÷ 125
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STEP 3
Dividing with a Remainder
Suppose you and your friend want to divide 9 polished rocks between you so that
you each get the same number of polished rocks. You will each get 4 rocks with 1
rock left over. You can say that the quotient 9 ÷ 2 has a remainder of 1.
EXAMPL 2
EXAMPLE
COMMON
CORE
Math On the Spot
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6.NS.2
Callie has 1,850 books. She must pack them into
boxes to ship to a bookstore. Each box holds 12
books. How many boxes will she need to pack all
of the books?
154 R2
⎯
12
1,850
⟌
Divide 1,850 by 12.
-12
65
-60
50
-48
_
2
My Notes
The quotient is 154, remainder 2. You can write 154 R2.
© Houghton Mifflin Harcourt Publishing Company
Reflect
5.
Interpret the Answer What does the remainder mean in this situation?
6.
Interpret the Answer How many boxes does Callie need to pack the
books? Explain.
YOUR TURN
Divide.
7. 5,796 ÷ 25
⎯
8. 67⟌ 3,098
9. A museum gift shop manager wants to put 1,578 polished rocks into
small bags to sell as souvenirs. If the shop manager wants to put
15 rocks in each bag, how many complete bags can be filled? How
many rocks will be left over?
Personal
Math Trainer
Online Assessment
and Intervention
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Lesson 5.1
109
Guided Practice
1. Estimate: 31,969 ÷ 488 (Explore Activity)
Round the numbers and then divide.
31,969 ÷ 488 =
÷
=
Divide. (Example 1, Example 2)
2. 3,072 ÷ 32 =
3. 4,539 ÷ 51 =
9
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
32 ⟌ 3, 0 7 2
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
51 ⟌ 4, 5 3 9
−
4. 9,317 ÷ 95 =
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
95 ⟌ 9, 3 1 7
−
−
1 9 2
−
4 5 9
−
5. 2,226 ÷ 53 =
6. Divide 4,514 by 74.
8. 2,001 ÷ 83 =
9. 39,751 ÷ 313 =
−
7. 3,493 ÷ 37 =
10. 35,506 ÷ 438 =
11. During a food drive, a local middle school collected 8,982 canned food
items. Each of the 28 classrooms that participated in the drive donated
about the same number of items. Estimate the number of items each
classroom donated. (Explore Activity)
12. A theater has 1,120 seats in 35 equal rows. How many seats are in each
row? (Example 1)
?
?
ESSENTIAL QUESTION CHECK-IN
14. What steps do you take to divide multi-digit whole numbers?
110
Unit 2
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13. There are 1,012 souvenir paperweights that need to be packed in boxes.
Each box will hold 12 paperweights. How many boxes will be needed?
(Example 2)
Name
Class
Date
5.1 Independent Practice
COMMON
CORE
6.NS.2
Personal
Math Trainer
my.hrw.com
Online
Assessment and
Intervention
Divide.
15. 44,756 ÷ 167 =
16. 87,628 ÷ 931 =
17. 66,253 ÷ 317 =
18. 76,255 ÷ 309 =
19. 50,779 ÷ 590 =
20. 97,156 ÷ 107 =
21. 216,016 ÷ 368 =
22. 107,609 ÷ 72 =
23. Emilio has 8,450 trees to plant in rows on his tree farm. He will plant
125 trees per row. How many full rows of trees will he have? Explain.
24. Camilla makes and sells jewelry. She has 8,160 silver beads and 2,880
black beads to make necklaces. Each necklace will contain 85 silver beads
and 30 black beads. How many necklaces can she make?
25. During a promotional weekend, a state fair gives a free admission to every
175th person who enters the fair. On Saturday, there were 6,742 people
attending the fair. On Sunday, there were 5,487 people attending the fair.
How many people received a free admission over the two days?
© Houghton Mifflin Harcourt Publishing Company
26. How is the quotient 80,000 ÷ 2,000 different from the quotient
80,000 ÷ 200 or 80,000 ÷ 20?
27. Given that 9,554 ÷ 562 = 17, how can you find the quotient
95,540 ÷ 562?
28. Earth Science The diameter of the Moon is about 3,476 kilometers.
The distance from Earth to the Moon is about 384,400 kilometers.
About how many moons could be lined up in a row between Earth
and the Moon? Round to the nearest whole number.
Diameter 3,476 km
Lesson 5.1
111
29. Vocabulary Explain how you could check the answer to a division
question in which there is a remainder.
30. Yolanda is buying a car with a base price of $16,750. She must also
pay the options, fees, and taxes shown. The car dealership will give
her 48 months to pay off the entire amount. Yolanda can only afford
to pay $395 each month. Will she be able to buy the car? Explain.
Jackson
Auto Dealer
4-door sedan
base price
$16,750
options
$
500
fees
$
370
taxes
$ 1,425
FOCUS ON HIGHER ORDER THINKING
Work Area
31. Check for Reasonableness Is 40 a reasonable estimate of a quotient for
78,114 ÷ 192? Explain your reasoning.
33. Make a Prediction In preparation for a storm, the town council buys
13,750 pounds of sand to fill sandbags. Volunteers are trying to decide
whether to fill bags that can hold 25 pounds of sand or bags that can
hold 50 pounds of sand. Will they have more or fewer sandbags if they fill
the 25-pound bags? How many more or fewer? Explain your reasoning.
112
Unit 2
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32. Critique Reasoning Harrison predicted that the actual quotient for
57,872 ÷ 305 will be less than the estimate 60,000 ÷ 300 = 200. Is
Harrison correct? Explain how Harrison arrived at his prediction (without
dividing the actual numbers).
LESSON
5.2
?
Adding and
Subtracting Decimals
COMMON
CORE
6.NS.3
Fluently add [and] subtract...
decimals using the standard
algorithm....
ESSENTIAL QUESTION
How do you add and subtract decimals?
EXPLORE ACTIVITY
COMMON
CORE
6.NS.3
Modeling Decimal Addition
You have probably used decimal grids to model decimals. For example,
25
the decimal 0.25, or ___
, can be modeled by shading 25 squares in a
100
10 × 10 grid. You can also use decimal grids to add decimal values.
A chemist combines 0.17 mL of water and 0.49 mL of hydrogen
peroxide in a beaker. How much total liquid is in the beaker?
A How many grid squares should you shade to represent
0.17 mL of water? Why?
B How many grid squares should you shade to represent
0.49 mL of hydrogen peroxide?
