3
Here’s What I
Need to Know
Standard ALG5.0 Students solve
multistep problems, including word
problems, involving linear equations in
one variable and provide justification
for each step (excluding inequalities).
Standard 7NS1.2 Add, subtract,
multiply, and divide rational numbers
(integers, fractions, and terminating
decimals) and take positive rational
numbers to whole-number powers.
Standard 7AF4.1 Solve two-step
linear equations and inequalities in
one-variable over the rational
numbers, interpret the solution or
solutions in the context from which
they arose, and verify the
reasonableness of the results.
Vocabulary Preview
fraction A number that
represents part of a whole
or group.
1
_
3
1
3
rational number A number
that can be written in the form
a
_ (b ≠ 0).
b
5
_
6
_ or 7
_
3.5
31
2
2
-3
_
-3
1
simplest form A fraction
that does not have any
common factor other than
one in its numerator and
denominator.
160 Chapter 3 Fractions
The
What
I will learn to:
• add, subtract, multiply, and divide fractions and
mixed numbers.
• solve equations and problems with rational solutions.
The
Why
Fractional pieces are used to form the rhythm of music.
The length of sound can be represented by whole, half,
quarter, eighth, and sixteenth notes.
Robert E Daemmrich/Getty Images
Fractions
Option 1
Are you ready for Chapter 3? Take the Online
Readiness Quiz at www.ca.algebrareadiness.com to find out.
Option 2
Complete the Quick Check below.
Vocabulary and Skills Review
Match each sentence with the correct vocabulary word.
1.
2.
? means to find the value of a variable in
an equation.
? .
x + 2 = 4 is an example of a(n)
Addition is the ? of subtraction.
4
4. The number _ is an example of a(n)
3.
5
? .
A.
equation
B.
expression
C.
fraction
D.
inverse operation
E.
solve
Find the missing number.
5.
1×
?
= 24
6.
2×
?
= 24
7.
3×
?
= 24
8.
6×
?
= 24
Solve.
9.
x+5=8
10.
y_
=2
5
11.
4a + 1 = 13
12.
3a - 2 = 7
Note-taking
T ip s
s or
Use abbreviation
for
sy mbols to stand
ons.
words or operati
Chapter Instruction
My Notes
A fraction, such as 1_,
3
represents part of a whole
or group. The fraction - 1_
3
is a negative fraction.
1
fraction _
3 = part of whole
– _1 = negative fraction
3
Recall that both -1 and 3 are
integers. However, - 1_ is not.
3
So a new group of numbers
is needed to define - 1_.
3
This group is called
the rational numbers.
Chapter 3
161
Explore
3-1a
Vocabulary
Math Lab
Represent Fractions
The
What:
I will color shapes to represent
different fractions.
The
Why:
Fractions are used to represent the
number of friends wearing jeans.
Pictures are useful tools for
illustrating fractions.
fraction (p. 162)
Materials
colored pencils or
markers (optional)
Standard 7MR2.5 Use a
variety of methods, such
as words, numbers,
symbols, charts, graphs, tables,
diagrams, and models to explain
mathematical reasoning.
Standard 6NS1.2 Interpret and
use ratios in different contexts
to show the relative size of two
quantities using appropriate
notations.
Fractions are numbers that represent parts of a set, or parts
of a whole. Four teenagers are standing in a cafeteria line.
Three teenagers are wearing jeans. What fraction represents
the number wearing jeans?
You can use set models, rectangular models, circle models,
and triangular models to represent fractions.
EXAMPLE
Use a Set Model
Create a set model that represents the fraction of
teenagers wearing jeans. Then, write the fraction.
Each block represents
one classmate. There are
4 classmates, so draw 4
blank blocks.
Shade 3 blocks to show the
3 classmates wearing jeans.
3 out of 4 classmates, or 3_, are wearing jeans.
4
162 Chapter 3 Fractions
EXAMPLE
Use a Rectangular Model
Create a rectangular model that represents the
fraction 3_.
4
Draw a rectangle.
Divide it into 4 equal parts.
EXAMPLE
Shade 3 sections to
show 3 out of 4 parts.
Use a Circle Model
Create a circle model that represents the fraction 3_.
4
Draw a circle.
Divide it into 4 equal parts.
EXAMPLE
Shade 3 sections to
show 3 out of 4.
Use Triangular Models
Create a triangular model that represents the
fraction 3_.
4
Draw a triangle with 3 equal sides.
Divide it into 4 equal parts.
Shade 3 sections to
show 3 out of 4.
Your Turn Pick a random group of 8 classmates. Create
set models, rectangular models, circle models, and
triangular models, to represent each of the following.
a.
What fraction of the group is wearing sneakers?
b.
What fraction of the group is wearing jeans?
c.
What fraction of the group has brown hair?
Lesson 3-1 Fractions
163
3-1
Vocabulary
Fractions
The
What:
I will compare and order fractions
using a number line.
The
Why:
Fractions can be used to show parts
of a whole, like the blue tiles in a floor.
fraction (p. 164)
rational number (p. 164)
additive inverse (p. 165)
Standard 6NS1.1
Compare and order
positive and negative
fractions, decimals, and mixed
numbers and place them on a
number line.
1
The fraction – _
3 can be
–1 or _
1
written as _
3
–3.
Mr. Chung tiled his kitchen floor. One out of every three tiles
was blue. The fraction 1_ represents the blue part of the floor.
3
A fraction, such as 1_, represents part of a whole or group.
3
Recall that integers like 5 and -5 are opposites, or additive
inverses. The opposite of 1_ is the fraction - 1_.
3
3
A new group of numbers is needed to define 1_ and - 1_. This
3
3
group is called the rational numbers. A rational number
can be written in the form _a where b ≠ 0.
b
WORK WITH A PARTNER Name three integers and
three fractions. Be sure that at least one of the fractions
is negative. Listen as your partner names three integers
and three fractions.
You can use a number line to show 1_ and - 1_.
3
1
-3
-1
164 Chapter 3 Fractions
1
3
0
1
3
EXAMPLE
Graph Fractions
Graph the fraction - _2 on a number line.
3
2
2
3
-3
-1
0
1
-1
Divide the distance
from 0 to -1
into thirds.
0
1
0
-1
Count back 2
sections from 0.
1
Label the point.
Your Turn Graph each fraction on a number line.
3
1
a. - _
b. - _
5
2
The fractions - 1_ and 1_ are opposites or additive inverses.
3
3
The sum of additive inverses is zero.
EXAMPLES
Find Additive Inverses
Write the additive inverse of each number.
1_
- 4_
2
3
The opposite of _1 is - 1_.
2
2
The opposite of - 4_ is 4_.
3 3
So, the additive inverse
of - 4_ is 4_.
So, the additive inverse
of 1_ is - 1_.
2
2
3
3
Your Turn Write the additive inverse of each number.
5
4
c. _
d. - _
5
6
The additive inverse of any number is the
same distance from 0 on the number line as
the number. The sign is different to indicate
the direction from 0.
1
1
3
-3
-1
0
1
EXAMPLE
Graph Additive Inverses
Write the additive inverse of _2 . Then graph the
3
number and its additive inverse.
2
2
3
-3
2
3
The additive inverse, - _, has the opposite sign.
-1
0
1
Your Turn Write the additive inverse of each number.
Then graph the number and its additive inverse.
3
1
e. _
f. - _
4
2
Lesson 3-1 Fractions
165
EXAMPLE
Compare Fractions
Use a number line to determine which number is
greater, - 1_ or - 3_.
4
A number to the right
on the number line is
greater than a number
to its left. Any number to
the left on the number
line is less than a number
to its right.
-1
0
4
1
- 1_ is greater than - 3_.
4
- 1_ > - 3_
4
4
4
Graph the fractions on the same
number line.
_ is to the right of - 3
_.
-1
4
4
Your Turn Use a number line to determine which
number is greater.
3
5
5
1
g. _ or _
h. - _ or - _
8
8
6
6
EXAMPLE
Order Fractions
Order the fractions - 5_, - 3_, - 7_ from least to greatest.
8
0
-1
8
8
Graph the fractions on the
same number line.
7 is farthest to the left.
-_
1
- 7_ is the least fraction.
8
8
- 3_ is the greatest fraction.
8
- _7 < - _5 < - _3
8
8
8
_ is farthest to the right.
-3
8
Your Turn Order each set of fractions from least to greatest.
3
2 5 3
4
2
i. _, _, _
j. - _, - _, - _
6 6 6
5
5
5
FASHION Marguerite changed the hem on her jeans
by - 1_8. She changed the hem on her slacks by - 3_8.
Which is the greater fraction?
-1
0
1
- _1 is greater than - _3.
8
8
3
1
_
_
- >8
8
Graph the fractions on the same
number line.
_ is to the right of - 3
_.
-1
8
8
Your Turn
k.
166 Chapter 3 Fractions
COOKING A chef changes the amount of tomatoes in a
recipe - 1_ cup and the amount of olives - 3_ cup. Which
4
4
is the greater fraction?
Voca b u la ry Re
vie w
Examples 1–7
(pages 163–164)
Example 1
(page 163)
fraction
rational numbe
r
additive inverse
VOCABULARY
1.
Explain how a positive fraction
and its opposite are alike and different.
Graph each fraction on a number line.
4
1
2. - _
3. _
5
8
Example 2
(page 163)
Write the additive inverse of each number.
5
1
4. _
5. - _
6
Example 3
(page 163)
9
Write the additive inverse. Then graph both fractions.
1
4
6. _
7. - _
8
Example 5
(page 164)
5
Use a number line to determine which number is greater.
3
1
2
2
8. _ or _
9. - _ or - _
3
Example 6
(page 164)
3
5
Order each set of fractions from least to greatest.
3 5 2
5
4
7
10. _, _, _
11. - _, - _, - _
7 7 7
Example 7 12.
(page 164)
9
F
For
See
Exercises Example(s)
15–18
19–22
23–26
27–30
31–34
37–38
1
2–3
4
5–6
7
8
F
unleaded fuel only
PLUMBER To install a new sink, Edna changed the length of one
pipe - 1_ , and the second pipe - 3_. Which is the greater fraction?
4
14.
9
E
unleaded fuel only
13.
9
TRANSPORTATION Winona has 2_ tank of gas in her car. Jackson
3
has 1_ tank of gas. Graph the numbers on the number line and
2
compare them.
E
HOMEWORK (%,0
5
4
Have a classmate tell you a number. Decide if the
number is an integer or a rational number.
Graph each fraction on a number line.
4
1
2
15. - _
16. - _
17. _
5
9
7
Write the additive inverse of each number.
5
3
2
19. _
20. _
21. - _
6
4
5
18.
_3
8
22.
- 6_
7
Write the additive inverse of each number. Then graph the number
and its additive inverse.
3
4
2
2
23. _
24. _
25. - _
26. - _
5
3
7
4
Lesson 3-1 Fractions
167
5
5
4
4
8
B.S.P.I./CORBIS
Use a number line to determine which number is greater.
3
3
2
4
1
1
27. _ or _
28. _ or _
29. - _ or - _
8
Order each set of fractions from least to greatest.
6
5
8
5
1 5 3
1
4
30. _ , _ , _
31. - _ , - _ , - _
32. - _ , - _ , - _
8 8 8
7
7
7
9
33.
Printed photographs are sensitive to
changes in temperature. A museum
keeps the storage room temperature
within a range of + 3_ °C to - 3_ °C. Graph
4
4
these fractions on a number line and
compare them.
34.
TEMPERATURE A scientist records
changes of - 7_ °C and - 3_ °C during a
8
8
science experiment. Which is the
greater fraction?
35.
36.
Write an
inequality comparing two negative
fractions, each with a 5 in the denominator.
9
9
PHOTOGRAPHY Many California
museums, including the Getty
Museum in Los Angeles, have
exhibits of photography collections.
H.O.T. Problem Identify any two fractions less than - 1_ and greater
8
than - 1_.
2
Choose the best answer.
37 Which rational number is greater
than - 3_ and less than -_1?
3
4
F - 5_
6
G - 1_
2
H - 1_
4
J
38 A scientist records these
temperature changes while cooling
a liquid. Which choice correctly
orders the numbers?
A - 7_ > - 4_ > - 2_ Hour Temperature
9
9
9
4
2
_
_
B - > - > -7_
9
9
9
2
7
4
C -_ >-_ >-_
9
9
9
D - _2 > -_4 > - _7
9
9
9
0
1
Change (°C)
- 4_
2
- 7_9
3
- 2_9
Solve. (Lesson 2-7)
1 = _k - 4
2x + 3 = 11
43.
SPORTS The Hawks scored 5 fewer than 3 times as many points as
the Tornadoes. The Hawks scored 37 points. How many points did
the Tornadoes score? (Lesson 2-7)
168 Chapter 3 Fractions
40.
5p - 1 = 19
_=8
2+n
39.
41.
3
42.
6
9
3-2
Fractions and Mixed Numbers
Vocabulary
The
What:
I will compare and order improper
fractions and mixed numbers.
The
Why:
You may use mixed numbers and
improper fractions when you
measure baking ingredients or build
a tree house.
proper fraction (p. 169)
improper fraction (p. 169)
mixed number (p. 169)
equivalent fraction
(p. 172)
Standard 6NS1.1
Compare and order
positive and negative
fractions, decimals, and mixed
numbers and place them on a
number line.
Paulina bakes banana bread. The recipe calls for 3_ or 1_1 cups
2
2
3_
1
_
of pecans. What kind of numbers are and 1 ?
2
2
In a proper fraction, the numerator is less than the
denominator. In an improper fraction, the numerator is
greater than or equal to the denominator.
1
4
3 halves or
proper fraction
3
2
improper fraction
PROPER FRACTIONS
IMPROPER FRACTIONS
5_
6
42
_
95
- 1_3
- 7_8
11
_
6
112
_
101
- 4_3
10
-_
10
You can rewrite an improper fraction as a mixed number.
Mixed numbers combine or “mix” an integer and a fraction.
Examples include 21_ and -6 2_.
2
5
3_
1
_
Paulina needs or 1 cups of pecans for her banana bread.
2
2
The number _3 is an example of an improper fraction. The
2
number 1_1 is an example of a mixed number.
2
Remember that any number that can be written in the form _a ,
b
where b ≠ 0, is a rational number. So improper fractions
and mixed numbers are rational numbers.
Lesson 3-2 Fractions and Mixed Numbers
169
WORK WITH A PARTNER Name five improper fractions.
Listen as your partner names five mixed numbers.
You can look at the numerator and denominator to decide if
a fraction is proper or improper.
EXAMPLES
Identify Proper and Improper Fractions
Identify each fraction as proper or improper.
5
7_
_
3
7
5>3
7=7
The numerator is greater than
the denominator.
The numerator is equal
to the denominator.
improper
improper
Your Turn Identify each fraction as proper or improper.
3
9
a. _
b. _
8
4
You can rewrite an improper fraction as a mixed number
using division.
EXAMPLES
You can also rewrite a
mixed number as an
improper fraction.
3 _21 is seven halves, or _27.
_.
4 _2 is fourteen thirds, or 14
3
3
Change Improper Fractions to
Mixed Numbers
Write each improper fraction as a mixed number.
