Download Rational Numbers

5.1
Rational Numbers
Goal: Write fractions as decimals and vice versa.
Vocabulary
Rational
number:
Terminating
decimal:
Repeating
decimal:
Example 1
Identifying Rational Numbers
Show that the number is rational by writing it as a quotient of two
integers.
2
3
b. 12
a. 3
1
4
d. 2 c. 4 Solution
a. Write the integer 3 as
.
b. Write the integer 12 as
are
or
. These fractions
.
2
3
c. Write the mixed number 4 as the improper fraction
1
4
d. Think of 2 as the opposite of
1
4
Then you can write 2 as
. First write
. To write
.
as
.
as a quotient
of two integers, you can assign the negative sign to either the
or the
. You can write
or
.
84
Chapter 5 • Pre-Algebra Notetaking Guide
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5.1
Rational Numbers
Goal: Write fractions as decimals and vice versa.
Vocabulary
Rational A rational number is a number that can be written as
number: the quotient of two integers.
Terminating
decimal:
Repeating
decimal:
Example 1
A decimal that has a final digit
A decimal that has one or more digits that repeat
without end
Identifying Rational Numbers
Show that the number is rational by writing it as a quotient of two
integers.
2
3
b. 12
a. 3
1
4
d. 2 c. 4 Solution
3
a. Write the integer 3 as 1 .
12
12
. These fractions
b. Write the integer 12 as 1 or 1
are equivalent .
2
14
c. Write the mixed number 4 as the improper fraction 3 .
3
1
1
1
9
d. Think of 2 as the opposite of 2 4 . First write 2 4 as 4 .
4
1
9
9
Then you can write 2 as 4 . To write 4 as a quotient
4
of two integers, you can assign the negative sign to either the
9
4
numerator or the denominator . You can write or
9
.
4
84
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Writing Fractions as Decimals
Example 2
5
16
4
9
a. Write as a decimal.
b. Write as a decimal.
a.
b.
0.3125
16.0
50
0
0
4
8
20
1
6
40
3
2
80
8
0
0
0.44. . .
9.0
40
36
40
3
6
Answer: Use a bar to
show the
in the
decimal:
4
0.4
.
9
Answer: The remainder
is
, so the decimal is a
decimal:
5
0.3125.
16
Checkpoint Write the fraction or mixed number as a decimal.
3
5
7
20
2. 1. 12
25
23
40
3. 2 4. Using Decimals to Compare Fractions
Example 3
42
48
27
30
Compare and .
42
48
27
30
and Answer: Compare the decimals.
27
30
so Copyright © Holt McDougal. All rights reserved.
Divide.
>
,
42
.
48
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85
Writing Fractions as Decimals
Example 2
4
9
5
16
a. Write as a decimal.
b. Write as a decimal.
a.
b.
0.3125
16.0
50
0
0
4
8
20
1
6
40
3
2
80
8
0
0
0.44. . .
9.0
40
36
40
3
6
Answer: Use a bar to
show the repeating digit
in the repeating decimal:
4
0.4
.
9
Answer: The remainder
is 0 , so the decimal is a
terminating decimal:
5
0.3125.
16
Checkpoint Write the fraction or mixed number as a decimal.
3
5
7
20
2. 1. 0.6
0.35
12
25
23
40
3. 2 4. 2.48
0.575
Using Decimals to Compare Fractions
Example 3
42
48
27
30
Compare and .
42
0.875
48
27
30
and 0.9
Answer: Compare the decimals.
Divide.
0.9
> 0.875 ,
27
42
so > .
30
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85
Example 4
Writing Terminating Decimals as Fractions
place,
3 is in
a. 0.03 so denominator is
place,
4 is in
b. 9.4 so denominator is
Example 5
.
.
Simplify fraction.
Writing a Repeating Decimal as a Fraction
To write 0.8
1
as a fraction, let x 0.8
1
.
1. Because 0.8
1
has 2 repeating digits, multiply each side of
1
by
x 0.8
, or
. Then
x
.
x
2. Subtract x from
x.
( x 0.8
1
)
x
3. Solve for x and simplify.
x
x
Answer: The decimal 0.8
1
is equivalent to the fraction
86
Chapter 5 • Pre-Algebra Notetaking Guide
.
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Example 4
Writing Terminating Decimals as Fractions
3 is in hundredths’ place,
3
a. 0.03 100
so denominator is 100 .
4
b. 9.4 9 10
2
9 5
Example 5
4 is in tenths’ place,
so denominator is 10 .
Simplify fraction.
Writing a Repeating Decimal as a Fraction
To write 0.8
1
as a fraction, let x 0.8
1
.
1. Because 0.8
1
has 2 repeating digits, multiply each side of
1
.
1
by 102 , or 100 . Then 100 x 81.8
x 0.8
1
100 x 81.8
2. Subtract x from 100 x.
( x 0.8
1
)
99 x 81
3. Solve for x and simplify.
99
81
x
99
99
x 9
11
Answer: The decimal 0.8
1
is equivalent to the fraction 9 .