© Houghton Mifflin Harcourt Publishing Company
C Use the grid at the right to model the addition. Use one
color for 0.17 mL of water and another color for 0.49 mL of
hydrogen peroxide.
D How much total liquid is in the beaker? 0.17 + 0.49 =
mL
Reflect
Multiple Representations Show how to shade each grid to represent the
sum. Then find the sum.
1. 0.24 + 0.71 =
2. 0.08 + 0.65 =
Lesson 5.2
113
Adding Decimals
Math On the Spot
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Adding decimals is similar to adding whole numbers. First align the numbers by
place value. Start adding at the right and regroup when necessary. Bring down
the decimal point into your answer.
EXAMPLE 1
COMMON
CORE
6.NS.3
Susan rode her bicycle 3.12 miles on Monday and 4.7 miles
on Tuesday. How many miles did she ride in all?
STEP 1
Align the decimal points.
STEP 2
Add zeros as placeholders
when necessary.
STEP 3
+
3
·
1
2
4
·
7
0
7
·
8
2
Add from right to left.
Susan rode 7.82 miles in all.
STEP 4
Use estimation to check that the answer is reasonable.
Round each decimal to the nearest whole number.
3
3.12
+
5
+
4.70
_
__
8
7.82
Reflect
3.
Why can you rewrite 4.7 as 4.70?
4.
Why is it important to align the decimal points when adding?
YOUR TURN
Add.
Personal
Math Trainer
5. 0.42 + 0.27 =
6. 0.61 + 0.329 =
7. 3.25 + 4.6 =
8. 17.27 + 3.88 =
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Unit 2
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Chabraszewski / Shutterstock
Since 8 is close to 7.82, the answer is reasonable.
Subtracting Decimals
The procedure for subtracting decimals is similar to the procedure for adding
decimals.
EXAMPL 2
EXAMPLE
COMMON
CORE
6.NS.3
Math On the Spot
my.hrw.com
A Mia is 160.2 centimeters tall. Rosa is 165.1 centimeters tall.
How much taller is Rosa than Mia?
STEP 1
Align the decimal points.
STEP 2
Add zeros as placeholders
when necessary.
STEP 3
Subtract from right to left,
regrouping when necessary.
4
-
My Notes
1
1
6
5
·
1
6
0
·
2
4
·
9
1
Rosa is 4.9 centimeters taller than Mia.
To check that your answer is reasonable, you can estimate.
Round each decimal to the nearest whole number.
165.1
160.2
__
4.9
165
160
__
5
Since 5 is close to 4.9, the answer is reasonable.
© Houghton Mifflin Harcourt Publishing Company
B Matthew throws a discus 58.7 meters. Zachary throws the discus
56.12 meters. How much farther did Matthew throw the discus?
STEP 1
Align the decimal points.
STEP 2
Add zeros as placeholders when
necessary.
STEP 3
Subtract from right to left,
regrouping when necessary.
-
5
8
·
5
6
2
6
10
7
0
·
1
2
·
5
8
Matthew threw the discus 2.58 meters farther
than Zachary.
Math Talk
Mathematical Practices
How can you
check a subtraction
problem?
To check that your answer is reasonable, you can estimate.
Round each decimal to the nearest whole number.
58.7
56.12
__
2.58
59
56
_
3
Since 3 is close to 2.58, the answer is reasonable.
Lesson 5.2
115
Guided Practice
Shade the grid to find each sum. (Explore Activity)
1. 0.72 + 0.19 =
2. 0.38 + 0.4 =
Add. Check that your answer is reasonable. (Example 1)
3.
54.87
+ 7.48
__
55
+ 7
_
4.
5.
2.19
+ 34.92
__
+
__
0.215
+ 3.74
__
+
__
Subtract. Check that your answer is reasonable. (Example 2)
6.
9.73
- 7.16
__
10
- 7
_
7.
8.
18.419
- 6.47
__
__
5.006
- 3.2
__
__
9. 17.2 + 12.9 =
12. 15.52 − 8.17 =
10. 28.341 + 37.5 =
11. 25.36 - 2.004 =
13. 25.68 + 12 =
14. 150.25 - 78 =
15. Perry connects a blue garden hose and a green garden hose to
make one long hose. The blue hose is 16.5 feet. The green hose is
14.75 feet. How long is the combined hose? (Example 1)
16. Keisha has $20.08 in her purse. She buys a book for $8.72.
How much does she have left? (Example 2)
?
?
ESSENTIAL QUESTION CHECK-IN
17. How is adding and subtracting decimals similar to adding and subtracting
whole numbers?
116
Unit 2
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Add or subtract. (Example 1, Example 2)
Name
Class
Date
5.2 Independent Practice
COMMON
CORE
Personal
Math Trainer
6.NS.3
my.hrw.com
Online
Assessment and
Intervention
Add or subtract.
18. 28.6 - 0.975 =
19. 5.6 - 0.105 =
20. 7.03 + 33.006 =
21. 57.42 + 4 + 1.602 =
22. 2.25 + 65.47 + 2.333 =
23. 18.419 - 6.47 =
24. 83 - 12.76 =
25. 102.01 - 95.602 =
26. Multiple Representations Ursula wrote the sum 5.815 + 6.021 as a sum
of two mixed numbers.
a. What sum did she write?
b. Compare the sum of the mixed
numbers to the sum of the decimals.
Use the café menu to answer 27–29.
27. Stephen and Jahmya are having lunch. Stephen buys a
garden salad, a veggie burger, and lemonade. Jahmya buys a
fruit salad, a toasted cheese sandwich, and a bottle of water.
Whose lunch cost more? How much more?
© Houghton Mifflin Harcourt Publishing Company
28. Jahmya wants to leave $1.75 as a tip for her server. She has a
$20 bill. How much change should she receive after paying
for her food and leaving a tip?
29. What If? In addition to his meal, Stephen orders a fruit salad
for take-out, and wants to leave $2.25 as a tip for his server.
He has a $10 bill and a $5 bill. How much change should he
receive after paying for his lunch, the fruit salad, and the tip?
30. A carpenter who is installing cabinets uses thin pieces of
material called shims to fill gaps. The carpenter uses four
shims to fill a gap that is 1.2 centimeters wide. Three of the
shims are 0.75 centimeter, 0.125 centimeter, and 0.09
centimeter wide. What is the width of the fourth shim?
Lesson 5.2
117
31. A CD of classical guitar music contains 5 songs. The length of each song is
shown in the table.