7
7
_
or 7 halves
2
2
3R1
_
2 7
3 1_
2
3
+ 1 =3 1
2
2
Divide 7 by 2. numerator ÷ denominator
Write the answer as a mixed number.
Place the remainder of 1 over the divisor 2.
Your Turn
Write each improper fraction as a mixed number.
8
5
c. _
d. _
3
170 Chapter 3 Fractions
2
You can use a number line to graph improper fractions and
mixed numbers.
EXAMPLE
Improper Fractions and Mixed Numbers
Graph 2 3_ and 5_ on a number line.
4
4
_ as 1 1
_
Rewrite 5
4
4
Divide each unit into fourths.
-3 -2 -1
0
1
3
2
3
Starting from zero, count to
the integer. Then count to the
fractional increment.
1
24
-1 4
-2
-1
-3 -2 -1
0
1
2
3
-3 -2 -1
0
1
2
3
Your Turn Graph each pair of numbers on a number line.
1 5
1 4
e. 4 _, _
f. -z _ , _
z z
3 3
To compare improper fractions and mixed numbers, graph
them on a number line. The number to the right is greater.
Compare Numbers
CONSTRUCTION Walt uses 1 _38 inch nails to attach
17 inch nails to attach paneling. Which
drywall and _
8
nails are longer?
Drywall
Paneling
-3
-2
–1
1 38
0
1
2 18
2
3
17
1
Write _
as 2 _
.
8
8
1
_ on a number line.
Graph 2 _
and 1 3
8
8
2 1_ > 1 3_
8
8
The paneling nails are longer.
Your Turn
g.
BAKING A recipe calls for 1 3_ cups of raisins and 2 1_ cups
4
4
of walnuts. Does the recipe use more raisins or walnuts?
Lesson 3-2 Fractions and Mixed Numbers
171
Suppose a recipe calls for 1_ teaspoon salt. Your only measuring
2
spoon is 1_ teaspoon. How many 1_ teaspoons of salt do you
4
4
need to equal 1_ teaspoon?
2
1
2
2
4
Notice the green and purple shaded portions are equal.
The fractions 1_ and 2_ are equivalent.
2
4
Equivalent fractions, such as 1_ and 2_, have the same value.
2
EXAMPLES
4
Show Equivalent Fractions
Use a model to show each pair of fractions is equivalent.
2
_, 4_
3 6
The fractions _2 and 4_ are equivalent.
3
6
10, 5
_
_
8 4
10
The fractions _
and 5_ are equivalent.
8
4
Your Turn Use a model to show each pair of fractions is
equivalent.
3 6
2 4
h. _, _
i. _, _
4 8
172 Chapter 3 Fractions
5 10
Voca b u la ry Re
vie w
Examples 1–9
(pages 169–172)
Examples 1–2
(page 170)
VOCABULARY
1.
Write an example of a proper
fraction and an improper fraction.
2.
What is a mixed number? How are
mixed numbers related to improper fractions?
Identify each fraction as proper or improper.
3
7
3. - _
4. _
2
Examples 3–4
(page 170)
(page 171)
8
Write each improper fraction as a mixed number.
8
9
5. _
6. _
5
5
7. _
4
Example 5
proper fraction
improper fraction
mixed number
equivalent fract
ion
2
12
8. _
5
Graph each pair of numbers on a number line.
1 5
1 7
9. 1 _, _
10. -2 _, _
3 3
3 5
11. 2 _, _
4 4
Example 6
(page 171)
5 5
1 3
12. -3 _, _
2 2
3
13. FITNESS On Saturday, Benito walked 2 _ miles. On Sunday, he
10
9
walked 1 _
miles. On which day did he walk farther?
10
5
14. BAKING A recipe calls for _ cups of peanut butter. What is this
2
amount written as a mixed number?
Examples 7–8
(page 172)
Use a model to show each pair of fractions is equivalent.
3 6
1 2
15. _, _
16. _, _
3 6
9 18
17. _, _
2 4
19.
HOMEWORK (%,0
For
See
Exercises Example(s)
20–23
24–31
32–39
40–41
42–45
1–2
3–4
5
6
7
4 8
5 10
18. _, _
3 6
Listen as your partner tells you a mixed number. Tell
your partner a mixed number that is greater than that number.
Identify each fraction as proper or improper.
11
4
20. _
21. _
5
8
22. _
3
7
27
23. _
19
Write each improper fraction as a mixed number.
5
11
24. _
25. _
3
9
26. _
4
2
12
27. _
5
Lesson 3-2 Fractions and Mixed Numbers
173
29.
Graph each pair of numbers on a number line.
3 7
1 3
32. 2 _, _
33. 1 _, _
2 2
1 7
34. -3 _, _
3 3
7 9
36. 1 _, _
8 8
4 10
38. -3 _, _
5 5
4 4
1 8
35. -1 _, _
5 5
5 8
37. -2 _, _
6 6
3 _
12
_
39. 3 ,
8 8
1
11
40. COOKING A soup recipe uses 3 _ cups of carrots and _ cups of
4
4
celery. Does the recipe use more carrots or celery?
41.
CAREER CONNECTION A small
business owner is responsible for hiring
and managing employees, collecting bills
and taxes, advertising, maintaining
records, and much more.
An owner of a bakery has 4 _1 pounds
8
37
of coffee and _
pounds of tea in stock.
8
Does he have more coffee or tea?
Use a model to show each pair of
fractions is equivalent.
3 6
1 2
42. _ , _
43. _ , _
5 10
3 9
44. _ , _
2 6
46.
In California, a small
business can be certified
as small or micro,
depending on income and
number of employees.
7 14
5 10
45. _ , _
5 10
GEOGRAPHY Different parts of the United States have different
time zones. California is in the Pacific time zone. Parts of
Newfoundland, Canada use Newfoundland time, which is
-3 1_ hours from standard Greenwich time. Graph this number.
2
Then compare it to -5 hours, the reference for the Eastern
time zone.
Newfoundland
Greenwich
Meridian
174 Chapter 3 Fractions
Dwayne Newton/PhotoEdit
_9
7
13
31. _
6
_7
4
15
30. _
7
28.
TEMPERATURE Lois finds that she can save $500 by setting the
store’s thermostat 1 1_ or 3_ degrees lower. Write this number as a
2
2
mixed number.
21
48. ARCHITECTURE An architect widens a doorway by _ inches.
8
Write this as a mixed number.
47.
49.
Write a paragraph explaining how to show
two fractions are equivalent.
50.
H.O.T. Problem Write a mixed number less than -2 and greater
than -3.
Choose the best answer.
51 Which mixed number is less than
-2 3_ and greater than -3 1_?
2
4
-2 _1
2
-2 7_
8
-3 5_
8
-3 3_
4
A
B
C
D
52 A scientist records these temperature
changes while studying a reaction.
Which choice correctly orders the
mixed numbers from least to greatest?
Hour Temperature
Change (°C)
1
-1 _1
2
-2 1_2
-1 3_4
2
3
F -1 3_, -1 1_, -2 1_
2
2
4
3_
1
_
G -1 , -1 , -2 1_
2
2
4
3
1
H -2 _, -1 _, -1 1_
2
2
4
J -2 _1, -1 _1, -1 _3
2
2
4
Use a number line to determine which number is greater. (Lesson 3-1)
6
5
5
3
8
7
54. - _ or - _
55. - _ or - _
53. - _ or - _
6
8
4
8
9
8
Find each quotient. (Lesson 2-5)
56.
54 ÷ -9
59.
WEATHER Suppose it rained 7 inches in April last year. The change
for the same month this year was -3 inches. How many inches did it
rain in April this year? (Lesson 2-3)
57.
-56 ÷ -8
58.
63 ÷ 7
Lesson 3-2 Fractions and Mixed Numbers
175
3
Progress Check 1
(Lessons 3-1 and 3-2)
Vocabulary and Concept Check
additive inverses (p. 165)
equivalent fraction (p. 172)
fraction (p. 164)
improper fraction (p. 169)
mixed number (p. 169)
proper fraction (p. 169)
rational number (p. 164)
Choose the term that best completes each statement. (Lessons 3-1 and 3-2)
1.
A(n)
numerator.
?
has a denominator that is greater than the
The fractions 1_ and - 1_ are
3
3
1
3. 3 _ is an example of a(n)
?
2.
2
?
.
.
Skills Check
Write the additive inverse. Then, graph the number and its additive
inverse. (Lesson 3-1)
4
1
4. - _
5. _
5
4
Use a number line to determine which number is greater. (Lesson 3-1)
3
5
2
1
6. - _ and - _
7. - _ and - _
3
3
8
8
Graph each pair of numbers on a number line. (Lesson 3-2)
3 6
5
2
8. 2 _, - _
9. -1 _, _
3
3
8
1
10. -1 _, _
3 3
4 4
3_
9
11. 3 , - _
4
4
Use a model to show each pair of fractions is equivalent. (Lesson 3-2)
2 4
7 14
12. _, _
13. _, _
5 10
2 4
Problem-Solving Check
TEMPERATURE During the first hour of a storm, the temperature
changes - 1_°F. During the second hour, the temperature changes - 3_°F.
4
4
Graph these fractions on a number line and compare them. (Lesson 3-1)
5
11
15. COOKING A chef uses 1 _ cups of onions and _ cups of peppers in a
8
8
sauce. Which of these amounts is greater? (Lesson 3-2)
3
16. REFLECT Explain how you would graph the fraction - _. (Lesson 3-1)
14.
4
176 Chapter 3 Fractions
3-3
Factors and
Simplifying Fractions
Vocabulary
The
What:
I will identify prime and composite
numbers, find greatest common
factors, and simplify fractions.
The
Why:
Simplifying fractions can make it
easier to compare statistics, such as
the number of students in a group
who belong to one class.
prime number (p. 177)
composite number
(p. 177)
factor (p. 177)
common factor (p. 179)
greatest common factor
(GCF) (p. 179)
equivalent form (p. 179)
simplest form (p. 179)
The principal of a local high school is writing about student
21 of student government members
activities. She writes that _
Standard 5NS1.4
Determine the prime
factors of all numbers
through 50 and write the
numbers as the product of their
prime factors by using exponents
to show multiples of a factor (e.g.,
2 4 = 2 × 2 × 2 × 3 = 2 3 × 3).
are seniors. How could this fraction be written in simplest form?
28
Any whole number greater than 1 is either a prime number or
a composite number. A prime number has exactly two
factors, 1 and itself. A composite number has more than two
factors. The numbers 0 and 1 are neither prime nor composite.
When you factor a number, you list all of the numbers that
can be multiplied together, two at a time, to get that number.
For example, 6 = 1 × 6 and 2 × 3, so the factors of 6 are
1, 2, 3, and 6.
PRIME NUMBERS
COMPOSITE NUMBERS
Numbers
Factors
Number
Factors
2
1, 2
6
1, 2, 3, 6
5
1, 5
9
1, 3, 9
7
1, 7
10
1, 2, 5, 10
11
1, 11
16
1, 2, 4, 8, 16
23
1, 23
21
1, 3, 7, 21
29
1, 29
28
1, 2, 4, 7, 14, 28
Lesson 3-3 Factors and Simplifying Fractions
177
EXAMPLES
Identify Prime and Composite Numbers
Identify each number as prime or composite.
3
12
factors: 1, 3
factors: 1, 2, 3, 4, 6, 12
The only factors are 1 and
the number.
There are more than
2 factors.
So, 3 is prime.
So, 12 is composite.
Your Turn Identify each number as prime or composite.
a.
6
b.
5
c.
11
d.
9
One way to find the factors for a number is to make an
organized list. List all of the numbers from 1 to half of the
number. If a number is a factor of the number, then write the
other factor. If it is not a factor of the number, or if that factor
is already listed, cross it off.
EXAMPLES
List Factors
List the factors of each number.
8
12
List the numbers from
1 to half of 8.
Decide if each number
is a factor.
List the numbers from
1 to half of 12.
Decide if each number
is a factor.
1
8
2
4
3 Not a factor
4 Already listed
The factors of 8 are
1, 2, 4, and 8.
1
12
2
6
3
4
4 Already listed
5 Not a factor
6 Already listed
The factors of 12 are
1, 2, 3, 4, 6, and 12.
Your Turn List the factors of each number.
e.
8
f.
9
g.
11
h.
14
WORK WITH A PARTNER List the factors of 18. Listen
as your partner lists the factors of 24.
178 Chapter 3 Fractions
If two or more numbers have the same factors, they are said to
have common factors. To find the common factors, list the
factors of each number. Then circle the factors common to
each list.
EXAMPLE
Find Common Factors
Find the common factors of 10 and 15.
10: 1, 2, 5, 10 List the factors of 10.
15: 1, 3, 5, 15 List the factors of 15.
10: 1, 2, 5, 10
15: 1, 3, 5, 15 Circle the common factors.
The common factors are 1 and 5.
Your Turn Find the common factors of each set of numbers.
i.
4 and 10
j.
6 and 18
The greatest of the common factors is called the greatest
common factor (GCF). To find the greatest common factor,
list the common factors for each number. Then circle the
greatest one.
EXAMPLE
Find the Greatest Common Factor
Find the greatest common factor (GCF) of 8 and 20.
Two or more numbers may
have many common
factors, but only one
greatest common factor.
8: 1, 2, 4, 8
20: 1, 2, 4, 5, 20
8: 1, 2, 4, 8
20: 1, 2, 4, 5, 20
8: 1, 2, 4, 8
20: 1, 2, 4, 5, 20
List the factors.
Underline the common factors.
Circle the greatest common factor.
Your Turn Find the greatest common factor (GCF) of the
numbers.
k.
12 and 18
l.
7 and 14
Equivalent fractions, such as 1_ and 2_, have the same value.
2
4
A fraction is in simplest form when the greatest common
1
2
2
4
factor of the numerator and denominator is 1.
Lesson 3-3 Factors and Simplifying Fractions
179
The fraction representing the number of seniors in the student
21, is not in simplest form. The greatest common
government, _
28
21.
factor of 21 and 28 can be used to simplify _
28
To write fractions in simplest form, start by finding the
greatest common factor of the numerator and denominator.
Then, divide the numerator and denominator by the GCF.
Factors of 21:
Factors of 28:
greatest common factor
EXAMPLE
Write Fractions in Simplest Form
8 in simplest form.
Write _
12
8: 1, 2, 4, 8
12: 1, 2, 3, 4, 6, 12 List the factors. Find the GCF.
Divide both numerator and denominator
8÷4
__
= 2_
12 ÷ 4
3
by the GCF.
8
So, _
= 2_ in simplest form.
12
3
Your Turn Write each fraction in simplest form.
6
12
m. _
n. _
9
20
SPORTS A football team made 18 of the last 24 extra
points it attempted. What was their success rate as a
fraction in simplest form?
18: 1, 2, 3, 6, 9, 18
List the factors. Find the GCF.
24: 1, 2, 3, 4, 6, 8, 12, 24
Divide both numerator and
18 ÷ 6 3_
__
=
24 ÷ 6
4
denominator by the GCF.
The football team’s success rate in simplest form for _3 or
4
3 out of 4.
Your Turn
o.
180 Chapter 3 Fractions
At Star Middle School, 60 out of 150 students are 8th
graders. Write the number of 8th graders as a fraction in
simplest form.