11
86
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Checkpoint Write the decimal as a fraction or mixed number.
5. 0.8
6. 3.75
7. 0.1
2
8. 5.3
Ordering Rational Numbers
Example 6
17
11
13
Order the numbers , 1.35, 5.67, 6, , from least
2
3
4
to greatest.
Graph the numbers on the number line. You may want to write
improper fractions as mixed numbers.
8
6
4
2
0
2
4
6
8
Answer: Read the numbers graphed on the number line from
left to right:
Copyright © Holt McDougal. All rights reserved.
,
,
,
,
,
.
Chapter 5 • Pre-Algebra Notetaking Guide
87
Checkpoint Write the decimal as a fraction or mixed number.
6. 3.75
5. 0.8
4
5
3
4
3 7. 0.1
2
8. 5.3
1
3
4
33
5 Ordering Rational Numbers
Example 6
17
11
13
Order the numbers , 1.35, 5.67, 6, , from least
2
3
4
to greatest.
Graph the numbers on the number line. You may want to write
improper fractions as mixed numbers.
17
2
8
6
6
13
4
4
11
3
1.35
2
0
2
4
5.67
6
8
Answer: Read the numbers graphed on the number line from
17
13
11
left to right: 2 , 6 , 4 , 1.35 , 3 , 5.67 .
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87
Focus On
Reasoning
Use after Lesson 5.1
Inductive and Deductive Reasoning
Goal: Determine whether mathematical statements are true or false.
Vocabulary
Inductive
reasoning
Conjecture
Deductive
reasoning
Example 1
Using Inductive Reasoning
Consider fractions between 0 and 1 whose denominators are
powers of 3, such as 3, 9, and 27. Make a conjecture about the
decimal form of such fractions.
Solution
Find a pattern using a few examples.
1
0.3
3
1
9
1
27
1
0.0
1
2
3
4
5
6
7
9
81
Conjecture: The decimal form of any fraction whose denominator
is a power of 3 is
Checkpoint Make a conjecture.
1. Make a conjecture about the difference of any two even integers.
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Chapter 5 • Pre-Algebra Notetaking Guide
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Focus On
Reasoning
Use after Lesson 5.1
Inductive and Deductive Reasoning
Goal: Determine whether mathematical statements are true or false.
Vocabulary
Inductive
reasoning
Making a conclusion based on several examples
Conjecture A statement that is thought to be true but not yet
shown to be true
Deductive The process of starting with one or more given facts
reasoning and using rules, definitions, or properties to reach a
conclusion
Example 1
Using Inductive Reasoning
Consider fractions between 0 and 1 whose denominators are
powers of 3, such as 3, 9, and 27. Make a conjecture about the
decimal form of such fractions.
Solution
Find a pattern using a few examples.
1
0.3
3
1
9
0.1
1
27
0.0
3
7
1
0.0
1
2
3
4
5
6
7
9
81
Conjecture: The decimal form of any fraction whose denominator
is a power of 3 is a repeating decimal.
Checkpoint Make a conjecture.
1. Make a conjecture about the difference of any two even integers.
Sample answer: The difference of any two even integers is even.
88
Chapter 5 • Pre-Algebra Notetaking Guide
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Example 2
Using Deductive Reasoning
Give a convincing argument to show that the conjecture in Example
1 is true.
Solution
Since the decimal system is based on powers of
form of any fraction with a
a
, the decimal
that is a
is
.
If the
of a fraction is a
factors will only consist of combinations of
, its prime
s and/or
other cases, the long division while converting the
would result in a pattern of
s. In all
to a
that continues
endlessly. So, if there are prime factors in the
than
or
other
, then the decimal form will be a
.
Consider fractions between 0 and 1 whose denominators are
powers of 3. The
in the
fractions will only include
s. Therefore, the decimal form of each
of these fractions will be a
Example 3
of these
.
Using a Counterexample
Show the conjecture is false by finding a counterexample.
Conjecture: The decimal form of any fraction whose denominator
is divisible by 10 is a terminating decimal.
Solution
Write fractions whose denominators are divisible by 10.
1
10
Because
1
0.05
20
is a
1
30
, the conjecture is
.
Checkpoint Show that the conjecture is false by finding a
counterexample.
2. Conjecture: The quotient of any two numbers is less than either number.
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Chapter 5 • Pre-Algebra Notetaking Guide
89
Using Deductive Reasoning
Example 2
Give a convincing argument to show that the conjecture in Example
1 is true.
Solution
Since the decimal system is based on powers of 10 , the decimal
form of any fraction with a denominator that is a power of 10 is
a terminating decimal .