Track 1
Track 2
6.5 minutes
8 minutes
Track 3
Track 4
Track 5
3.93 minutes 4.1 minutes 5.05 minutes
a. Between each song is a 0.05-minute break. How long does it
take to listen to the CD from the beginning of the first song
to the end of the last song?
b. What If? Juan wants to buy the CD from an Internet music site.
He downloads the CD onto a disc that can hold up to 60 minutes
of music. How many more minutes of music can he still buy after
downloading the CD?
FOCUS ON HIGHER ORDER THINKING
Work Area
33. Communicate Mathematical Ideas The Commutative Property of
Addition states that you can change the order of addends in a sum. The
Associative Property of Addition states that you can change the grouping
of addends in a sum. Use an example to show how the Commutative
Property of Addition and the Associative Property of Addition apply to
adding decimals.
34. Critique Reasoning Indira predicts that the actual difference of
19 - 7.82 will be greater than the estimate of 19 - 8 = 11. Is Indira
correct? Explain how Indira might have arrived at that prediction without
subtracting the actual numbers.
118
Unit 2
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Getty Images
32. Analyze Relationships Use the decimals 2.47, 9.57, and 7.1 to write two
different addition facts and two different subtraction facts.
LESSON
5.3 Multiplying Decimals
?
COMMON
CORE
6.NS.3
Fluently …multiply…
multi-digit decimals using
the standard algorithm… .
ESSENTIAL QUESTION
How do you multiply decimals?
EXPLORE ACTIVITY
COMMON
CORE
6.NS.3
Modeling Decimal Multiplication
Use decimal grids or area models to find each product.
0.3
A 0.3 × 0.5
0.3 × 0.5 represents 0.3 of 0.5. Shade 5 rows
of the decimal grid to represent 0.5.
0.5
Shade 0.3 of each 0.1 that is already shaded
to represent 0.3 of
.
square(s) are double-shaded.
This represents
hundredth(s), or 0.15.
0.3 × 0.5 =
© Houghton Mifflin Harcourt Publishing Company
B 3.2 × 2.1
3.2
Use an area model. In the model, the large
squares represent wholes, the small rectangles
along the right and lower edges represent tenths,
and the small squares at the lower right represent 2.1
hundredths. The model is 3 and 2 tenths units long,
and 2 and 1 tenth unit wide.
The area of the model is
whole(s) +
tenth(s) +
hundredth(s) square units.
3.2 × 2.1 =
Reflect
1.
Analyze Relationships How are the products 2.1 × 3.2 and 21 × 32
alike? How are they different?
Lesson 5.3
119
Multiplying Decimals
Math On the Spot
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To multiply decimals, first multiply as you would with whole numbers. Then
place the decimal point in the product. The number of decimal places in the
product equals the sum of the number of decimal places in the factors.
EXAMPLE 1
COMMON
CORE
6.NS.3
Delia bought 3.8 pounds of peppers. The peppers cost $1.99
per pound. What was the total cost of Delia’s peppers?
1.99
←
2 decimal places
×
3.8
←
+
1 decimal place
__
1592
+
5970
__
7.562
←
3 decimal places
Round the answer to hundredths
to show a dollar amount.
The peppers cost $7.56.
Reflect
Communicate Mathematical Ideas How can you use estimation to
check that you have placed the decimal point correctly in your product?
YOUR TURN
Multiply.
3.
12.6 ←
decimal place(s) 4.
9.76 ←
decimal place(s)
× 0.46 ← _____________________
decimal place(s) __________
+ + decimal place(s)
×
15.3 ← _____________________
______
378
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120
Unit 2
+
____________
+
____________
←
decimal place(s)
←
decimal place(s)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©PhotoDisc/
Getty Images
2.
Estimating to Check Reasonableness
In Example 1, you used estimation to check whether the decimal point was
placed correctly in the product. You can also use estimation to check that your
answer is reasonable.
EXAMPL 2
EXAMPLE
COMMON
CORE
Blades of grass grow 3.75 inches per month. If the grass continues
to grow at this rate, how much will the grass grow in 6.25 months?
3.75 ← 2 decimal places
+ 2 decimal places
×
6.25 ← _________________
__
1875
7500
+
225000
__
23.4375
←
Math On the Spot
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6.NS.3
Animated
Math
my.hrw.com
My Notes
4 decimal places
The grass will grow 23.4375 inches in 6.25 months.
Estimate to check whether your answer is reasonable.
Round 3.75 to the nearest whole number.
Round 6.25 to the nearest whole number.
Multiply the whole numbers.
×
= 24
The answer is reasonable because 24 is close to 23.4375.
YOUR TURN
Multiply.
© Houghton Mifflin Harcourt Publishing Company
5.
7.14
×
6.78
__
5712
6.
11.49
×
8.27
__
____________
+
+
____________
7.
Rico bicycles at an average speed of 15.5 miles per hour.
What distance will Rico bicycle in 2.5 hours?
8.
miles
Use estimation to show that your answer to 7 is reasonable.
Personal
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Online Assessment
and Intervention
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Lesson 5.3
121
Guided Practice
1. Use the grid to multiply 0.4 × 0.7.
(Explore Activity)
2. Draw an area model to multiply 1.1 × 2.4.
(Explore Activity)
0.4 × 0.7 =
1.1 × 2.4 =
Multiply. (Example 1 and Example 2)
3. 0.18 × 0.06 =
4. 35.15 × 3.7 =
5. 0.96 × 0.12 =
6. 62.19 × 32.5 =
7. 3.4 × 4.37 =
8. 3.762 × 0.66 =
9. Chan Hee bought 3.4 pounds of coffee that cost $6.95 per pound.
How much did he spend on coffee? $
10. Adita earns $9.40 per hour working at an animal shelter.
Catherine tracked her gas purchases for one month.
11. How much did Catherine spend on gas in week 2?
$
12. How much more did she spend in week 4 than
Week
1
2
3
4
Gallons
10.4
11.5
9.72
10.6
in week 1? $
?
?
ESSENTIAL QUESTION CHECK-IN
13. How can you check the answer to a decimal multiplication problem?
122
Unit 2
Cost per gallon ($)
2.65
2.54
2.75
2.70
© Houghton Mifflin Harcourt Publishing Company
How much money will she earn for 18.5 hours of work? $
Name
Class
Date
5.3 Independent Practice
COMMON
CORE
6.NS.3
Personal
Math Trainer
my.hrw.com
Online
Assessment and
Intervention
Make a reasonable estimate for each situation.