Voca b u la ry Re
vie w
Examples 1–7
(pages 178–180
Examples 1–2
(page 178)
Examples 3–4
(page 178)
Example 5
(page 179)
Example 6
(page 179)
Example 7
(page 180)
VOCABULARY
1.
Write an example of a prime
number and explain how you
know it is a prime number.
Repeat for a composite number.
2.
What is the greatest common
factor of two numbers?
3.
How do you know when a fraction is in simplest form?
Identify each number as prime or composite.
4.
16
5.
(page 180)
8.
For
See
Exercises Example(s)
25–32
33–40
41–46
47–52
53–60
61–64
1–2
3–4
5
6
7
8
6.
21
7.
27
2
9.
10
10.
16
11.
27
Find the common factors of each set of numbers.
12.
6 and 12
13.
5 and 8
14.
12 and 14
Find the greatest common factor (GCF) of each set of numbers.
15.
4 and 16
16.
7 and 10
17.
Write each fraction in simplest form.
6
5
12
18. _
19. _
20. _
8
16
3 and 18
21.
8
_
24
SPORTS Della shot 16 free throws in her last basketball game.
She made 10 of them. What is her success rate as a fraction in
simplest form?
3
23. PACKAGING A company finds that 3 of 18 boxes it ships, or _ ,
18
arrive damaged. What is this fraction in simplest form?
22.
24.
HOMEWORK (%,0
19
List the factors of each number.
10
Example 8
prime number
composite num
ber
factor
common factor
greatest comm
on factor
(GC F )
simplest form
Have a partner say a number. Decide if it is prime or
composite. Name another number that is in the same category.
Then challenge your partner to do the same.
Identify each number as prime or composite.
25.
2
26.
10
27.
15
28.
19
29.
24
30.
25
31.
29
32.
33
List the factors of each number.
33.
8
34.
14
35.
18
36.
21
37.
23
38.
25
39.
30
40.
32
Lesson 3-3 Factors and Simplifying Fractions
181
41.
3 and 9
42.
12 and 16
43.
15 and 20
44.
18 and 33
Find the greatest common factor (GCF) of the numbers.
45.
6 and 12
46.
8 and 15
47.
15 and 30
Write each fraction in simplest form.
6
12
12
49. _
50. _
51. _
15
53.
36
18
48.
16 and 24
52.
8
_
32
POPULATION In one city, there are 4 middle schools
4 . What is this fraction
and 18 elementary schools or _
18
in simplest form?
FLAGS Zack is making an American flag. The white
cloth is 9 inches wide. The red cloth is 12 inches wide.
The stripes must all be the same width. If he does not
want to waste cloth, what is the widest the stripes can be? During the 2005FPO
school year, there
55. ART Tara has 18 red and 12 blue beads with which
were over 6 million
she wants to make bracelets. She wants to use all of
students studying in
the 9,372 California
her beads. She decides to put an equal number of
public schools.
beads and only one color on each bracelet. What is the
greatest number of beads she can use on each bracelet?
54.
Write three numbers that have a GCF of 4.
56.
57.
H.O.T. Problem Find the GCF of any two prime numbers.
Demonstrate why this pattern is true.
Choose the best answer.
58 The chart shows the results of a
school district survey. What
fraction, in simplest form, of the
total number of students in the
district return to the same school
as the year before?
13
A 2_
C _
3
B 3_
4
Status
Number
New to district
600
New school in district
200
Same school in district
2400
Total
3200
15
24
D _
32
Use a number line to determine which number is greater. (Lesson 3-2)
3
2
1
1
59. -2 _ or -3 _
60. -1 _ or -1 _
3
8
4
3
Solve each equation. (Lesson 2-7)
n
_+1=2
3
61.
2x - 5 = 3
63.
FINANCE A stock changed -$8 in the last 4 months. The price
changed the same amount each month. How much did it change the
first month? (Lesson 2-5)
182 Chapter 3 Fractions
62.
Amos Morgan/Getty Images; Tim Pannell/CORBIS
Find the common factors of each set of numbers.
Ed Murray/Star Ledger/CORBIS
3-4
Problem-Solving Strategy:
Draw a Diagram
Vocabulary
draw a diagram (p. 183)
Standard MR2.0
Students use strategies,
skills, and concepts in
finding solutions.
Standard MR2.5 Use a variety of
methods, such as words,
numbers, symbols, charts, graphs,
tables, diagrams, and models, to
explain mathematical reasoning.
Standard 6NS1.2 Interpret and
use ratios in different contexts
to show the relative size of two
quantities using appropriate
notations.
Use the Strategy
Seth is playing in a school chess
tournament. The tournament
starts with 8 players. The winner
of each match continues on to the
next round. What fraction of the
players (in simplest form) will
still be in the tournament in the
third round?
5NDERSTAND
What does Seth know?
• The number of chess players starting the tournament.
• Only winners advance.
Plan
What does Seth want to know?
Seth wants to compare the number of players in the third
round to the number that start the tournament. He wants to
write this comparison as a fraction in simplest form. Seth can
draw a diagram.
Seth creates a diagram for 8 players.
A diagram can be any
sketch, graph, chart or
picture that helps you
understand the
information given in
a problem.
Round 1
Player A
Player B
Player C
Player D
Player E
Player F
Player G
Player H
Round 2
Winner
Winner
Winner
Winner
Round 3
Round 4
Winner
Champion
Winner
So, 2_ of the players will be left. The fraction 2_ simplifies to 1_.
8
8
Check
4
Solve another way.
Each day, 1_ of the players are eliminated. On Day 1, there are 8
2
players. So on Day 2, there are 4 players and on Day 3 there
are 2 players.
2 of 8 is 2_ , or 1_.
8
4
The answer is correct.
Lesson 3-4 Problem-Solving Strategy: Draw a Diagram
183
Draw a diagram as you solve the problem.
1.
GAMES How many total games will be played in a tournament of
16 players if players are eliminated after losing one game?
5NDERSTAND
Plan
What do you know?
What are you trying to find out? How can you figure it out?
Draw a diagram to solve the problem.
Check
2.
How can you be sure that your answer is correct?
Gabriel is organizing a chess tournament for
64 players. Players will be eliminated after losing one game. Each
player plays only one game per day. Gabriel wants to know how
many days the tournament will last. Why is drawing a diagram a
good strategy for solving this problem?
Solve using the draw a diagram strategy.
3.
PHYSICS A ball is dropped from 32
meters above the ground. Each time it
hits the ground, it bounces up 1_ as high
2
as it fell. What fraction (in simplest
form) of the original height is the last
bounce up when it hits the ground the
third time?
4.
ENTERTAINMENT Abe, Curtis, and
Chet stand in line to buy movie tickets.
In how many different ways can they
stand in line?
5.
GENEALOGY Shawna made a list of her
mother’s parents, grandparents, great
grandparents, and great-great
grandparents. If none of these people
are stepparents, how many people are
listed? What fraction of the total, in
simplest form, are great grandparents?
Father
Solve using any strategy.
6.
SCHOOL Jaime scored 89 on his math
test. Questions are worth 5 points or
2 points. There is no partial credit.
Suppose Jaime answered 37 questions
correctly. How many 5-point questions
are correct?
7.
GEOMETRY The numbers shown here
are called rectangular numbers. How
many dots make up the eighth
rectangular number?
1
184 Chapter 3 Fractions
3
4
8.
NATURE A snail at the bottom of a
10-foot well climbs up 3 feet each day,
but slips back 2 feet at night. How many
days will it take the snail to reach the
top of the well and escape?
9.
BAKING A cake that is 9 inches by
9 inches will serve 9 people. How many
cakes that measure 12 inches by
12 inches will serve 48 people?
(Hint: the answer is not 4.)
Mother
Shawna
2
3
Progress Check 2
(Lesson 3-3)
Vocabulary and Concept Check
common factor (p. 179)
composite number (p. 177)
equivalent fraction (p. 179)
factor (p. 177)
greatest common factor
(GCF) (p. 179)
prime numbers (p. 177)
simplest form (p. 179)
Choose the term that best completes each statement.
1.
The
?
of 4 and 8 is 4.
?
When you
a number, you list all the numbers that can
be multiplied together, two at a time, to get that number.
1 4
?
3. The fraction _ is _ written in
.
2.
2
4.
8
The integers 2, 3, and 5 are examples of
?
.
Skills Check
Identify each number as prime or composite. (Lesson 3-3)
5.
7
6.
11
7.
9
8.
10
Find the greatest common factor (GCF) of the numbers. (Lesson 3-3)
9.
2 and 9
10.
2 and 12
11.
8 and 12
12.
6 and 9
Write each fraction in simplest form. (Lesson 3-3)
5
3
9
13. _
14. _
15. _
10
9
12
16.
8
_
20
Problem-Solving Check
17.
LANDSCAPING In Alma’s garden, 9 of the 12 trees are oak trees.
What fraction of the trees, in simplest form, are oaks? (Lesson 3-3)
ART Keira makes bracelets. She has 20 square beads and 15 round
beads. She puts one kind of bead on each bracelet. She puts the same
number of beads on each one. If she makes 7 bracelets, what is the
greatest number of beads she can use on each bracelet? (Lesson 3-3)
8
19. REFLECT In your own words, explain how to write _ in simplest
12
form. (Lesson 3-3)
18.
Chapter 3 Progress Check
185
3-5
Multiplying Fractions
Vocabulary
The
What:
I will multiply fractions and raise
fractions to positive powers.
The
Why:
You often multiply fractions when
you work with measurements, such
as when cooking or designing
interior spaces.
product (p. 186)
Standard 7NS1.2 Add,
subtract, multiply, and
divide rational numbers
(integers, fractions, and
terminating decimals) and take
positive rational numbers to
whole-number powers.
Standard 6NS2.1 Solve
problems involving addition,
subtraction, multiplication, and
division of positive fractions
and explain why a particular
operation was used for a given
situation.
Standard 6NS2.2 Explain the
meaning of multiplication, and
division of positive fractions and
perform the calculations.
About 1_ of Earth’s land can be used for farming. About 2_
3
5
of this farmland is used to grow grain crops. What part of
Earth’s land is used to grow grain?
Solve this problem by multiplying 1_ and 2_.
3
5
1
2 , about _
2 of
Since the product of the fractions _ and 2_ is _
3
5 15
15
Earth’s land is used to grow grain.
2_ of 1_ is _
2.
5
Teaching another person
is one of the best ways to
learn the skill yourself.
Explain the rule for
multiplying fractions to a
classmate.
3
15
RULES FOR MULTIPLYING FRACTIONS
Words
To multiply fractions multiply the numerators, then multiply the
denominators.
1×2
1
2
_=_
×2
=_
Numbers _
3
5
3×5
15
Pictures
2
5
2
5
1
3
1
3
Symbols
_.
For fractions a_ and _c (b ≠ 0, d ≠ 0), a_ · _c = ab
b
d
b d
cd
WORK WITH A PARTNER You and your partner each
name a proper fraction. Then, work together to find the
product of the two fractions.
186 Chapter 3 Fractions
EXAMPLES
Multiply Fractions
Find each product.
3_ × 3_
4 8
3_ 3_ _
9
× =3×3=_
4 8 4 × 8 32
Multiply numerators.
Multiply denominators.
_1 × _3 × 1_
5 4 2
1×3×1 _
_1 × _3 × 1_ = __
= 3 Multiply numerators.
5 4 2 5 × 4 × 2 40 Multiply denominators.
Your Turn Find each product.
1 1
a. _ × _
2
6
b.
_3 × 1_ × 3_
5 2 4
In previous examples, both products were in simplest form.
This is not true when common factors are involved. In these
cases, you have two options.
Option 1: Multiply, then simplify.
Option 2: Simplify, then multiply.
EXAMPLE
Write Products in Simplest Form
Multiply. Write each product in simplest form.
5_ × _
3
9
10
Option 1: Multiply first.
3
_5 × _
9 10
5×3 _
_
= 15
Multiply numerators.
Multiply denominators.
15 1_
_
=
Simplify.
9 × 10
90
90
6
Option 2: Simplify first.
5_ _
× 3
9 10
1
3
_5 × _
9 10 2
The GCF of 3 and 9 is 3.
Divide by 3.
1_ × 3_ 1
2
39
1 × 1 = 1_
_1 × 1_ = _
3 2 3×2 6
Multiply numerators.
Multiply denominators.
_1
6
Simplify.
Your Turn Multiply. Write each product in simplest form.
3 2
2 5
2 3
c. _ × _
d. _ × _
e. _ × _
3
6
4
5
9
4
Lesson 3-5 Multiplying Fractions
187
The rule you use for multiplying positive fractions also
applies to negative fractions.
EXAMPLES
Remember to also use
the sign rules for
multiplying integers.
positive (+) × positive (+)
= positive (+)
negative (–) × negative (–)
= positive (+)
positive (+) × negative (–)
= negative (–)
negative (–) × positive (+)
= negative (–)
Multiply Negative Fractions
Multiply. Write each product in simplest from.
- 4_ × 2_
3
5
8
4 × 2 = -_
- _4 × 2_ = - _
5 3
5×3
15
Negative × Positive = Negative
Multiply numerators.
Multiply denominators.
- 1_2 × - 3_
4
Negative × Negative = Positive
Multiply numerators.
Multiply denominators.
× 3 3_
- 1_ × - 3_ = 1_
=
2
4 2×4 8
Your Turn Multiply. Write each product in simplest form.
3
5
4
1
f. _ × - _
g. - _ × - _
5
3
4
8
To multiply positive or negative mixed numbers, change them
to improper fractions first. Then multiply.
EXAMPLES
Multiply Improper Fractions
Multiply. Write each product in simplest form.
1
_ × 5_
2
3
× 5 5_
1
_ × 5_ = 1_
=
2 3 2×3 6
Multiply numerators.
Multiply denominators.
9
_ × 5_
4 3
9×5 _
= 45
=_
Multiply numerators.
Multiply denominators.
45 ÷ 3
= __
Simplify.
4×3
12
12 ÷ 3
_
= 15 or 3 3_
4
4
Write the answer as a mixed number.
Your Turn Multiply. Write each product in simplest form.
3 5
3
9
7
7
h. _ × _
i. _ × - _
j. - _ × - _
4
188 Chapter 3 Fractions
4
2
4
3
4
NUTRITION Berta eats 2 1_ energy bars. Each bar has _12
4
of its calories from fat. What fraction of the total number
of bars she ate represents calories from fat? Draw a
diagram that illustrates the problem. Then, explain
how your diagram can be used to solve the problem.
Mixed numbers can be
written as improper
fractions.
2 _41 is nine fourths, or 9_4.
1 bar
1
bar
4
1 bar
The diagram shows the
total 2 1_ bars. The green
4
and purple sections each
show 1_ of the total bars.
2
You can see that the mixed
number 1 1_ represents half
8
of the bars.
1 1_ bars represent calories come from fat.
8
FITNESS Leon runs 1_2 lap around the track on Saturday.
How many laps does he run on Sunday if he runs 1_ the
4
distance? Explain how you solved the problem.
×1
= 1_
2×4
= 1_
8
Multiply numerators.
Multiply denominators.
Simplify.
Leon runs 1_ lap on Sunday.
8
Your Turn
k.
NUTRITION Cala eats 3 energy bars. Each bar has 1_ of its
4
calories from protein. What fraction of the total number
of bars she ate represents calories from protein? Draw a
diagram that illustrates the problem. Then, explain how
your diagram can be used to solve the problem.
()
You have learned x 3 = x · x · x. Likewise, the expression 3_
4
equals 3_ · 3_ · 3_.
4 4 4
3_ 3 3_ 3_ 3_
= · ·
(4)
EXAMPLE
3
4 4 4
Evaluate Fractions Raised to Powers
3
Evaluate _23 .
3
()
(_) = _ × _ × _
2
3
2
3
2
3
2
3
Use the base as a factor 3 times.
8
×2×2=_
= 2__
3×3×3
Your Turn Evaluate.
1 4
l. _
(2)
27
Multiply numerators.
Multiply denominators.
m.
( 3_4 )
3
Lesson 3-5 Multiplying Fractions
189
Voca b u la ry Re
vie w
product
Examples 1–10
(pages 187–189)
VOCABULARY
1.
Explain how to find each product of two proper fractions.
Multiply. Write each product in simplest form.
3
7
2 1 2
2 5
Examples 1–2 2. _ × _
3. _ × _ × _
4. _ × _
10
(page 187)
Example 3
3
5.
_3 × 4_
4 9
6.
_7 × - 3_
8
4
8.
_7 × - 2_
4
3
9.
_3 × 5_
2 3
(page 187)
Examples 4–5
5
4
(page 188)
Examples 6–7 11.
(page 188)
12.
3
5
_3 × 3_
2 2
COOKING A banana bread recipe uses 3_ cup of flour. How many
4
cups of flour are needed to make 1_ the recipe? Explain how you
2
solved this problem.
(3)
(page 189)
15.
14.
( 1_2 )
7
Discuss with a classmate how you would find the
product of the fractions _a and _c .
b
1–2
3
4–5
6–7
8–9
10
10.
- 2_ × - 1_
INTERIOR DESIGN The basement of a house is going to be 3_
4
1
_
finished as a recreation room. Carpeting will cover of that space.
Evaluate.
2 4
Example 10 13. _
15–18
19–22
23–26
27–30
31–34
35–38
7.
8
What fraction of the whole basement will be carpeted? Draw a
diagram that illustrates this situation. Then, explain how your
diagram can be used to solve the problem.
(page 189)
For
See
Exercises Example(s)
3
2
Examples 8–9
HOMEWORK (%,0
3
d
Multiply. Write each product in simplest form.
3 1
2 4
4 1 2
16. _ × _
17. _ × _
18. _ × _ × _
19.
22.
25.
28.
31.
4 8
3
_ × 1_ × 3_
4 2 4
9
_
× 2_
10 3
- _3 × _5
4 8
_7 × _1
4 2
20.
23.
26.
29.
3 5
9
2_ × _
3 10
3_ _
× 11
4 12
- 4_ × - 2_
5
3
_1 × _5
2 3
21.
24.
27.
30.
5 3 3
3_ 5_
×
5 9
1_ × - 3_
2
4
- 7_ × - 3_
8
5
11
_
3×
6
NUTRITION A new energy bar has 1_ of its calories from fat. 2_ of
4
3
those calories are unsaturated fat. What fraction of the total number
of calories comes from unsaturated fat? Draw a diagram that
illustrates this situation. Then, explain how your diagram can be
used to solve the problem.
190 Chapter 3 Fractions
CAREER CONNECTION Crop farmers grow, store, package, and
market crops. A farmer plants 3_ of his farm with apple trees.
4
Granny Smith apple trees make up 1_ of the apple trees. What
3
fraction of the entire farm is planted with Granny Smith apple
trees? Explain how you solved this problem.
2
33. BAKING A farmer uses _ cup of his apples to bake an apple cake.
3
How many cups of apples are needed to make 1_ the recipe? Explain
David Wells/The Image Works
32.
2
how you solved this problem.
California farmers
and ranchers
produce $73 million
in food, fiber, and
flowers every day.
Evaluate.
5
1
34. _
()
2
38.
35.
()
_2
3
4
()
_4
5
36.
4
( )
3
_
37.
3
10
Write a multiplication sentence with two
fractions that have a product of 1_.
2
39.
H.O.T. Problem One factor is _2. The product is 1_. Explain how to find
3
3
the other factor.
Choose the best answer.
40 Half of a garden will be planted
with flowers. Roses will make up 2_
3
of the flowers, and 1_ of the roses
2
will be yellow roses. What fraction
of the entire garden will be yellow
roses?
A 1_
C 1_
3
B 1_
6
8
1
D _
12
41 How many cups of blueberries are
needed to make 18 muffins?
Blueberry Muffins
Yield 16 muffins
Ingredients:
3
14
cup flour
1
cup sugar
tsp baking powder
3
4
cup milk
tsp salt
1
egg
cup butter
12
1
22
1
2
3
4
3
cup blueberries
Sift together flour, baking powder, and salt. Cream the butter
and sugar. Add milk and egg, beating well. Add to butter
mixture. Fold in the blueberries and flour. Pour the batter into
muffin tins. Bake at 400°F for 20 minutes.
F 3_
H 1 1_
2
4
G 1 1_
J
4
2
Find the greattest common factor of the numbers. (Lesson 3-3)
42.
4 and 10
46.
SPORTS This season, Jack scored 15 of 20 penalty shots in soccer.
Write his success rate as a fraction in simplest form. (Lesson 3-3)
43.
8 and 12
44.
2 and 5
45.
6 and 18
Lesson 3-5 Multiplying Fractions
191
3-6
Vocabulary
Dividing Fractions
The
What:
I will divide fractions and mixed
numbers.
The
Why:
Division of fractions is used for tasks
that involve measurements, like
building bookcases or making
stuffed animals.
reciprocal (p. 192)
multiplicative inverse
(p. 192)
Inverse Property of
Multiplication (p. 192)
Standard 7NS1.2 Add,
subtract, multiply, and
divide rational numbers
(integers, fractions, and
terminating decimals) and take
positive rational numbers to
whole-number powers.
Standard 6NS2.1 Solve
problems involving addition,
subtraction, multiplication, and
division of positive fractions
and explain why a particular
operation was used for a given
situation.
Liz has 3_ yard of wood to make a bookcase. Suppose she
4
divides the wood into 1_ -yard pieces. How many 1_ -yard
8
8
pieces are there in 3_ -yard of wood?
4
In this lesson you will learn how to use multiplication when
dividing fractions.
To divide fractions you need to use the reciprocal, or
multiplicative inverse. The reciprocal of 5_ is 7_.
7
5
Standard 6NS2.2 Explain the
meaning of multiplication and
division of positive fractions
and perform the calculations.
fraction
1
_
2
_
-3
4
reciprocal
2
_
1
_
-4
3
definition
_ (a, b ≠ 0),
For every number a
b
b
_
the reciprocal is a .
The Inverse Property of Multiplication states that the
product of a number and its multiplicative inverse is 1.
fraction
reciprocal
product
For every number _a (a, b ≠ 0), there is exactly one number
b
a
b
b
_
_
_
a such that b · a = 1.
192 Chapter 3 Fractions
EXAMPLES
All integers can be written
as rational numbers.
4 = 4_1
Find Reciprocals
Write the reciprocal of each fraction.
- 5_
4
6
Find the number that gives a
product of 1.
Write the integer as a
_
fraction. 4 = 4
1
- 5_ × - 6_ = 1
_4 × 1_ = 1
1 4
6
5
The reciprocal of - 5_ is - 6_.
6
The reciprocal of 4_ is 1_.
1
5
4
Your Turn Write the reciprocal of each fraction.
7
1
a. _
b. - _
8
6
WORK WITH A PARTNER Name a fraction. Have your
partner name its reciprocal. Repeat four more times.
Reciprocals and the Inverse Property of Multiplication are
used to divide fractions. To solve the example at the beginning
of the lesson, multiply 3_ by the reciprocal of 1_.
8
4
RULES FOR DIVIDING FRACTIONS
Words
To divide fractions, multiply the dividend by the reciprocal of the
divisor. In other words, multiply by the reciprocal of the second
fraction.
3 1 3 8 24
Numbers _4 ÷ _8 = _4 × _1 = _
4 =6
Pictures
3
yard
4
1
yard
4
1
yard
8
Symbols
1
yard
4
1
yard
8
1
yard
8
1
yard
4
1
yard
8
1
yard
8
1
yard
8
a·d
For fractions a_ and _c (b, c, d ≠ 0), a_ ÷ _c = a_ · _dc = _
b
d
b
d
b
b·d
Liz will have six 1_ -yard pieces of wood for her bookcase.
8
Lesson 3-6 Dividing Fractions
193
EXAMPLE
Divide Fractions
5 . Write each quotient in simplest form.
Find 7_8 ÷ _
12
5
7
12
_7 ÷ _
_
= ×_
8 12 8
5
12
= 7_ × _
28
3
5
21
=_
10
1
= 2_
Multiply by the reciprocal of the
second fraction.
The GCF of 8 and 12 is 4.
Divide 8 and 12 by 4
Multiply numerators.
Multiply denominators
Write as a mixed number.
10
Your Turn Divide. Write each quotient in simplest form.
1 1
2 5
c. _ ÷ _
d. _ ÷ _
3
2
3
EXAMPLE
6
Divide Negative Fractions
Find 2_5 ÷ - 3_. Write each quotient in simplest form.
When multiplying integers:
positive × positive
= positive
negative × negative
= positive
positive × negative
= negative
negative × positive
= negative
4
3
2
2
_ ÷ - _ = _ × - 4_
5
3
4 5
8
= -_
15
Multiply by the reciprocal of the
second fraction.
Multiply numerators.
Multiply denominators.
Your Turn Divide. Write each quotient in simplest form.
2
1
4
7
e. _ ÷ - _
f. - _ ÷ - _
7
3
5
8
When working with mixed numbers, change them to improper
fractions first. Then follow the process for dividing fractions.
EXAMPLE
Divide Improper Numbers
Find 5_2 ÷ 3_. Write each quotient in simplest form.
4
5
4
_
_
= ×
2 3
_
= 20
6
10
_
= = 3 1_
3
3
Multiply by the reciprocal of the second fraction.
Multiply numerators. Multiply denominators.
Simplify. Write the answer as a mixed number.
Your Turn Divide. Write each quotient in simplest form.
4 9
11 3
g. _ ÷ _
h. _ ÷ _
5
4
4
4
WORK WITH A PARTNER In your own words, explain
the process for dividing fractions. Listen as your partner
explains the process in his/her own words.
194 Chapter 3 Fractions
SEWING Clara has 5_ yard of fabric to make teddy
4
bears. Each bear requires 1_8 yard of fabric. How many
bears can Clara make? Draw a diagram that illustrates
the problem. Then, explain how your diagram can be
used to solve the problem.
5
yard
4
1
yard
4
1
yard
8
1
yard
4
1
yard
8
1
yard
8
1
yard
4
1
yard
8
1
yard
8
1
yard
8
1
yard
4
1
1
yard 8 yard
8
1
yard
4
1
yard
8
1
yard
8
The diagram shows the total length of fabric is 5_ yard.
4
It shows the 1_ -yard pieces needed for each bear. Count the
8
number of 1_ -yard pieces to solve the problem. Clara can
8
make 10 bears.
MASONRY A mason installs a 122-inch brick curb.
61 inches long. Suppose the bricks
Each brick is 7 5_8 or _
8
are laid end-to-end. How many bricks will the mason
use? Explain how you solved the problem.
Divide the curb length by the length
of one brick. Write the integer as a
122
fraction. 122 = _
1
61 _
61
122 ÷ _
= 122 ÷ _
8
1
8
8
122 × _
=_
1
61
2
8
122 × _
=_
1
61
Multiply by the reciprocal of the
second fraction.
The GCF of 122 and 61 is 61. Divide
122 and 61 by 61.
2×8 _
=_
= 16
Multiply numerators.
Multiply denominators.
= 16
Write the answer as an integer.
1×1
1
61
Divide the total 122 inches of the curb by the _
-inch brick.
8
61 _
8
122 by the reciprocal _
To divide, multiply _
. 122 × _
= 16
8
1
1
61
The mason will use 16 bricks.
Your Turn
i.
FITNESS A bike path is 2 kilometers long. There are
distance markers every 1_ kilometer. How many distance
4
markers are there? Draw a diagram that illustrates the
problem. Then, explain how your diagram can be used
to solve the problem.
Lesson 3-6 Dividing Fractions
195
Voca b u la ry Re
vie w
Examples 1–7
1.
What is a multiplicative inverse?
Explain using words.
2.
Explain the Inverse Property of
Multiplication in your own words.
(pages 193–195)
Examples 1–2
(page 193)
multiplicative in
verse
reciprocal
Inverse Propert
y of
Multiplication
Write the reciprocal of each fraction.
5
7
3. _
4. - _
10
4
Divide. Write each quotient in simplest form.
3
3
1 7
Example 3 5. _ ÷ _
6. _ ÷ _
2
(page 193)
Example 4
9
7.
- 3_ ÷ - 2_
9.
2 3_ ÷ 1 1_
7
(page 194)
Example 5
(page 194)
4
5
10
5
3
8.
5 ÷ 3_
-_
10.
-1 2_ ÷ 5_
12
3
4
6
PLUMBING A plumber has a 4 1_ -meter or 9_ -meter pipe. A new sink
2
2
requires a 1_ -meter pipe. How many 1_ -meter pipes can be cut from
2
2
the 9_ -meter pipe? Draw a diagram that illustrates the problem.
2
Then, explain how your diagram can be used to solve the problem.
3
12. INDUSTRY A sheet of paper is 6 _ inches wide. The sheet is divided
4
into 3 columns. How many inches wide is each column? Explain
how you solved the problem.
Examples 6–7 11.
(page 195)
13.
HOMEWORK (%,0
For
See
Exercises Example(s)
14–17
18–21
22–25
26–29
30–33
1–2
3
4
5
6–7
Write a division problem involving two fractions. Have
your classmate change your problem into a multiplication problem.
Listen as your classmate gives you a division problem to change.
Write the reciprocal of each fraction.
5
4
14. _
15. - _
12
7
16. - _
3
2
17.
19
Divide. Write each quotient in simplest form.
18.
20.
22.
24.
26.
28.
196 Chapter 3 Fractions
_2 ÷ 4_
7 7
5
_3 ÷ _
4 12
- 3_ ÷ - 1_
4
2
- _7 ÷ _1
8 3
15 _
_
÷5
8
4
_7 ÷ _7
3 2
19.
21.
23.
25.
27.
29.
_4 ÷ 2_
9 3
9
_3 ÷ _
5 10
5_
÷ - 5_
6
8
9 ÷ _3
-_
20 4
_3 ÷ _2
2 3
29 _7
_
÷
6
3
Michael Newman/PhotoEdit
30.
The construction
industry employs
approximately
800,000 workers
statewide.
CAREER CONNECTION Carpenters, plumbers, and masons make
and install materials for new buildings.
A mason cuts cement slabs into fourths to make blocks. The orignal
slab is 6 feet long. How long is each block? Draw a diagram that
illustrates the problem. Then, explain how your diagram can be
used to solve the problem.