14. A gallon of water weighs 8.354 pounds. Simon uses 11.81 gallons
of water while taking a shower. About how many pounds of water
did Simon use?
15. A snail moves at a speed of 2.394 inches per minute. If the snail
keeps moving at this rate, about how many inches will it travel
in 7.489 minutes?
16. Tricia’s garden is 9.87 meters long and 1.09 meters wide. What is the
area of her garden?
Kaylynn and Amanda both work at the same store. The table shows
how much each person earns, and the number of hours each person
works in a week.
Wage
Kaylynn
Amanda
$8.75 per hour
$10.25 per hour
Hours worked
per week
37.5
30.5
17. Estimate how much Kaylynn earns in a week.
18. Estimate how much Amanda earns in a week.
© Houghton Mifflin Harcourt Publishing Company
19. Calculate the exact difference between Kaylynn and Amanda’s weekly
salaries.
20. Victoria’s printer can print 8.804 pages in one minute. If Victoria prints
pages for 0.903 minutes, about how many pages will she have?
A taxi charges a flat fee of $4.00 plus $2.25 per mile.
21. How much will it cost to travel 8.7 miles?
22. Multistep How much will the taxi driver earn if he takes one passenger
4.8 miles and another passenger 7.3 miles? Explain your process.
Lesson 5.3
123
Kay goes for several bike rides one week. The table shows her speed and
the number of hours spent per ride.
Monday
Tuesday
Wednesday
Thursday
Friday
Speed (in miles per hour)
8.2
9.6
11.1
10.75
8.8
Hours Spent on Bike
4.25
3.1
2.8
1.9
3.75
23. How many miles did Kay bike on Thursday?
24. On which day did Kay bike a whole number of miles?
25. What is the difference in miles between Kay’s longest bike ride and her
shortest bike ride?
26. Check for Reasonableness Kay estimates that Wednesday’s ride was
about 3 miles longer than Tuesday’s ride. Is her estimate reasonable?
Explain.
FOCUS ON HIGHER ORDER THINKING
Work Area
28. Represent Real-World Problems A jeweler buys gold jewelry and resells
the gold to a refinery. The jeweler buys gold for $1,235.55 per ounce, and
then resells it for $1,376.44 per ounce. How much profit does the jeweler
make from buying and reselling 73.5 ounces of gold?
29. Problem Solving To find the weight of the gold in a 22 karat gold object,
multiply the object’s weight by 0.917. To find the weight of the gold in
an 18 karat gold object, multiply the object’s weight by 0.583. A 22 karat
gold statue and a 14 karat gold statue both weigh 73.5 ounces. Which one
contains more gold? How much more gold does it contain?
124
Unit 2
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27. Explain the Error To estimate the product 3.48 × 7.33, Marisa multiplied
4 × 8 to get 32. Explain how she can make a closer estimate.
LESSON
5.4 Dividing Decimals
?
COMMON
CORE
6.NS.3
Fluently …divide multi-digit
decimals using the standard
algorithm… .
ESSENTIAL QUESTION
How do you divide decimals?
EXPLORE ACTIVITY
COMMON
CORE
6.NS.3
Modeling Decimal Division
Use decimal grids to find each quotient.
A 6.39 ÷ 3
Shade grids to model 6.39. Separate the model into 3 equal groups.
How many are in each group?
6.39 ÷ 3 =
B 6.39 ÷ 2.13
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Shade grids to model 6.39. Separate the model into groups of 2.13.
How many groups do you have?
6.39 ÷ 2.13 =
Reflect
1.
Multiple Representations When using models to divide decimals,
when might you want to use grids divided into tenths instead of
hundredths?
Lesson 5.4
125
Dividing Decimals by Whole Numbers
Dividing decimals is similar to dividing whole numbers. When you divide
a decimal by a whole number, the placement of the decimal point in the
quotient is determined by the placement of the decimal in the dividend.
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EXAMPLE 1
COMMON
CORE
A A high school track is 9.76 meters wide. It is divided
into 8 lanes of equal width for track and field events.
How wide is each lane?
My Notes
Divide using long division as with whole numbers.
Place a decimal point in the quotient directly above
the decimal point in the dividend.
6.NS.3
1.22
⎯⎯⎯
8⟌ 9.76
- 8
17
-1 6
16
-16
0
Each lane is 1.22 meters wide.
B Aerobics classes cost $153.86 for 14 sessions.
What is the fee for one session?
Math Talk
Mathematical Practices
How can you check
to see that the answer
is correct?
Divide using long division as with whole numbers.
Place a decimal point in the quotient directly above
the decimal point in the dividend.
The fee for one aerobics class is $10.99.
Reflect
Check for Reasonableness How can you estimate
to check that your quotient in A is reasonable?
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2.
YOUR TURN
Divide.
⎯⎯⎯
3. 5⟌ 9.75
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Unit 2
10.99
⎯⎯⎯⎯
14⟌ 153.86
- 14
13
-0
13 8
-1 2 6
1 26
-1 2 6
0
4.
⎯⎯⎯
7⟌ 6.44
Dividing a Decimal by a Decimal
When dividing a decimal by a decimal, first change the divisor to a whole
number by multiplying by a power of 10. Then multiply the dividend by
the same power of 10.
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EXAMPL 2
EXAMPLE
COMMON
CORE
6.NS.3
A Ella uses 0.5 pound of raspberries in each raspberry cake that
she makes. How many cakes can Ella make with 3.25 pounds
of raspberries?
STEP 1
The divisor has one decimal
place, so multiply both the
dividend and the divisor
by 10 so that the divisor
is a whole number.
⎯⎯⎯
⎯⎯⎯
0.5⟌ 3.25
0.5⟌ 3.25
STEP 2
Divide.
6.5
⎯⎯⎯
5⟌ 32.5
−30
25
−2 5
0
0.5 × 10 = 5
3.25 × 10 = 32.5
Math Talk
Mathematical Practices
Ella can make 6 cakes.
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Getty Images
B Anthony spent $11.52 for some pens that were on sale
for $0.72 each. How many pens did Anthony buy?
STEP 1
The divisor has two decimal
places, so multiply both the
dividend and the divisor by
100 so that the divisor is a
whole number.
⎯⎯⎯
⎯⎯⎯
0.72⟌ 11.52
0.72⟌ 11.52
STEP 2
The number of cakes Ella
can make is not equal to the
quotient. Why not?
Divide.
16
⎯⎯⎯
⟌
72 1152
−72
432
− 432
0
0.72 × 100 = 72
11.52 × 100 = 1152
Anthony bought 16 pens.