2
12
31. PLUMBING Suppose a copper pipe is 2 _ or _ meters long. This pipe
5
5
is cut to make 4 equal pieces. How long is each piece? Draw a diagram
that illustrates the problem. Then, explain how your diagram can
be used to solve the problem.
1
32. COOKING Mary needs _ cup of hot sauce for a stew. Suppose each
2
bottle holds 1_ cup. How many bottles does she need? Explain how
6
you solved the problem.
33.
25
HORTICULTURE Kayla waters a tree with _
gallons of water once a
4
5_
week. Her watering can holds gallons of water. How many times
2
does she have to fill the watering can in a week? Explain how you
solved the problem.
34.
Write an equation involving two fractions
that have a quotient of 2_.
35.
H.O.T. Problem Let n represent a rational number between 0 and 1.
A number x is multiplied by n. The number x is then divided by n.
Which is greater: the product or the quotient? Explain.
3
Choose the best answer.
15
37 A newspaper is 7 1_ or _
inches wide
2
36 Payat needs 12 cups of
pretzels to make a
snack mix. Each bag
contains 2 3_ cups of
4
pretzels. How much
will the pretzels cost?
A $6.15
C $10.25
B $8.20
D $12.30
2
It is divided into 2 columns. The
left column is divided in half again
to list daily stock prices. How wide
is that column?
15 inches
15 inches
F _
H _
16
15 inches
G _
4
J
8
15 inches
_
2
Multiply. Write each product in simplest form. (Lesson 3-5)
3 2
3 7
3 4
1 5
38. _ × _
39. _ × _
40. - _ × _
41. _ × _
2
42.
6
4
9
8
5
2
3
TRANSPORTATION Half of Greg’s class is going on a field trip. The
bus will hold 2_ of those students. What fraction of the entire class
3
will ride the bus? (Lesson 3-5)
Lesson 3-6 Dividing Fractions
197
3
Progress Check 3
(Lessons 3-5 and 3-6)
Vocabulary and Concept Check
multiplicative inverse
product (p. 162)
(p. 192)
reciprocal (p. 192)
Inverse Property of
Multiplication (p. 192)
Choose the term that best completes each statement.
?
1.
Numbers that have a product of 1 are
?
or
.
2.
?
The
states that the product of a number and its
multiplicative inverse is 1.
Skills Check
Multiply. Write each product in simplest form. (Lesson 3-5)
3
8
2 3
3. _ × _
4. - _ × - _
3
5.
8
4
- 1_ × 4_
2
9
4 5
6. _ × _
3 2
5
Evaluate. (Lesson 3-5)
1 5
7. _
(2)
8.
( 3_4 )
3
Divide. Write each quotient in simplest form. (Lesson 3-6)
5 2
3
3
9. _ ÷ _
10. _ ÷ _
6 3
3
7
11. _ ÷ - _
12
4
10 5
3
7
12. _ ÷ - _
4
8
Problem-Solving Check
13.
LANDSCAPING A gardener is planting _2 of a garden with daisies. Of
3
1
_
the daisies, of them are yellow. What fraction of the whole garden is
2
yellow daisies? Explain how you solved the problem. (Lesson 3-5)
5
14. CONSTRUCTION A carpenter is using _ -inch long pieces of plastic
8
tubing. How many pieces of plastic tubing can be cut from a 5-inch long
tube of plastic? Draw a diagram that illustrates this situation. Then,
explain how your diagram can be used to solve the problem. (Lesson 3-6)
15.
REFLECT Explain how to divide a fraction by another fraction. (Lesson 3-6)
198 Chapter 3 Fractions
Adding and Subtracting
Fractions with Like
Denominators
3-7
Vocabulary
The
What:
I will add and subtract fractions with
like denominators.
The
Why:
Musicians use fractions when reading
music and playing instruments.
like denominators
(p. 199)
Standard 7NS1.2 Add,
subtract, multiply, and
divide rational numbers
(integers, fractions, and
terminating decimals) and take
positive rational numbers to
whole-number powers.
Ike plays two quarter notes on his trumpet. What is the total
value of the notes?
You can solve this problem by adding 1_ + 1_.
Standard 6NS2.1 Solve
problems involving addition,
subtraction, multiplication, and
division of positive fractions and
explain why a particular
operation was used for a given
situation.
4
1_ + 1_ = 1_
4 4 2
1
4
4
1
4
1
2
To add fractions easily, the denominators should be the same.
In other words, the fractions should have like denominators.
like denominators
unlike denominators
ADDING FRACTIONS WITH LIKE DENOMINATORS
Words
To add fractions with like denominators, add the numerators. Write
the sum over the common denominator. Then simplify.
1 1
+1 2
_
_
Numbers _
+_=1
=_
=1
4 4
4
4 2
Pictures
Symbols
1
4
1
4
1 1
_
+_
4 4
a+b
_ and b
_ (c ≠ 0), a
_+b
_ = _.
For fractions a
c
c
c
c
c
Lesson 3-7 Adding and Subtracting Fractions with Like Denominators
199
WORK WITH A PARTNER Name two fractions with like
denominators of 6 and add them. Listen as your partner
names two fractions with like denominators of 7, and
adds them.
EXAMPLE
Add Fractions
Add. Write each sum in simplest form.
5
_ + 2_
Never add the
denominators when you
add fractions.
8 8
5+2
5
_ + 2_ = _
8 8
8
The fractions have like denominators.
Add the numerators.
= 7_
8
9
7 +_
_
Write the sum over the denominator.
7+9
9
7 +_
_
=_
The fractions have like denominators.
Add the numerators.
10
10
10
10
10
16
=_
10
= 8_ or 13_
5
5
Write the sum over the denominator.
Simplify.
Your Turn Add. Write each sum in simplest form.
3 1
5 5
4 2
a. _ + _
b. _ + _
c. _ + _
9
9
4
4
6
6
SUBTRACTING FRACTIONS WITH LIKE DENOMINATORS
Words
To subtract fractions with like denominators, subtract the
numerators. Write the difference over the common denominator.
Then simplify.
3
-1
_=3
_
_=1
_
Numbers _
-1
=2
4
4
4
4
2
Pictures
Symbols
1
4
1
4
1
4
a-b
_ and b
_ (c ≠ 0), a
_-b
_ = _.
For fractions a
c
c
c
c
c
200 Chapter 3 Fractions
EXAMPLE
Subtract Fractions
Find 7_8 - _38. Write the difference in simplest form.
7-3
_7 - 3_ = _
8 8
8
The fractions have like denominators.
Subtract the numerators.
= 4_
Write the difference over the denominator.
8
= 1_
2
Simplify.
Your Turn Subtract. Write each difference in simplest
form.
9
3
7 2
d. _ - _
e. _ - _
9
9
10
10
You can also add and subtract improper fractions with like
denominators. Write the answer as a mixed number.
EXAMPLE
Add Improper Fractions
19 + _
15. Write the sum in simplest form.
Find _
8
8
19 + 15
19 _
_
+ 15 = __
8
8
_
= 34
8
8
17 or 4 1_
=_
4
4
Add the numerators.
Write the sum over the like
denominator.
Simplify. Write the answer as a
mixed number.
Your Turn Add. Write each sum in simplest form.
13 24
7 11
f. _ + _
g. _ + _
4
4
5
5
Lesson 3-7 Adding and Subtracting Fractions with Like Denominators
201
EXAMPLE
Subtract Improper Fractions
31 - _
17. Write the difference in simplest form.
Find _
10
10
31 _
31 - 17 Subtract the numerators.
_
- 17 = __
10
10
10
Write the difference over the like
denominator.
14
=_
10
= 7_ or 1 2_
5
5
Simplify. Write the answer as a mixed
number.
Your Turn Subtract. Write each difference in
simplest form.
8 4
29 15
h. _ - _
i. _ - _
3
4
Which operation will you use to find how long he
played altogether? Explain.
15 7_
You can find how long he played by adding _
+ .
4
4
12
_
_ = 15
_
= 4 +3
4 4
3
3
1_
=1+_
4
4
4
_
_ = _7
=4+3
4 4
00:00
0
4
2
4
5
Change the mixed numbers to improper
fractions.
15 7_
+
3 3_ + 1 3_ = _
4
encore
3
4
How long did Dominic play altogether? Write your
answer in simplest form.
00:06
1
8
MUSIC Dominic played a 3 3_ -minute song on the piano.
4
3
_
Then, he played a 1 -minute encore.
Mixed numbers can be
written as improper
fractions.
_=3+3
_
33
song
8
3
4
4
4
+7
_
= 15
Add the numerators.
_
= 22
Write the sum over the like
denominator.
4
6
4
Simplify. Write the answer as a mixed
number.
11 = 5 1_
=_
2
2
Your Turn
FASHION Gloria cut 1_ feet of fabric from a 9_ -foot roll.
4
202 Chapter 3 Fractions
4
j.
Which operation will you use to find how many feet of
fabric are left? Explain.
k.
How many feet of fabric does Gloria have left? Write
your answer in simplest form.
Voca b u la ry Re
vie w
like denominato
rs
Examples 1–6
(pages 200–202)
Examples 1–2
(page 200)
VOCABULARY
1.
Write two fractions with like denominators.
2.
Use the fractions in Exercise 1 to explain how to add fractions with
like denominators.
Add. Write each sum in simplest form.
5
3 5
1
3. _ + _
4. _ + _
12
Example 3
(page 201)
(page 201)
(page 202)
(page 202)
9
10
10
8
4
8
Subtract. Write each difference in simplest form.
15 5
13 11
9. _ - _
10. _ - _
6
6
9
_ -foot roll.
FASHION Katie cut _ feet of fabric from a 23
4
4
4
Example 6
8
Add. Write each sum in simplest form.
15 21
7 9
7. _ + _
8. _ + _
4
Example 5
8
Subtract. Write each difference in simplest form.
8 7
9
1
5. _ - _
6. _ - _
9
Example 4
12
11.
4
Which operation will you use to find how many feet of fabric are
left? Explain.
How many feet of fabric does Katie have left? Write your answer in
simplest form.
1
13. MUSIC Jade played two _ notes on her bassoon. How long were the
2
notes altogether?
12.
14.
HOMEWORK (%,0
For
See
Exercises Example(s)
15–18
19–22
23–26
27–30
31–34
1–2
3
4
5
6–7
You and a classmate write a pair of fractions with like
denominators. Without solving, discuss how you would find both
the sum and difference of the fractions.
Add. Write each sum in simplest form.
5
7 1
7
15. _ + _
16. _ + _
9 9
5 1
17. _ + _
6 6
16 16
7
1
18. _ + _
12 12
Subtract. Write each difference in simplest form.
6 4
7 3
19. _ - _
20. _ - _
7
7
11
1
21. _ - _
12
12
8 8
9
7
22. _ - _
16 16
Lesson 3-7 Adding and Subtracting Fractions with Like Denominators
203
Add. Write each sum in simplest form.
10 5
7 11
23. _ + _
24. _ + _
4
4
12 8
25. _ + _
5
5
3
3
23 27
26. _ + _
8
8
Subtract. Write the difference in simplest form.
23 13
13 11
27. _ - _
28. _ - _
6
6
33
13
29. _ - _
8
8
4
4
15
17
30. _ - _
8
8
FASHION Mia knits a scarf with 3_ inch of trim. She decides to add
4
another 3_ inch of trim.
4
31.
Which operation will you use to find how many inches of trim
there are altogether? Explain.
How many inches of trim does Mia have altogether? Write your
answer in simplest form.
7
33. CONSTRUCTION Vincent uses a _ -inch wrench to tighten a bolt.
8
How much wider is the opening of a 7_ -inch wrench than a 3_ -inch
32.
8
wrench?
34.
8
NATURE A local park is 4 7_ acres. If 2 1_ of the acres are closed to the
8
8
public, how many acres are open to the public?
35.
Write an equation that involves addition of
fractions and has a sum of 1_. Write an equation that involves
2
subtraction of fractions and has a difference of 1_.
36.
H.O.T. Problems Each of these fraction addition problems
has a pattern.
1_ + 1_ = 1_
2
4 4 2
3_ 3_ 3_
+ =
4 4 2
_3 + _3 = 3_
8 8 4
_5 + _5 = _5
6 6 3
The addends in each equation are the same. Notice that the
denominator of the sum is one-half the denominator of one of the
addends. Explain why this pattern is true.
204 Chapter 3 Fractions
Masterfile
37.
MUSIC The longest song on a new CD lasts 4 7_ minutes. The shortest
8
song lasts 3 3_ minutes. How much longer is the longest song than
8
the shortest one?
FPO
Musicians write, teach, and perform musical
compositions. “I Love You, California” was
written by F.B. Silverwood.
38.
MUSIC A musician plays three eighth notes in a row. What fraction
of a whole note do the notes make altogether?
Choose the best answer.
40 This chart records the number of
miles Otto runs each day. How
many more miles did he run on
Saturday and Sunday together than
on Friday?
39 To make a jacket, a designer uses
3
_ bolt of red fabric, 1_ of blue fabric,
8
8
1
_
and of black fabric. What fraction
8
of a full bolt of fabric does she use?
5
A _
C 5_
8
D 3_
4
24
B 1_
2
Sun
Mon
Tues
Wed
Thur
Fri
Sat
2 3_4
3 1_4
3
2 3_4
2 1_2
3 1_4
2
H 1 3_
F 1 1_
4
G 1 1_2
4
J
2
Divide. Write each quotient in simplest form. (Lesson 3-6)
5 3
3 3
3 1
2 5
41. _ ÷ _
42. _ ÷ _
43. _ ÷ _
44. _ ÷ _
9
45.
3
4
6
8
4
4
2
CONSTRUCTION Suppose you have a piece of metal 6 inches long.
How many 3_ -inch bars can be cut from the piece of metal? (Lesson 3-6)
4
Lesson 3-7 Adding and Subtracting Fractions with Like Denominators
205
3-8
Adding Fractions with
Unlike Denominators
Vocabulary
The
What:
I will add fractions with unlike
denominators.
The
Why:
You add fractions with unlike
denominators to find how much
material is needed to build a wall or
sew an outfit.
unlike denominators
(p. 206)
least common
multiple (LCM) (p. 206)
least common
denominator (LCD)
(p. 206)
Standard 7NS1.2 Add,
subtract, multiply, and
divide rational numbers
(integers, fractions, and
terminating decimals) and take
positive rational numbers to
whole-number powers.
Mateo and Tamika volunteer to build houses for a charitable
group. They build walls with 3_ inch of drywall and 1_ inch
8
2
of insulation. How thick are the walls?
You can solve this problem by adding 3_ + 1_.
8
1
8
Standard 6NS2.1 Solve
problems involving addition,
subtraction, multiplication, and
division of positive fractions and
explain why a particular
operation was used for a given
situation.
Standard 7MR2.1 Use
estimation to verify the
reasonableness of calculated
resulted.
1
8
3
8
1
8
2
1
2
+
1
2
To add fractions with unlike denominators, replace each
fraction with equivalent fractions that have like denominators.
To do this, find a common multiple among the multiples of
the denominators. The least common multiple (LCM) of
two or more numbers is the smallest multiple common to the
numbers.
EXAMPLE
Find the Least Common Multiple
Find the of 8 and 12.