YOUR TURN
Divide.
⎯⎯⎯
5. 0.5⟌ 4.25
6.
⎯⎯⎯
0.84⟌ 15.12
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Lesson 5.4
127
Guided Practice
Divide. (Explore Activity, Examples 1 and 2)
⎯⎯⎯
1. 4⟌ 29.5
⎯⎯⎯⎯
2. 3.1⟌ 10.261
⎯
3. 2.4⟌ 16.8
⎯⎯⎯
4. 0.96⟌ 0.144
5. 38.5 ÷ 0.5 =
6. 23.85 ÷ 9 =
7. 5.6372 ÷ 0.17 =
8. 8.19 ÷ 4.2 =
9. 66.5 ÷ 3.5 =
10. 0.234 ÷ 0.78 =
11. 78.74 ÷ 12.7 =
12. 36.45 ÷ 0.09 =
13. 90 ÷ 0.36 =
14. 18.88 ÷ 1.6 =
15. Corrine has 9.6 pounds of trail mix to divide into 12 bags. How many
pounds of trail mix will go in each bag?
16. Michael paid $11.48 for sliced cheese at the deli counter. The cheese cost
$3.28 per pound. How much cheese did Michael buy?
17. A four-person relay team completed a race in 72.4 seconds. On average,
what was each runner’s time?
18. Elizabeth has a piece of ribbon that is 4.5 meters long. She wants to cut
it into pieces that are 0.25 meter long. How many pieces of ribbon will
she have?
19. Lisa paid $43.95 for 16.1 gallons of gasoline. What was the cost per gallon,
rounded to the nearest hundredth?
?
?
ESSENTIAL QUESTION CHECK-IN
21. When you are dividing two decimals, how can you check whether you
have divided the decimals correctly?
128
Unit 2
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20. One inch is equivalent to 2.54 centimeters. How many inches are there in
50.8 centimeters?
Name
Class
Date
5.4 Independent Practice
COMMON
CORE
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Use the table for 22 and 23.
Custom Printing Costs
Quantity
25
50
75
100
Mugs
$107.25
$195.51
$261.75
$329.00
T-shirts
$237.50
$441.00
$637.50
$829.00
22. What is the price per mug for 25 coffee mugs?
23. Find the price per T-shirt for 75 T-shirts.
A movie rental website charges $5.00 per month for membership and
$1.25 per movie.
24. How many movies did Andrew rent this month if this month’s bill was
$16.25?
25. Marissa has $18.50 this month to spend on movie rentals.
a. How many movies can she view this month?
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b. Critique Reasoning Marisa thinks she can afford 11 movies in one
month. What mistake could she be making?
Victoria went shopping for ingredients to make a stew. The table shows the
weight and the cost of each of the ingredients that she bought.
Ingredient
Weight (in pounds)
Cost
Potatoes
6.3
$7.56
Carrots
8.5
$15.30
4
$9.56
2.50
$1.25
Beef
Bell peppers
26. What is the price for one pound of bell peppers?
27. Which ingredient costs the most per pound?
28. What If? If carrots were $0.50 less per pound, how much would Victoria
have paid for 8.5 pounds of carrots?
Lesson 5.4
129
29. Brenda is planning her birthday party. She wants to have 10.92 liters of
punch, 6.5 gallons of ice cream, 3.9 pounds of fudge, and 25 guests at the
birthday party.
a. Brenda and each guest drink the same amount of punch. How many
liters of punch will each person drink?
b. Brenda and each guest eat the same amount of ice cream. How many
gallons of ice cream will each person eat?
c. Brenda and each guest eat the same amount of fudge. How many
pounds of fudge will each person eat?
To make costumes for a play, Cassidy needs yellow and white fabric that
she will cut into strips. The table shows how many yards of each fabric she
needs, and how much she will pay for those yards.
Fabric
Yards
Cost
Yellow
12.8
$86.40
White
9.5
$45.60
30. Which costs more per yard, the yellow fabric or the white fabric?
31. Cassidy wants to cut the yellow fabric into strips that are 0.3 yards wide.
How many strips of yellow fabric can Cassidy make?
FOCUS ON HIGHER ORDER THINKING
33. Analyze Relationships Constance is saving money to buy a new bicycle
that costs $195.75. She already has $40 saved and plans to save $8 each
week. How many weeks will it take her to save enough money to purchase
the bicycle?
34. Represent Real-World Problems A grocery store sells twelve bottles of
water for $13.80. A convenience store sells ten bottles of water for $11.80.
Which store has the better buy? Explain.
130
Unit 2
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32. Problem Solving Eight friends purchase various supplies for a camping
trip and agree to share the total cost equally. They spend $85.43 on
food, $32.75 on water, and $239.66 on other items. How much does each
person owe?
LESSON
5.5
?
Applying Operations
with Rational
Numbers
ESSENTIAL QUESTION
COMMON
CORE
6.NS.3
Fluently add, subtract,
multiply, and divide
multi-digit decimals….
How can you solve problems involving multiplication
and division of fractions and decimals?
Interpreting a Word Problem
When you solve a word problem involving rational numbers, you often need
to think about the problem to decide which operations to use.
EXAMPL 1
EXAMPLE
Problem
Solving
COMMON
CORE
6.NS.3
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Naomi earned $54 mowing lawns in two days. She worked 2.5 hours
yesterday and 4.25 hours today. If Naomi was paid the same amount for
every hour she works, how much did she earn per hour?
Analyze Information
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Alamy
Identify the important information.
• Naomi made $54 mowing lawns.
• Naomi worked 2.5 hours yesterday and 4.25 hours today.
• You are asked to find how much she earned per hour.
Formulate a Plan
• The total amount she earned divided by the total hours she worked
gives the amount she earns per hour.
• Use the expression 54 ÷ (2.5 + 4.25) to find the amount she earned
per hour.
Justify and Evaluate
Solve
Follow the order of operations.
(2.5 + 4.25) = 6.75
Add inside parentheses.
54 ÷ 6.75 = 8
Divide.
Naomi earned $8 per hour mowing lawns.
Justify and Evaluate
You added 2.5 and 4.25 first to find the total number of hours worked.
Then you divided 54 by the sum to find the amount earned per hour.
Lesson 5.5
131
YOUR TURN
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1. Casey buys 6.2 yards of blue fabric and 5.4 yards of red fabric. If the blue
and red fabric cost the same amount per yard, and Casey pays $58 for
all of the fabric, what is the cost per yard?