8: 8, 16, 24
List multiples of each number until you find
12: 12, 24
a common multiple. Identify the LCM.
The LCM is 24.
Your Turn Find the LCM of each pair of numbers.
a.
206 Chapter 3 Fractions
9 and 12
b.
5 and 6
When you add fractions with unlike denominators, rename
one or both of the fractions so all fractions have the same
denominator.
EXAMPLE
Rename Fractions
Rename 1_2 with a denominator of 6.
×3
_1 = _?
2 6
Multiplying both the
numerator and the
denominator by 3 does
not change the value of
the fraction. It is the
same as multiplying by
3
_ or 1.
3
The factor is 3 because 2 × 3 = 6.
×3
×3
_1 = 3_
2 6
Multiply the numerator by the same factor.
×3
Your Turn Rename each fraction with the given
denominator.
1 ?
2
?
c. _ = _
d. _ = _
2
8
3
12
When adding fractions, the LCM of the denominators can be
used as the least common denominator (LCD).
EXAMPLE
Find the Least Common Denominator
Find the LCD of _1 and 5_.
2
9
2: 2, 4, 6, 8, 10, 12, 14, 16, 18
9: 9, 18
The denominators of the
fractions are 2 and 18. List
multiples of each denominator
until you find a common
multiple. Identify the LCM.
The LCD of 1_ and 5_ is 18.
2
9
Your Turn Find the LCD of each pair of fractions.
3
2
1
7
e. _ and _
f. _ and _
3
6
8
10
Lesson 3-8 Adding Fractions with Unlike Denominators
207
EXAMPLE
Add Fractions with Unlike
Denominators
Find 1_3 + 3_5. Write the sum in simplest form.
3: 3, 6, 9, 12, 15
5: 5, 10, 15
×5
List multiples of each
denominator. The LCD is 15.
×3
5
? =_
_1 = _
3 15 15
9
? =_
_3 = _
5 15 15
×5
×3
5+9
5
9
_
+_
=_
15
15
15
14
=_
15
Rename the fractions using the
LCD. Identify the factor
multiplying the denominator.
Multiply the numerator by the
same factor.
Add the numerators.
Write the sum over the
denominator.
Your Turn Add. Write each sum in simplest form.
3
1
7
2
g. _ + _
h. _ + _
8
5
12
10
To add fractions with unlike denominators, rename the
fractions with a common denominator. Then add and
simplify.
Refer to the begining of the lesson. Use this process to find the
thickness of Mateo and Tamika’s walls.
drywall
Check your sums by
estimating. Round
fractions to 0, _21 , or 1.
Then add to get an
estimate.
_1 rounds to 0.
3
3 is a little more than _1 ,
_
5
but not quite 1.
Compare this to your
14
answer of _
15 , which is
almost 1.
2
208 Chapter 3 Fractions
insulation
The wall will be 7_ inch thick.
8
wall
WORK WITH A PARTNER Discuss with your partner
how to estimate the sums of: 1_ + 7_ and 3_ + 5_.
9
8
4
6
To add mixed numbers with unlike denominators, write the
mixed numbers as improper fractions. Then rename the
improper fractions with a common denominator.
Add Improper Fractions
ADVERTISING Sara sold ads for her school’s yearbook.
_ pages of ads to restaurants and 2 1_ or 5_
She sold 3 1_ or 13
2
2
4
4
pages to grocery stores.
What operation will you use to find how much
advertising she sold in all? Explain.
You can find how much advertising she sold by adding
3 1_ and 2 1_. Add to find the total pages of ads.
2
4
How much advertising did Sara sell in all? Write your
answer as a mixed number in simplest form.
13 5_
3 _1 + 2 1_ = _
+
Write the mixed numbers to
improper fractions.
4: 4, 8, 12, 16, 20
2: 2, 4, 6, 8, 10
List the multiples of each
denominator. The LCD is 2.
2
4
2
4
×1
×2
13 _? _
_
= = 13
4
4
10
_5 = _? = _
2 4
4
4
×1
×2
13 _
+ 10
_
__
+ 10 = 13
4
Rename the fractions using the
LCD.
4
Add the numerators.
4
23
=_
or 5 3_
4
4
Write as a mixed number.
Your Turn
BAKING A baker makes a cake and puts icing on top. The
cake is 2 1_ or 9_ inches high. The icing is 1_ inch thick. How
2
4
4
tall is the cake at its center?
i.
What operation will you use to find the total height of
the cake? Explain.
j.
What is the total height of the cake? Write your answer
as a mixed number in simplest form.
Lesson 3-8 Adding Fractions with Unlike Denominators
209
Examples 1–8
(pages 206–208)
Example 1
(page 206)
Example 2
(page 207)
VOCABULARY
1.
Give an example of fractions with
unlike denominators.
2.
Explain how you use the LCM
to find the LCD of two fractions.
Find the LCM of each pair of numbers.
3.
3 and 12
(page 207)
4.
4 and 10
Find the LCD of each pair of fractions.
5
3
1
7
5. _ and _
6. _ and _
2
Example 3
unlike denomin
ators
least common m
ultiple
(LC M)
least common
denominator (L
C D)
8
8
12
Rename the fraction with the given denominator.
3 ?
?
1
7. _ = _
8. _ = _
4
8
2
10
Add. Write each sum in simplest form.
5 1
1 1
Example 4
9. _ + _
10. _ + _
(page 208)
Example 6
(page 208)
Examples 6–7
8 4
3
2
11. _ + _
10 5
3 7
13. _ + _
2 6
15.
(page 209)
2 3
1 3
12. _ + _
6 8
11 5
14. _ + _
3
4
ADVERTISING Mr. Roth buys a _1 -page ad on the back cover of a
4
magazine and a 1_ -page ad in the middle. How much advertising
2
does he buy in all?
ENGINEERING An architect
designs a floor that has a 1 1_
4
or 5_4 -inch base and 7_8 -inch
tile on top.
16.
What operation will you
use to find how thick the
floor is altogether?
Explain.
17.
How thick is the floor
altogether? Write your
answer as a mixed number
in simplest form.
18.
Say or write
multiples of 6. Have your partner say or write multiples of 8.
Work together to find the LCM of 6 and 8.
210 Chapter 3 Fractions
California architectural landmarks
include Louis Kahn’s Salk Institute in
La Jolla and Frank Lloyd Wright’s
Hollyhock House in Hollywood.
Eddie Brady/Lonely Planet Images
Voca b u la ry Re
vie w
HOMEWORK (%,0 Find the LCM of each pair of numbers.
For
See
Exercises Example(s)
19–22
1
23–26
2
27–30
3
31–38
4–5
39–41
5–6
43–48
6–7
19.
2 and 6
20.
5 and 15
21.
4 and 5
22.
6 and 8
Rename the fraction with the given denominator.
_1 = _?
2 6
3
?
25. _ = _
4 12
_2 = _?
3 6
?
2
26. _ = _
3 24
23.
24.
Find the LCD of each pair of fractions.
2
_ and 5_
3
6
3
1
29. _ and _
3
4
7_ and 3_
8
4
5
5
30. _ and _
6
8
27.
28.
Add. Write each sum in simplest form.
_1 + 1_
2 8
2 1
33. _ + _
3 6
7
1
35. _ + _
10 4
3 5
37. _ + _
4 6
_1 + 1_
4 3
4
1
34. _ + _
5 10
3
1
36. _ + _
8 12
2 3
38. _ + _
3 8
31.
32.
3 or 23
_ miles and walks for another 1 1_ or
FITNESS Leah jogs for 2 _
2
10
3
_ miles. How far does she jog10
and
run?
2
39.
What operation will you use to find how far she exercises in
all? Explain.
How far does Leah jog and run? Write your answer as a mixed
number in simplest form.
1
41. HORTICULTURE A gardener mixes _ cup of liquid fertilizer with
3
5
1
2_ or _ cups of water. How much liquid is there in all?
40.
2
42.
2
ROOFING A roofer attaches roofing
felt and shingles to plywood when
laying down a new roof. How thick
are the felt and shingles altogether?
Add. Write the sum in simplest form.
11 3
4 17
43. _ + _
44. _ + _
8
2
7
7
_
_
45. +
3 4
3
12
11 17
46. _ + _
4
Sheathing
1
Roofing felt
16
1
Asphalt shingles
8
6
Lesson 3-8 Adding Fractions with Unlike Denominators
211
ENGINEERING An architect drew the
following plan.
47.
How thick is the sheathing and siding
altogether?
48.
How thick is the drywall and insulation
altogether?
49.
Write a fraction
addition sentence with an approximate sum of
2. Use fractions with unlike denominators.
50.
H.O.T. Problem Theo says the LCM of two
numbers is always the product of the
numbers. Do you agree? Explain your
thinking.
Choose the best answer.
51 Refer to the blueprint in Exercise 47.
How thick is the wall, including the
drywall, insulation, sheathing, and
siding?
A 4 1_ inches
8
B 4 _1 inches
2
C 5 1_ inches
8
D 5 _3 inches
4
5
-inch drywall
8
1
3 2 -inch insulation
3 -inch wall sheathing
4
7 -inch siding
8
52 A newspaper is printed with the
margins and other dimensions
shown here. What is the total
height of the
1 1 inch
6
page?
F 8 _2 inches
3
G 8 3_ inches
4
_
H 8 11 inches
12
J
7 1 inches
12
9 inches
Find each sum. Write the answer in simplest form. (Lesson 3-7)
_5 + 2_
9 9
5 1
55. _ + _
6 6
53.
57.
_1 + _3
8 8
9
7
56. _ + _
10 10
54.
FARMING Pia plants 1_ of her farm with corn and 1_ with soybeans.
3
3
What fraction of her farm is planted with corn or soybeans? (Lesson 3-7)
212 Chapter 3 Fractions
3
inch
4
3-9
Subtracting Fractions with
Unlike Denominators
Vocabulary
The
What:
I will subtract fractions with unlike
denominators.
The
Why:
Cooking involves using measurements
with unlike denominators.
like denominators
(p. 213)
unlike denominators
(p. 213)
Standard 7NS1.2 Add,
subtract, multiply, and
divide rational numbers
(integers, fractions, and
terminating decimals) and take
positive rational numbers to
whole-number powers.
Standard 6NS2.1 Solve
problems involving addition,
subtraction, multiplication, and
division of positive fractions and
explain why a particular
operation was used for a given
situation.
A chef needs 5_ cup of tomatoes to make a pasta sauce. He has
8
1
_ cup of tomatoes. How much more does the chef need?
2
You can solve this problem by subtracting 1_ from 5_.
2
_5 - 1_
8 2
1
8
1
8
1
2
1
8
1
8
8
1
8
To add fractions, the fractions should have like denominators.
The same is true when you subtract fractions.
To subtract fractions with unlike denominators,
rename the fractions with a common denominator.
Then subtract and simplify.
Use this process to find how much more tomatoes the chef needs.
chef’s tomatoes
_5 - 1_
8 2
= _5 - 4_
8
total tomatoes needed
8
5-4
= _
8
= 1_
8
total still needed
The chef needs another 1_ cup of tomatoes.
8
WORK WITH A PARTNER You and a partner each name
a fraction with a denominator 8 or less. Choose fractions
with unlike denominators. Work together to subtract the
fractions.
Lesson 3-9 Subtracting Fractions with Unlike Denominators
213
EXAMPLE
Subtract Fractions with Unlike
Denominators
Find 5_6 - _13 . Write the difference in simplest form.
6: 6, 12, 18, 24, 30
3: 3, 6, 9, 12, 15
×1
List multiples of each
denominator. The LCD is 6.
×1
5
_ = _? = 5_
6 6 6
1_ = _? = 2_
3 6 6
×1
Rename the fractions using the
LCD. Subtract the numerators.
×1
5
-2
_ - 2_ = 5_
6 6
6
Subtract the numerators.
Write the difference over the like
denominator. Simplify.
= 3_ or 1_
6
2
Your Turn Subtract. Write each difference in simplest
form.
5
9
5 3
1
2
a. _ - _
b. _ - _
c. _ - _
12
8
10
5
6
4
To subtract mixed numbers with unlike denominators,
rename the fractions using a common denominator.
EXAMPLE
To change a mixed
number to an improper
fraction:
Multiply the integer by
the denominator of the
fraction.
Add the numerator of
the fraction to that
product.
Write the sum over the
denominator.
To change an improper
fraction to a mixed
number:
Divide the numerator by
the denominator.
Write the remainder
over the denominator.
Use the whole number
and fraction to make
the mixed number.
214 Chapter 3 Fractions
Subtract Improper Fractions
17 - 4_. Write the difference in simplest form.
Find _
6
3
List multiples of each
6: 6, 12, 18, 24, 30
3: 3, 6, 9, 12, 15
denominator. The LCD is 6.
×1
×2
17 = _? = _
17
_
6
6
6
×1
_4 = _? = 8_
3 6 6
×2
17 - 8
17 - 8_ = __
_
6
Rename the fractions using the
LCD. Subtract the numerators.
6
Subtract the numerators.
6
= _9 = 3_ or 1 1_
6 2
2
Write the difference over the like
denominator. Simplify.
Write as a mixed number.
Your Turn Subtract. Write each difference in simplest
form.
23 5
5
17 23
d. _ - _
e. _ - _
f. 3 - _
6
4
4
8
3
ADVERTISING A television station has 6 hours of
_ hours of the
advertising each day. Mara has sold 3 1_ or 13
4
4
advertising time for next Wednesday.
2
1
3
4
5
6
Hours
Which operation will you use to find how many more
hours she needs to sell? Explain.
Subtract to find how much more time is needed.
How many more hours of advertising does Mara need
to sell? Write your answer in simplest form.
6 hours per day - 3 _1 hours sold
4
6 = _6
1
Change the whole number to a fraction.
13
3 1_ = _
Change the mixed number to an
improper fraction
4
4
×4
6_ _? _
= = 24
1 4
4
_ with the same denominator.
Rename 6
1
×4
13 __
24 - _
_
= 24 - 13
4
4
Subtract the numerators.
4
11 or 2 3_
=_
4
4
Write the difference over the like
denominator. Write as a mixed number.
Mara needs to sell 2 3_ more hours of advertising.
4
Your Turn
BIOLOGY A biologist collects salamanders to study
them. The average length of salamanders from the east
coast is 2 1_ or 9_ inches. The average length of salamanders
4
4
_ inches.
from the west coast is 3 1_8 or 25
8
g.
Which operation will you use to find the difference
in average length? Explain.
h.
What is the difference in average length of the
salamanders? Write the answer in simplest form.
Lesson 3-9 Subtracting Fractions with Unlike Denominators
215
Voca b u la ry Re
vie w
Examples 1–6
(pages 187–188)
like denominato
rs
unlike denomin
ators
VOCABULARY
1.
How do you recognize fractions
with unlike denominators?
2.
How do you rewrite fractions with unlike denominators so that
they have like denominators?
Subtract. Write each difference in simplest form.
Examples 1–2
(page 187)
Example 3–4
(page 188)
Example 5–6
(page 188)
7
_ - 1_
8 4
7
1
5. _ - _
12 4
13 13
7. _ - _
8
4
1
9. 3 - 1 _
2
3.