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Converting Fractions and Decimals
to Solve Problems
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Recall that you can use a number line to find equivalent fractions and decimals.
If a fraction and a decimal are equivalent, they are represented by the same
point on a number line.
EXAMPLE 2
COMMON
CORE
6.NS.3
Each part of a multipart question on a test is worth the same number of
points. The whole question is worth 37.5 points. Roz got _12 of the parts of
a question correct. How many points did Roz receive?
Solution 1
Convert the decimal to a fraction greater than 1.
75
1
_
× 37.5 = _12 × __
2
2
STEP 2
75
1
, or ___
.
Write 37.5 as 37__
2
2
Multiply. Write the product in simplest form.
75
1 __
_
× 75
= __
= 18 _34
4
2
2
3
Roz received 18__
points.
4
Solution 2
Math Talk
STEP 1
Mathematical Practices
Do the solutions give
the same result?
Explain.
Convert the fraction to a decimal.
1
_
× 37.5 = 0.5 × 37.5
2
STEP 2
Multiply.
0.5 × 37.5 = 18.75
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Unit 2
Roz received 18.75 points.
YOUR TURN
2. The bill for a pizza was $14.50. Charles paid for _53 of the bill. Show two ways
to find how much he paid.
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STEP 1
Name
Class
Date
5.5 Guided Practice
1. Bob and Cheryl are taking a road trip that is 188.3 miles. Bob drove _57 of
the total distance. How many miles did Bob drive? (Example 1)
2. The winner of a raffle will receive _34 of the $530.40 raised from raffle ticket
sales. How much money will the winner get? (Example 2)
5.5 Independent Practice
COMMON
CORE
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6.NS.3
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3. Chanasia has 8.75 gallons of paint. She wants to use _25 of the paint to
paint her living room. How many gallons of paint will Chanasia use?
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4. Harold bought 3 pounds of red apples and 4.2 pounds of green
apples from a grocery store, where both kinds of apples are $1.75
a pound. How much did Harold spend on apples?
Samuel and Jason sell cans to a recycling center that
pays $0.40 per pound of cans. The table shows the
number of pounds of cans that they sold for several days.
5. Samuel wants to use his earnings from Monday
and Tuesday to buy some batteries that cost
$5.60 each. How many batteries can Samuel buy?
Show your work.
Samuel’s cans
(pounds)
Jason’s cans
(pounds)
Monday
16.2
11.5
Tuesday
11.8
10.7
Wednesday
12.5
7.1
Day
6. Jason wants to use his earnings from Monday and Tuesday for online
movie rentals. The movies cost $2.96 each to rent. How many movies can
Jason rent? Show your work.
7. Multistep Samuel and Jason spend _34 of their combined earnings from
Wednesday to buy a gift. How much do they spend? Is there enough left
over from Wednesday’s earnings to buy a card that costs $3.25? Explain.
Lesson 5.5
133
8. Multiple Representations Give an example of a problem that could be
solved using the expression 9.5 × (8 + 12.5). Solve your problem.
Tony and Alice are trying to reduce the amount
of television they watch. For every hour they watch
television, they have to put $2.50 into savings. The
table shows how many hours of television Tony and
Alice have watched in the past two months.
Tony
Hours watched
in February
35.4
Hours watched
in March
18.2
Alice
21.8
26.6
9. Tony wants to use his savings at the end of March to buy video games.
The games cost $35.75 each. How many games can Tony buy?
10. Alice wants to use her savings at the end of the two months to buy
concert tickets. If the tickets cost $17.50 each, how many can she buy?
FOCUS ON HIGHER ORDER THINKING
Work Area
11. Represent Real-World Problems A caterer prepares three times as
many pizzas as she usually prepares for a large party. The caterer usually
prepares 5 pizzas. The caterer also estimates that each party guest will
eat _13 of a pizza. Write an expression that represents this situation. How
many party guests will the pizzas serve?
12. Explain the Error To find her total earnings for those two weeks, Nadia
writes 7.5 × 18.5 + 20 = $158.75. Explain her error. Show the correct
solution.
13. What If? Suppose Nadia raises her rate by $0.75 an hour. How many
hours would she need to work to earn the same amount of money she
made in the first two weeks of the month? Explain.
134
Unit 2
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Nadia charges $7.50 an hour for babysitting. She babysits 18.5 hours the
first week of the month and 20 hours the second week of the month.
MODULE QUIZ
Ready
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and Intervention
5.1 Dividing Whole Numbers
1. Landon is building new bookshelves for his
bookstore’s new mystery section. Each shelf can
hold 34 books. There are 1,265 mystery books.
How many shelves will he need to build?
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5.2 Adding and Subtracting Decimals
2. On Saturday Keisha ran 3.218 kilometers. On
Sunday she ran 2.41 kilometers. How much farther
did she run on Saturday than on Sunday?
5.3 Multiplying Decimals
3. Marta walked at 3.9 miles per hour for 0.72 hours.
How far did she walk?
Multiply.
4. 0.07 × 1.22
5. 4.7 × 2.65
5.4 Dividing Decimals
© Houghton Mifflin Harcourt Publishing Company
Divide.
6. 64 ÷ 0.4
7. 4.7398 ÷ 0.26
8. 26.73 ÷ 9
9. 4 ÷ 3.2
5.5 Applying Multiplication and Division
of Rational Numbers
10. Doors for the small cabinets are 11.5 inches long.
Doors for the large cabinets are 2.3 times as long as
the doors for the small cabinets. How many large
doors can be cut from a board that is 10 _12 feet long?
ESSENTIAL QUESTION
11. Describe a real-world situation that could be modeled by dividing two
rational numbers.
Module 5
135
MODULE 5 MIXED REVIEW
Personal
Math Trainer
Assessment
Readiness
Selected Response
1. Delia has 493 stamps in her stamp
collection. She can put 16 stamps on each
page of an album. How many pages can
she fill completely?
6. Nelson Middle School raised $19,950 on
ticket sales for its carnival fundraiser last
year at $15 per ticket. If the school sells
the same number of tickets this year but
charges $20 per ticket, how much money
will the school make?
A 30 pages
C
31 pages
A $20,600
C
B 32 pages
D 33 pages
B $21,600
D $30,600
2. Sumeet uses 0.4 gallon of gasoline each
hour mowing lawns. How much gas does
he use in 4.2 hours?
A 1.68 gallons
B 3.8 gallons
13 gallons
B 3.85 kilometers
3. Sharon spent $3.45 on sunflower seeds.
The price of sunflower seeds is $0.89 per
pound. How many pounds of sunflower
seeds did Sharon buy?
C
4.29 kilometers
D 5.39 kilometers
Mini-Task
A 3.07 pounds
B 3.88 pounds
4.15 pounds
D 4.34 pounds
4. How many 0.4-liter glasses of water does it
take to fill up a 3.4-liter pitcher?
A 1.36 glasses
C
B 3.8 glasses
D 8.5 glasses
8. To prepare for a wedding, Aiden bought 60
candles. He paid $0.37 for each candle. His
sister bought 170 candles at a sale where
she paid $0.05 less for each candle than
Aiden did.
a. How much did Aiden spend on candles?
8.2 glasses
5. Each paper clip is _34 of an inch long and
costs $0.02. Exactly enough paper clips are
laid end to end to have a total length of
36 inches. What is the total cost of these
paper clips?
A $0.36
C
B $0.54
D $1.20
Unit 2
7. Keri walks her dog every morning. The
length of the walk is 0.55 kilometer on each
weekday. On each weekend day, the walk
is 1.4 times as long as a walk on a weekday.
How many kilometers does Keri walk in
one week?
A 2.75 kilometers
D 16 gallons
C
$26,600
$0.96
b. How much did Aiden’s sister spend on
candles?
c. Who spent more on candles? How
much more?
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C
136
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UNIT 2
Study Guide
MODULE
?
4
Review
Operations with Fractions
Key Vocabulary
reciprocals (recíprocos)
ESSENTIAL QUESTION
How can you use operations with fractions to solve real-world
problems?
EXAMPLE 1
Add.
Subtract.
7 __
_
The GCF of 9 and 12 is 36.
+ 5
9 12
5 × 3 Use the GCF to make fractions
7×4
____
+ _____
9×4
12 × 3 with common denominators.
28 __
43
__
+ 15
= __
Simplify.
36
36 36
9
__
- _5
10 6
9×3
5×5
_____
- _____
10 × 3
6× 5
The GCF of 10 and 6 is 30.
27 __
2
__
- 25
= __
30
30 30
Simplify.
43
7
__
= 1__
36
36
1
2
__
= __
15
30
Use the GCF to make fractions
with common denominators.
EXAMPLE 2
Multiply.
A. _45 × _18
B. 2_14 × _15
4 × 1 __
____
= 4
5 × 8 40
Multiply numerators.
Multiply denominators.
4 ÷ 4
1
_____
= __
40 ÷ 4 10
Simplify by dividing by the GCF.
9 _
_
×1
4 5
Rewrite the mixed number as
a fraction greater than 1.
9 × 1 __
____
= 9
4 × 5 20
Multiply numerators.
Multiply denominators.
© Houghton Mifflin Harcourt Publishing Company
EXAMPLE 3
Divide.
A. _27 ÷ _12
2 _
_
×2
7 1
Rewrite the problem as
multiplication using the
reciprocal of the second
fraction.
2 × 2 _
____
=4
7 × 1 7
Multiply numerators.
Multiply denominators.
B. 2_13 ÷ 1_34
Write both mixed numbers
as improper fractions.
7 _
_
÷7
3 4
1
7×4
_____
= _4
3 × 71 3
Multiply by the reciprocal
of the second fraction.
1_13
4 = 1__
1
Simplify: __
3
3
EXERCISES
Add. Write the answer in simplest form. (Lesson 4.1)
1. _38 + _45
9
2. 1__
+ _34
10
6
3. _28 + __
12
Subtract. Write the answer in simplest form. (Lesson 4.1)
4.
1_37 - _45
5.
5
7 __
_
- 12
8
6.
5
3__
- _48
10
Unit 2
137
Multiply. Write the answer in simplest form. (Lesson 4.1)
7. _17 × _45
8. _56 × _23
14
9. _37 × __
15
10. 1_13 × _58
11. 1_29 × 1_12
12. 2_17 × 3_23
Divide. Write the answer in simplest form. (Lessons 4.2, 4.3)
13. _37 ÷ _23
14. _18 ÷ _34
15. 1_15 ÷ _14
16. On his twelfth birthday, Ben was 4 _34 feet tall. On his thirteenth
birthday, Ben was 5 _38 feet tall. How much did Ben grow between his
twelfth and thirteenth birthdays? (Lesson 4.1)
17. Ron had 20 apples. He used _25 of the apples to make pies. How many
apples did Ron use for pies? (Lesson 4.4)
18. The area of a rectangular garden is 38 _14 square meters. The width of
the garden is 4 _12 meters. Find the length of the garden. (Lesson 4.4)
MODULE
?
5
Operations with Decimals
Key Vocabulary
order of operations (orden
de las operaciones)
ESSENTIAL QUESTION
EXAMPLE 1
To prepare for a race, Lloyd ran every day for two weeks. He ran a
total of 67,592 meters. Lloyd ran the same distance every day. He
took a two-day rest and then started running again. The first day
after his rest, he ran the same distance plus 1,607.87 meters more.
How far did Lloyd run that day?
Step 1 Divide to see how far Lloyd ran every day during the two weeks.
4,828
⎯
14⟌ 67,592
Lloyd ran 4,828 meters a day.
Step 2 Add 1,607.87 to 4,828 to find out how far Lloyd ran the first day after his rest.
1,607.87
+ 4,828.00
6,435.87
138
Unit 2
Lloyd ran 6,435.87 meters that day.
© Houghton Mifflin Harcourt Publishing Company
How can you use operations with decimals to solve real-world
problems?
EXAMPLE 2
Rebecca bought 2.5 pounds of red apples. The apples cost $0.98 per
pound. What was the total cost of Rebecca’s apples?
2.5 ← 1 decimal place
× .98 ← + 2 decimal places
200
+ 2250
2.450 ← 3 decimal places
The apples cost $2.45.
EXAMPLE 3
Rashid spent $37.29 on gas for his car. Gas was $3.39 per gallon.
How many gallons did Rashid purchase?
Step 1 The divisor has two decimal places,
so multiply both the dividend and
the divisor by 100 so that the divisor
is a whole number:
⎯
⎯
3.39⟌ 37.29
339⟌ 3729
Step 2 Divide:
11
⎯
339⟌ 3729
−339
339
−339
0
Rashid purchased 11 gallons of gas.