BUSINESS
worked 1 3_
4
5_ 1_
6 2
1
1
6. _ - _
6 10
23 3
8. _ - _
10 2
29
10. 4 - _
12
Rick needs to work 4 1_2 or 9_2 hours on Saturday. He has
or _7 hours so far.
4
4.
11.
Which operation will you use to find how many more hours he
needs to work? Explain.
12.
How many more hours does Rick need to work?
BIOLOGY A biologist collects snails from two different islands.
The average length of the snails from one island is 3_ inch. The
4
average length of the other snails is 2_ inch. What is the difference
3
in average length?
14.
Talk about how to find the sum of 5_ and 1_ and the
8
4
difference of 5_ and 1_. Discuss the similarities and differences of
13.
8
4
adding and subtracting fractions.
HOMEWORK (%,0 Subtract. Write each difference in simplest form.
For
See
Exercises Example(s)
15–22
1–2
23–30
3–4
31–36
5–6
39–40
_7 - _3
8 4
5
1
17. _ - _
12 4
11 7
19. _ - _
12 8
3 1
21. _ - _
8 3
15.
216 Chapter 3 Fractions
_2 - 1_
3 6
3 1
18. _ - _
8 3
5 5
20. _ - _
6 8
8 5
22. _ - _
9 6
16.
Subtract. Write each difference in simplest form.
23.
11 - 3
_
_
24.
2
4
8 13
25. _ - _
3
6
53 11
27. _ - _
12
4
11
_
29. 4 8
29 7_
_
-
6
3
31 7
26. _ - _
8
4
11 13
28. _ - _
6
8
31
_
30. 3 12
HOBBIES Opal uses 3_ -inch beads to make a necklace and 5_8 -inch
4
beads for a bracelet.
31.
Which operation will you use to find how much longer the necklace
beads are than the bracelet beads? Explain.
32.
How much longer are the necklace beads than the bracelet beads?
Write your answer in simplest form.
33.
11 -mile track.
FITNESS Omar plans to run once around the 2 3_ or _
4
4
He has run 1 1_ miles so far. How much farther does he have to go?
2
34.
35.
FASHION Esteban wants to use 5_ -inch buttons. His buttonholes
8
11 -inch across. How much
measure _
wider is the buttonhole
12
than the button?
NATURE This map shows the area of different parts of the local
nature preserve. How many more acres of flower gardens are there
than woods?
Area
Ponds and Lakes
21
Flower Gardens
28
Woods
36.
37.
Acres
2
3
13
4
13
7
ENGINEERING A 6 1_ or _
-mile road is being repaved. So far 4 _
2
2
10
of the miles have been repaired. How many miles are left to be
repaved?
Write a fraction subtraction sentence with
like denominators so that the difference is 1_. Then write a fraction
3
subtraction sentence with unlike denominators so that the
difference is also 1_.
3
38.
H.O.T. Problems Seth says that the following equation is correct. Is
he correct? Explain why or why not.
5 1_ - 1 3_ = 3 3_
2
4
4
Lesson 3-9 Subtracting Fractions with Unlike Denominators
217
39.
40.
COOKING A chef has chopped 3_ cup of
4
vegetables for a recipe. She needs
2 1_ cups in all. How many more cups
2
does she need?
COOKING A fruit salad recipe includes
2
_ cup of pineapple and 3_ cup of grapes.
3
4
How many more cups of grapes are
there than pineapple?
Choose the best answer.
different size ads in a newspaper.
Suppose you want to place a _1 -page
4
4
1 or _
11 mile so far. How much
1_
10
ad and a 1_ -page ad. How much
10
3
farther does she have to run?
more space will you have than if
you placed just a _1 -page ad?
3
C 1_
1
A _
2
20
9
D 3_
20
20
B 3 2_
5
Chefs plan menus,
purchase ingredients,
and create dishes. The
restaurant industry is the
largest private employer
in California, with over
1.4 million employees.
42 This chart shows the cost for
41 Solana plans to run twice around
the 2 1_ or 9_ -mile track. She has run
4
Jeff Greenberg/PhotoEdit
CAREER CONNECTION A chef runs a
kitchen in a restaurant, hotel, resort, ship or
other location. His or her responsibilities
include planning the menu, cooking,
purchasing ingredients, and managing the
staff and budget.
Ad Size
_1 page
4
_1 page
3
_1 page
2
Full page
Price
$40
$65
$80
$140
1 page
F _
12
G 1_ page
4
Add. (Lesson 3-8)
1 1
43. _ + _
3
47.
2
44.
3
_ + 1_
4 6
45.
3_ 1_
+
8 2
46.
H _1 page
J
3_ 2_
+
4 3
SPORTS Hank spends _1 of his practice time shooting free throws
2
1
_
and of the time running drills. What fraction of his practice time
4
does he spend on free throws and drills? (Lesson 3-8)
218 Chapter 3 Fractions
6
1_ page
3
3
Progress Check 4
(Lessons 3-7, 3-8, and 3-9)
Vocabulary and Concept Check
least common denominator
(LCD) (p. 182)
least common multiple
(LCM) (p. 181)
like denominators (p. 175)
unlike denominators (p. 181)
Choose the term that best completes each statement.
3
3
?
1. The fractions _ and _ have
.
4
2.
?
The
8
of the fractions 1_ and 1_ is 12.
3
4
Skills Check
Add. Write each sum in simplest form. (Lesson 3-7)
3
3
1
1
3. _ + _
4. 1 _ + 2 _
10
10
4
4
Subtract. Write each difference in simplest form. (Lesson 3-7)
5
7 5
11
5. _ - _
6. _ - _
8
8
12
12
Add. Write each sum in simplest form. (Lesson 3-8)
1 3
1 2
7. _ + _
8. _ + _
4 8
5 1
9. _ + _
6 8
2
3
2
1
_
10. 1 + 2 _
3
4
Subtract. Write each difference in simplest form. (Lesson 3-9)
7 3
7
1
11. _ - _
12. _ - _
8 4
7
1
13. _ - _
10 4
12 2
5
1
14. 2 _ - 1 _
3
6
Problem-Solving Check
15.
COOKING Ray puts _1 cup of olive oil in a pan. Then he decides to add
3
another 1_ cup to the recipe. How much olive oil is in there in all?
3
(Lesson 3-7)
16.
17.
FITNESS Ella is biking 4 1_ miles today. She has biked 2 1_ miles so far.
2
4
How much farther does she have to go? (Lesson 3-9)
REFLECT Explain how you decide to add or subtract when solving a
word problem with fractions or mixed numbers. (Lesson 3-8)
Chapter 3 Progress Check
219
3-10
Vocabulary
Fractions in Expressions
and Equations
The
What:
I will simplify expressions and solve
equations involving fractions.
The
Why:
You can solve equations with
fractions to find the target heart rate
when you exercise.
expression (p. 192)
equation (p. 193)
solve
(p. 193)
inverse operations
(p. 193)
Standard ALG5.0
Students solve multistep
problems, including
word problems, involving linear
equations in one variable and
provide justification for each step
[excluding inequalities].
Standard 7AF4.1 Solve two-step
linear equations and inequalities
in one-variable over the rational
numbers, interpret the solution or
solutions in the context from
which they arose, and verify the
reasonableness of the results.
Standard 6AF1.1 Write and solve
one-step linear equations in
one-variable.
Fitness experts want adults to reach a target heart rate when
exercising. Measuring the target heart rate helps determine if
a person is exercising at the proper pace. The expression
below is used to find the rate. The variable a represents the
person’s age. What is the target rate for a 20-year-old adult?
You can find the answer by substituting 20 for the variable a.
3
_ (220 · a) = 3_ (220 – 20)
4
4
= _3 (200)
4
= 150
Substitute 20 for the variable a.
Simplify inside the parentheses first.
Subtract 20 from 220.
_ and 200.
Multiply 3
4
The target heart rate for a 20-year-old adult is 150 beats
per minute.
WORK WITH A PARTNER Use the heart rate expression
to find the target heart rate for a 40-year-old adult and a
60-year-old adult.
You can use the Associative, Commutative, and Distributive
Properties to simplify expressions. These properties allow you
to change the order and grouping of numbers when you
combine like terms.
220 Chapter 3 Fractions
EXAMPLE
Simplify the expression.
Simplify 2_ + 3_ w + 1_ - 1_ w.
3
8
6
8
(3 6 ) (8 8 )
= ( 4_ + 1_ ) + ( 3_ w - 1_ w )
6 6
8
8
Use the commutative property to
group like terms.
= 5_ + 2_ w
Add and subtract the numerators
of fractions with like denominators.
= 5_ + 1_ w
Simplify.
= _2 + 1_ + _3 w - 1_ w
6
8
6
4
Use the LCD to rename fractions
with unlike denominators.
Your Turn Simplify each expression.
3
3 4
1 7
4
2
a. _ + _ a - _ a + _
b. _ + _ x + _
2
9
9
10
4
5
4
An equation is a mathematical sentence stating two expressions
are equal. You can solve an equation by finding the value or
values of the variable that makes the equation true.
When solving equations with fractions, use inverse operations
as you did with integers. Addition undoes subtraction.
Multiplication undoes division.
EXAMPLE
Solve One-Step Equations with Fractions
Solve k + 2_ = 3_.
3
4
_ from each side to
Subtract 2
3
_.
“undo” the addition of 2
3
k + 2_ - 2_ = 3_ - 2_
3
An equation is a math
sentence with an = sign.
Whatever you do to one
side of an equation, you
must do to the other side.
3
3
4
9
8
k=_
-_
12
1
k=_
12
Check: k
12
The LCD of 3 and 4 is 12.
Subtract the numerators.
?
+ 2_ = 3_
3
4
? 3
1 + 2_ =
_
_
3
12
4
Substitute the answer in the
original equation.
8 ? _
1 +_
_
= 9
12
12 12
9
9
_
=_
✓
12 12
The answer checks.
Your Turn Solve each equation.
1
2
c. _ + x = _
3
3
d.
3_ 1_
= -f
8 2
Lesson 3-10 Fractions in Expressions and Equations
221
EXAMPLE
Solve One-Step Equations
with Fractions
Solve 3_ t = 2_3.
4
_3 t ÷ 3_ = 2_ ÷ 3_
4
4 3 4
3
Divide each side by _
to “undo” the
4
multiplication.
t = 2_ × 4_
_ multiply by its reciprocal of
To divide by 3
4
3
t = 8_
9
3
4
_.
3
Multiply numerators. Multiply denominators.
3 ? 2
Check: _ t = _
3
4
?
3
8
2
_×_=_
4 9 3
? 2
24 =
_
_
36 3
_2 = _2 ✓
3 3
Substitute the answer in the original
equation.
Your Turn Solve the equation.
2
1
e. _ n = _
3
6
NATURE A town preserves open space for nesting birds
based on its shoreline. The area in square miles of
protected land a is related to the miles of shoreline m.
_1 a = m
3
The town has 4 miles of shoreline. How much land is
in the protected area?
1
_a = 4
3
Substitute 4 for m.
_1 a ÷ 1_ = 4 ÷ 1_
3
3
3
_ to “undo” the
Divide each side by 1
3
multiplication.
a=4×3
1
_
To divide by _
, multiply by its reciprocal of 3
3
1
or 3.
a = 12
Simplify.
There are 12 square miles of protected land.
Your Turn
f.
222 Chapter 3 Fractions
BUSINESS The equation 2_ p = d shows the price of
3
clothing after it goes on sale. The variable p is related to
the discount rate of d. Find the original price of a belt if
the discount price is $12.
Two-step equations have two operations in the equation. Solve
using inverse operations.
Use inverse operations:
Undo addition or subtraction first.
Then undo multiplication or division.
EXAMPLES
Solve Two-Step Equations with Fractions
Solve 2x + _13 = 3_.
4
2x + _1 = 3_
3 4
2x + 1_ - 1_ = 3_ - 1_
3
3
4
3
_ from each side to “undo”
Subtract 2
4
addition.
9
4
2x = _
-_
The LCD of 4 and 3 is 12.
5
2x = _
Subtract the numerators.
Write the integer as a fraction.
12
12
12
5
2x ÷ 2 = _
÷2
12
5
x=_
× 1_
To divide by 2, multiple by its
_.
reciprocal, 1
2
5
x=_
Multiply numerators. Multiply
denominators.
12
2
24
Check: 2x
Divide each side by 2 to “undo” the
multiplication.
?
+ 1_ = 3_
3 4
?
5
2 × _ + 1_ = 3_
24 3 4
10 _1 ? _3
_
+ =
24 3 4
?
10 _
_
+ 8 = _3
24 24 4
18 ? 3_
_
=
24 4
3_ 3_
= ✓
4 4
Substitute the answer in the original
equation.
The answer checks.
Your Turn Solve each equation.
3
1
1
1
g. _ + 3c = _
h. 2 - _ s = _
2
4
2
3
i.
_3 = 3_ j - 1_
8 4
2
Lesson 3-10 Fractions in Expressions and Equations
223
Voca b u la ry Re
vie w
Examples 1–5
VOCABULARY
(pages 193–194)
Example 1
expression
equation
solve
inverse operation
s
1.
Use the term expression to
describe an equation.
2.
Explain how you can use
inverse operations to solve an equation.
Simplify each expression.
(page 193)
3.
2
_ + 3_ f + 1_
3 4
4
4.
1_ k + 3_ - 1_ k
2
4 6
5.
1_ + 3_ b + 1_ + 1_ b
3 8
2 4
7.
z + 7_ = 1 1_
8.
z - 1_ = 5_
Solve each equation.
Example 2
6.
(page 193)
k + 5_ = 7_
8
8
8
4
3
6
Solve each equation.
Example 3
9.
(page 193)
Example 4
10.
LANDSCAPING A landscaper decides to plant red and yellow roses
according to the equation y = 2_ r. In this equation, r represents red
3
roses and y represents yellow roses. How many red roses should
she plant if she has 24 yellow roses?
12.
COMMUNITY SERVICE The sophomores and freshmen at Adams
High perform service hours each week. The number of hours
performed is related by the equation 2_ s + 1 = f. In this equation, the
3
number of sophomore hours is represented by s and the number of
freshman hours is represented by f. If the freshmen perform 9 hours
a week, how many hours will the sophomores perform?
Solve each equation.
(page 194)
13.
16.
2w + 2_ = 1
3
14.
6–7
224 Chapter 3 Fractions
_2 c + 1_ = 2
3
2
15.
_3 t - 5_ = 3_
8
8 4
What is the difference between an expression and an
equation? How are the two alike?
HOMEWORK (%,0 Simplify each expression.
3
1 1
For
See
17. _ d + _ + _ d
Exercises Example(s)
8
4 8
3
1
2
19–24
1
19. _ t + _ t - _
6
3
4
25–32
2–3
1
2 1
1
33–36
4
21. _ m + _ - _ m - _
2
3
6
4
37–40
5
41–48
2_ n = 1_
3
2
11.
(page 193)
Example 5
3
_ x = 1_
3
4
_1 + 1_ + 1_ a
6 4 2
3
5
1
20. _ k - _ k + _
8
2
4
3
7
11 3
22. _ f + _ - _ f - _
8
12 8
4
18.
Ed Honowitz/Getty Images
Solve each equation.
1 6
23. x + _ = _
8
5
_
25.
= y + 2_
6
12
5
27. d - _ = 1
6
7
1
29. _ = b - _
5
10
4
Solve each equation.
3
1
31. _ m = _
8
4
1
2
33. _ n = _
3
5
_7 + b = 1 _3
8
4
9
1
_
_
26.
=r+
12
4
3_ 1_
28. r - =
4 8
5
1
30. _ = q - _
8
16
24.
_2 z = _6
3
7
1 1
34. _ = _ t
4 2
32.
35.
NATURE A town votes to preserve open space for parks. It uses the
equation p = 1_ a, where p is each new acre of park and a is each new
3
acre of land that can be developed. If the town creates a 4 1_ or
4
17 -acre park, how many acres can it develop?
_
36.
HEALTH Fitness experts usually suggest that adults exercise to
meet a target heart rate. This rate is about 3_ (220 - a), where a is
4
the person’s age, depending on fitness level and other heath
considerations. What age matches a target heart rate of 120?
37.
CAREER CONNECTION
A company uses the equation
s = 2_ p + 2 to determine the
5
sale price of its products. In this
equation, s represents the
sale price and p represents the
original price. Suppose the sale
price is $4. What was the
original price?
38.
MARKET RESEARCH A
FPO
_ million Californians
In 2002, over 1 1
salesperson discovers that the
2
worked as retail salespersons.
relationship of shoppers to
purchasers in her store follows
the equation p = 3_ s. In this
4
equation, p is purchasers and s is shoppers. If the store had
60 purchasers on Saturday, about how many shoppers were
in the store?
4
Solve each equation.
39.
2k + 1_ = 2
2
_5 + 2_ m = 1
6 3
1
2 1
43. _ b - _ = _
2
3 6
41.
40.
7 p + 3_ = 1
4
_5 = 1_ d + 3_
8 4
8
2
1
7
_
_
_
44. q - =
3
4 8
42.
Lesson 3-10 Fractions in Expressions and Equations
225
Write a two-step equation that includes at
least two fractions. Exchange problems with a classmate and solve.
3
1 5
2
46. H.O.T. Problems Kanita solves _ k + _ = _. She gets an answer of _.
2 6
3
4
Do you agree or disagree with her solution? Why or why not?
45.
47.
EARTH SCIENCE Below Earth’s crust, there are three layers—the
mantle, inner core, and outer core. The interior of the Earth is onehalf mantle, one-third outer core, and one-sixth inner core. Simplify
the expression 1_ m + 1_ c + 1_ c. Identify the fraction that represents
2
3
6
the core.
Crust
Mantle
Outer Core
Inner Core
Choose the best answer.
49 How many solutions are there to
the two-step equation 2_ x - 1_ = 3_ ?
48 Which equation has the same
solution as 1_ x = 3_?
2
8
8
_
A 2x =
C 3_ x = 1_
3
B 1_ x = 1_
4
3
3
2
8
D 2_ x = 3_
8
3
F 0
H 1
G 2
J
Subtract. (Lesson 3-9)
50.
54.
_1 - 1_
2 3
51.
_7 - 1_
8 2
52.
_3 - 2_
4 3
53.
_5 - 3_
6 8
PACKAGING There was 5_ of a pizza left in a box. Doug ate 1_ of the
8
4
pizza for lunch. How much pizza was there after Doug ate lunch?
(Lesson 3-9)
226 Chapter 3 Fractions
4
6
4
Stud y T ip s
3
Use abbreviation
s or
sy mbols to stand
for
words or operati
ons.
Study Guide
Understanding and Using
the Vocabulary
After completing the chapter, you should be able to define each term,
property, or phrase and give an example of each.
additive inverse (p. 153)
common factor (p. 167)
composite number (p. 165)
draw a diagram (p. 171)
equation (p. 208)
equivalent fraction (p. 169)
expression (p. 208)
factor (p. 165)
fraction (p. 152)
greatest common factor (GCF) (p. 167)
improper fraction (p. 159)
inverse operations (p. 209)
Inverse Property of Multiplication (p. 180)
least common denominator (LCD)
least common multiple (LCM)
like denominators (p. 189)
mixed number (p. 157)
multiplicative inverse (p. 180)
prime number (p. 165)
product (p. 174)
proper fraction (p. 157)
rational number (p. 152)
reciprocal (p. 180)
simplest form (p. 167)
solve (p. 209)
unlike denominators (p. 194)
(p. 194)
(p. 195)
Complete each sentence with the correct mathematical term or phrase.
3
?
1. 2 _ is a
.
4
2.
The
?
of 1_ and 3_ is 12.
3.
A(n)
?
can be written in the form _a (b ≠ 0).
3
4
b
3
7
4. The fractions _ and _ have
8
8
3
5
5. The fractions _ and _ have
6
4
5
?
6. _ is an
.
3
?
7.
The
8.
2, 3, 5, and 11 are examples of a
9.
4, 8, 14, and 20 are examples of a
10.
The
?
?
.
?
.
of 6 and 10 is 2.
?
.
?
.
of 5_ is 8_.
8
5
Chapter 3 Study Guide
227
3
Study Guide
Skills and Concepts
Objectives and Examples
LESSON 3-1
pages 152–156
Compare and order fractions using a
number line.
0
-1
1
Graph - _2 and - 1_ on the same number
3
2
line.
- 1_ > - 2_
2
LESSON 3-2
Review Exercises
3
Graph each fraction on a number line.
3
1
11. _
12. - _
4
6
Order each set of fractions from least to
greatest.
5 7 3
1 3 2
13. _, _, _
14. _, _, _
5 5 5
6
3
1
15. - _, - _, - _
7
7
7
8 8 8
8
5
2
16. - _, - _, - _
9
9
9
pages 157–163
Compare and order fractions and mixed
numbers.
Write 5_ as a mixed number.
3
1R2
_
35
-3
2
5
_ = 1 2_
3
3
Write each mixed number as an
improper fraction.
2
1
17. 2 _
18. -3 _
3
4
Write each improper fraction as a mixed
number.
8
7
19. _
20. - _
2
8
21. _
3
5
13
22. - _
5
Graph each pair of numbers on a
number line.
3 2
1
7
23. -2 _, -1 _
24. 1 _, 1 _
2
LESSON 3-3
8
4
3
pages 165–170
Simplify fractions using the greatest
common factor (GCF).
Identify the greatest common factor
(GCF) of the numbers.
Find the greatest common factor of 6 and 15.
23.
6: 1, 2, 3, 6 List the factors of each
15: 1, 3, 5, 15 number. The GCF is 3.
Write each fraction in simplest form.
6
8
4
26. _
27. _
28. _
The GCF of 6 and 15 is 3.
228 Chapter 3 Fractions
4 and 10 24. 6 and 18
8
10
25.
5 and 9
12
Skills and Concepts
Objectives and Examples
LESSON 3-5
pages 174–179
Multiply fractions.
Multiply numerators.
Multiply denominators.
8
2×4=_
_2 × 4_ = _
3 5 3 × 5 15
Raise fractions to positive powers.
3
8
2
_ = 2_ × 2_ × 2_ = _
(3)
3
3
3
LESSON 3-6
27
7 ÷3
7 ×5
_
_=_
_
5
3
10
35 = 7
_ = 11_
_
6
Multiply by the reciprocal of
the second fraction.
Simplify. Write as a mixed
number.
7
30 6
6
LESSON 3-7
Add the numerators. Write
the sum over the like
denominator.
7+1
7 +_
1 =_
_
10
10
8 =4
_
=_
10
5
10
10
6
=_
= 3_
10
Simplify.
Subtract the numerators.
Write the difference over
the like denominator.
7 -_
1 =_
7-1
_
10
_2 × _3
3 4
5
4
31. _ × - _
5
8
29.
Evaluate.
1 5
33. _
(2)
30.
3
_
× 5_
34.
( 3_4 )
10 6
5 2
32. _ × _
2 3
3
Divide. Write each quotient in simplest
form.
_3 ÷ 2_
5 3
5
7
37. - _ ÷ _
12 6
35.
_1 ÷ _3
8 4
14 ÷ _4
_
38.
3
3
36.
pages 175–180
Add and subtract fractions with like
denominators.
10
Multiply. Write each product in simplest
form.
pages 180–185
Divide fractions.
10
Review Exercises
5
Add. Write each sum in simplest form.
1_ + 1_
3 3
11
7
41. _ + _
12 12
39.
3_ 1_
+
8 8
7 11
42. _ + _
4
4
40.
Subtract. Write each difference in
simplest form.
3 1
7 5
43. _ - _
44. _ - _
4 4
5
1
45. _ - _
12 12
8 8
10 5
46. _ - _
3
3
Simplify.
Chapter 3 Study Guide
229
Study Guide
3
Objectives and Examples
LESSON 3-8
Review Exercises
pages 194–200
Add fractions with unlike denominators.
Add. Write each sum in simplest form.
7 .
Add 1_ + _
7
_1 + _
5 10
1 5
48. _ + _
2 6
2 3
49. _ + _
3 4
11 + _9
_
50.
8
4
3
47.
12
×4
×1
4
_1 = _
3 12
7 =_
7
_
×4
×1
12
Rename the fractions
using the LCD.
12
+7 _
4 +_
7 =4
_
_
= 11
10 12
12
12
LESSON 3-9
Add the numerators.
Simplify.
pages 201–206
Subtract fractions with unlike
denominators.
Subtract. Write each difference in
simplest form.
Subtract 5_ - 1_ .
_5 - 1_
8 4
1 1
52. _ - _
2 3
5 1
53. _ - _
6 2
11 7
54. _ - _
3
4
6
51.
3
×1
×2
5_ 5
=_
6 6
1
_ = 2_
3 6
×1
×2
Rename the fractions
using the LCD.
5-2 3
_5 - 2_ = _
= _ = 1_
6 6
6
6 2
LESSON 3-10
Subtract the
numerators.
Simplify.
pages 208–214
Solve equations with fractions.
Solve each equation.
Solve 2_ x - 1_ = 5_ .
55.
3
2
6
_2 x - 1_ + 1_ = 5_ + 1_
3
2 2 6 2
2
_ x ÷ 2_ = 4_ ÷ 2_
3
3 3 3
x=2
230 Chapter 3 Fractions
1
Add _
to each side to
2
“undo” subtraction.
_
Divide each side by 2
3
to “undo” the
multiplication.
1
k - 4_ = _
5 10
3
2
56. z ÷ _ = _
3
4
2
1
57. _ c - _ = 1
3
2
3
1
7
58. _ y + _ = _
4
4 12
(
)
3
Chapter Test
Vocabulary and Concept Check
1.
Write two examples each of improper fractions and mixed numbers.
2.
Write a pair of fractions with like denominators. Write a pair of fractions
with unlike denominators.
Skills Check
Use a number line to determine which number is greater.
3.
5
7 and _
_
12
4.
12
1 5_ and 2 1_
6
6
Identify each number as prime or composite.
5.
15
6.
9
Identify the greatest common factor (GCF) of the numbers.
7.
8 and 16
8.
8 and 10
Write each fraction in simplest form.
9.
5
_
20
10.
12
-_
18
11.
3
_
15
12.
14
-_
35
Multiply divide, add, or subtract. Write each answer in simplest form.
1
_ × 4_
4 5
3
5
17. _ + _
10 10
1 3
21. n + _ = _
2 4
13.
3_ 2_
×
8 3
5 1
18. _ + _
8 4
5
2
22. _ g = _
3
6
14.
7_ ÷ 1
_
8 4
11
7
19. _ - _
12 12
15.
5_ 1_
÷
3 2
7 1
20. _ - _
8 2
16.
Problem-Solving Check
AGRICULTURE A farmer plans to keep _18 of her crop and sell 7_8 to
customers. She will sell 2_3 of the total at local markets.
23.
Which operation will you use to find the total that will be sold at local
markets? Explain.
What fraction of the total crop will be sold at local markets? Write your
answer in simplest form.
5
1
25. MUSIC Carmen practices the flute 2 _ or _ hours each day. On Saturday,
2
2
3
7
_
_
she practiced 1 or hours. How much time did she have left to practice?
24.
4
4
Chapter 3 Chapter Test
231
3
Standards Practice
PART 1 Multiple Choice
6 How many cups of orange juice and
apple juice are needed for the punch?
Choose the best answer.
Punch Recipe
1 What number is the multiplicative
inverse of 2_?
3
A 1_
C 1
3
B 3_
2
D 2_
1
2 Which mixed number is less than -1 _2?
3
F -1 _3
H -1
4
G -1 1_
J 0
3
3 How can you tell that a fraction is in its
simplest form?
A The numerator is less than the
denominator.
B The denominator is less than the
numerator.
C The GCF of the numerator and
denominator is 1.
D The GCF of the numerator and
denominator is greater than 1.
4 Which equation has the same value
for x as 3_ x = 1_?
4
2
F 1_ x = 3_ cups
4
2
2
_
G x = _1 cups
6
3
Cups
Apple
33
Grape
14
Mango
1
3
Orange
42
F 5 3_ cups
2
1
1
H 7 5_ cups
6
J 8 1_ cups
6
3
1
_
_
7 How do you divide ÷ ?
8 4
3_
1
A Multiply and _.
8
4
B Multiply _3 and _4.
8
1
8
_
C Multiply and _1.
3
4
8
_
D Multiply and _4.
3
1
4
G 7 1_ cups
2
8 What is the GCF of 6 and 12?
F 1
H 6
G 3
J
12
H x + 1_ = 3_ cups
J
2 4
1 cups
_1 x - _
12
8
5 Which of the following numbers
is prime?
A 15
C 17
B 16
D 18
232 Chapter 3 Fractions
Juice
PART 2 Short Answer
Record your answers on the answer sheet
prodived by your teacher or on a separate
sheet of paper.
9 ENGINEERING A new road will run
15 miles through a park and then
10 miles to a town. Five stop signs are
placed at equal distances along the
route. What is the distance between
each sign?
10 This chart shows the number of minutes
Elias played in each quarter of the last
basketball game. What fraction of the
whole game did Elias play?
Quarter
13 Solve for z.
1_ z - 2_ = 1_
3
3 6
PART 3 Extended Response
Total
Minutes
Minutes Elias Played
1
15
9
2
15
6
3
15
3
4
15
12
Record your answers on the answer sheet
provided by your teacher or on a separate
sheet of paper.
14 A chef is making stock to use for
11 In Ms. Alvarez’s class, 3_ of the students
8
are running for a position in the student
government. Only 1_ of them will win
3
their elections. What fraction of the
class will hold a position in the
government?
recipes in her kitchen. She combines
4 1_ or 9_ cups of vegetable broth with
2
2
9_
1
_
2 or cups of water.
4
4
a. How much liquid is there in all?
b. The chef then boils the liquid
mixture until it reduces by half.
How much liquid is left?
c. The chef uses 1 1_ or 3_ cups of the
12 This chart shows the number of miles
Salil hiked each day on his trip. How
many more miles did he hike on
Saturday and Sunday than on Monday
and Tuesday?
Day
Saturday
Miles Hiked
3 _3
Sunday
1
3_
Monday
2 1_2
Tuesday
3
2
2
reduced liquid for a sauce and
freezes the rest. How much stock
does she freeze?
5
10
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Chapter 3 Standards Practice
233