EXERCISES
Add. (Lesson 5.2)
1. 12.24 + 3.9
2. 0.986 + 0.342
3. 2.479 + 0.31
5. 7.285 - 0.975
6. 14.31 - 13.41
Subtract. (Lesson 5.2)
© Houghton Mifflin Harcourt Publishing Company
4. 6.19 - 3.05
Multiply. (Lesson 5.3)
7.
12
×0.4
8.
0.15
× 9.1
9.
3.12
×0.25
Divide. (Lessons 5.1, 5.4)
10. 78,974 ÷ 21
11. 19,975 ÷ 25
12. 67,396 ÷ 123
⎯
13. 5⟌ 64.5
⎯
14. 0.6⟌ 25.2
⎯
15. 2.1⟌ 36.75
16. A pound of rice crackers costs $2.88. Matthew purchased _14 pound
of crackers. How much did he pay for the crackers? (Lesson 5.5)
Unit 2
139
Unit 2 Performance Tasks
1.
Chef Chef Alonso is creating a recipe called
Spicy Italian Chicken with the following ingredients: _34 pound chicken,
2_12 cups tomato sauce, 1 teaspoon oregano, and _12 teaspoon of his
special hot sauce.
CAREERS IN MATH
a. Chef Alonso wants each serving of the dish to include _12 pound of
chicken. How many _12 pound servings does this recipe make?
b. What is the number Chef Alonso should multiply the amount of
chicken by so that the recipe will make 2 full servings, each with
_1 pound of chicken?
2
c. Use the multiplier you found in part b to find the amount of all the
ingredients in the new recipe.
d. Chef Alonso only has three measuring spoons: 1 teaspoon,
_1 teaspoon, and _1 teaspoon. Can he measure the new amounts of
4
2
oregano and hot sauce exactly? Explain why or why not.
2. Amira is painting a rectangular banner 2_14 yards wide on a wall
in the cafeteria. The banner will have a blue background. Amira
has enough blue paint to cover 1_12 square yards of wall.
b. The school colors are blue and yellow, so Amira wants to add yellow
rectangles on the left and right sides of the blue rectangle. The
yellow rectangles will each be _34 yard wide and the same height as
the blue rectangle. What will be the total area of the two yellow
rectangles? Explain how you found your answer.
c. What are the dimensions of the banner plus yellow rectangles? What
is the total area? Show your work.
140
Unit 2
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a. Find the height of the banner if Amira uses all of the blue
paint. Show your work.
UNIT 2 MIXED REVIEW
Personal
Math Trainer
Assessment
Readiness
6. What is the absolute value of -36?
Selected Response
1. Each paper clip is _78 of an inch long and
costs $0.03. Exactly enough paper clips are
laid end to end to have a total length of
56 inches. What is the total cost of these
paper clips?
A $0.49
C $1.47
B $0.64
D $1.92
2. Which of these is the same as _89 ÷ _23 ?
8
3
A _÷_
9
2
2 _
_
÷ 89
B
3
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8
2
C _×_
9
3
8 _
3
_
×2
D
9
3. A rectangular tabletop has a length of 4_34
feet and an area of 11_7 square feet. What is
A -36
B 0
C 6
D 36
7. Noelle has _56 of a yard of purple ribbon
9
and __
of a yard of pink ribbon. How much
10
ribbon does she have altogether?
11
A 1__ yards
15
4
_
B 1 yards
5
1
C 2_ yards
5
14
__
D 1 yards
16
8. Apples are on sale for $1.20 a pound. Logan
bought _34 of a pound. How much money did
he spend on apples?
8
the width of the tabletop?
A $0.75
C $0.90
1
A 1__ feet
16
1
B 2_ feet
2
1
C 4_ feet
4
1
D 8_ feet
2
B $0.80
D $1.00
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4. Dorothy types 120 words per minute. How
many words does Dorothy type in 1.75
minutes?
A 150 words
B 180 words
9. Samantha bought 4.5 pounds of pears.
Each pound cost $1.68. How much did
Samantha spend in all?
A $7.52
C $8.40
B $7.56
D $75.60
10. Gillian earns $7.50 an hour babysitting on
the weekends. Last week she babysat for
2.2 hours on Saturday and 3.5 hours on
Sunday. How much did Gillian earn?
C 200 words
A $4.25
C $42.75
D 210 words
B $40.25
D $427.50
5. What is the opposite of 17?
A -17
1
B -__
17
1
C __
17
D 17
11. Luis made some trail mix. He mixed 4_23 cups
of popcorn, 1_14 cups of peanuts, 1_13 cups of
raisins, and _34 cup of sunflower seeds. He
gave 5 of his friends an equal amount of
trail mix each. How much did each friend
get?
3
A 1_5 cups
2
B 1_ cups
3
3
C 1_4 cups
D 2 cups
Unit 2
141
A 5.01 miles
C 5.60 miles
B 5.06 miles
D 5.65 miles
13. Landon drove 103.5 miles on Tuesday,
320.75 miles on Wednesday, and 186.30
miles on Thursday. How far did Landon
drive all three days combined?
A 61.55 miles
C 610.55 miles
B 610.055 miles
D 6,105.5 miles
a. On Saturday, _18 of the people who visited
were senior citizens, _18 were infants, _14
were children, and _12 were adults. How
many of each group visited the zoo on
Saturday?
Senior Citizens:
Infants:
Children:
Adults:
1
b. On Sunday, __
of the people who visited
16
3
were senior citizens, __
were infants, _38
16
3
were children, and _8 were adults. How
many of each group visited the zoo on
Sunday?
Mini Task
Senior Citizens:
14. Carl earns $3.25 per hour walking his
neighbor’s dogs. He walks them _13 of an
hour in the morning and _12 of an hour in the
afternoon.
Infants:
a. How much time does Carl spend dog
walking every day?
b. How much time does Carl spend dog
walking in a week?
c. Ten minutes is equal to _16 of an hour.
How many minutes does Carl work dog
walking each week?
Children:
Adults:
c. The chart shows how much each type of
ticket costs.
Type of Ticket
Cost
Infants
Free
Children Over 2
$4.50
Adults
$7.25
Senior Citizens
$5.75
d. How much money did the zoo make on
Saturday? Show your work.
d. How much money does Carl earn each
week?
15. The city zoo had an equal number of
visitors on Saturday and Sunday. In
all, 32,096 people visited the zoo that
weekend. How many visited each day?
142
Unit 2
e. How much did the zoo make on Sunday?
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12. Emily cycled 20.25 miles over 4 days last
week. She cycled the same amount each
day. How many miles did Emily cycle each
day to the nearest hundredth?