If the denominator of a fraction is a power of 10 , its prime
factors will only consist of combinations of 2 s and/or 5 s. In all
other cases, the long division while converting the fraction to a
decimal would result in a pattern of remainders that continues
endlessly. So, if there are prime factors in the denominator other
than 2 or 5 , then the decimal form will be a repeating decimal .
Consider fractions between 0 and 1 whose denominators are
powers of 3. The prime factors in the denominators of these
fractions will only include 3 s. Therefore, the decimal form of each
of these fractions will be a repeating decimal .
Using a Counterexample
Example 3
Show the conjecture is false by finding a counterexample.
Conjecture: The decimal form of any fraction whose denominator
is divisible by 10 is a terminating decimal.
Solution
Write fractions whose denominators are divisible by 10.
1
0.1
10
1
0.05
2
0
20
1
0.03
30
Because 1 is a counterexample , the conjecture is false .
30
Checkpoint Show that the conjecture is false by finding a
counterexample.
2. Conjecture: The quotient of any two numbers is less than either number.
Sample answer: 15 5 3, and 3 > 15
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89
5.2
Adding and Subtracting
Like Fractions
Goal: Add and subtract like fractions.
Adding and Subtracting Like Fractions
Words To add or subtract fractions with the same denominator,
write the sum or difference of the numerators over the denominator.
4
9
1
9
9
2
11
11
11
Numbers a
c
9
b
c
Algebra , c 0
Example 1
c
a
b
, c 0
c
c
c
Adding Like Fractions
A survey asked 100 students
ages 7 to 11 what sports
apparel they prefer to wear.
The circle graph at the right
summarizes their responses.
What fraction of the students
responded either major league
baseball or NBA?
Apparel Survey
Other
9
NFL
30
Pro wrestling
17
NBA
21
Major league
baseball
23
Solution
To find the fraction of the students who responded either major
league baseball or NBA, find the sum of
.
Write sum of numerators
over denominator.
and
Add. Then simplify.
Answer: The fraction of the students who prefer either major
league baseball or NBA apparel is
90
Chapter 5 • Pre-Algebra Notetaking Guide
.
Copyright © Holt McDougal. All rights reserved.
5.2
Adding and Subtracting
Like Fractions
Goal: Add and subtract like fractions.
Adding and Subtracting Like Fractions
Words To add or subtract fractions with the same denominator,
write the sum or difference of the numerators over the denominator.
Numbers 7
9
2
11
11
11
ab
a
b
Algebra , c 0
c
c
c
ab
a
b
, c 0
c
c
c
4
9
Example 1
1
9
5
9
Adding Like Fractions
A survey asked 100 students
ages 7 to 11 what sports
apparel they prefer to wear.
The circle graph at the right
summarizes their responses.
What fraction of the students
responded either major league
baseball or NBA?
Apparel Survey
Other
9
NFL
30
Pro wrestling
17
NBA
21
Major league
baseball
23
Solution
To find the fraction of the students who responded either major
23
21
and .
league baseball or NBA, find the sum of 100
100
23
21
100
100
23 21
100
Write sum of numerators
over denominator.
11
44
25
100
Add. Then simplify.
Answer: The fraction of the students who prefer either major
11
league baseball or NBA apparel is 2
.
5
90
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Subtracting Like Fractions
Example 2
When you perform
operations with
negative fractions, be
sure to assign a
negative sign in front
of a fraction to the
numerator of the
fraction.
4
9
Write difference of numerators
over denominator.
3
9
a. 2
11
Subtract.
4
11
b. 4
11
To subtract , add
Write sum of numerators
over denominator.
Add.
.
Checkpoint Find the sum or difference.
2
7
4
7
3
13
1. Adding and Subtracting Mixed Numbers
Example 3
3
7
6
7
a. 4 3 Write mixed numbers as
improper fractions.
Write sum of numerators
over denominator.
3
10
Add. Then write fraction
as mixed number.
9
10
b. 11 8 Write mixed numbers as
improper fractions.
Write difference of numerators
over denominator.
Copyright © Holt McDougal. All rights reserved.
8
13
2. Subtract. Then write fraction
as mixed number.
Chapter 5 • Pre-Algebra Notetaking Guide
91
Subtracting Like Fractions
Example 2
When you perform
operations with
negative fractions, be
sure to assign a
negative sign in front
of a fraction to the
numerator of the
fraction.
4 3
a. 9
4
9
3
9
7
9
2
4
Write difference of numerators
over denominator.
Subtract.
2
4
b. 1
1 11
11
11
4
4
To subtract , add 1
.
1
11
24
11
Write sum of numerators
over denominator.
6
1
1
Add.
Checkpoint Find the sum or difference.
2
7
4
7
3
13
1. 6
7
5
13
Adding and Subtracting Mixed Numbers
Example 3
3
6
31
27
a. 4 3 7 7
7
7
3
Write mixed numbers as
improper fractions.
31 27
7
Write sum of numerators
over denominator.
58
2
7 8 7
Add. Then write fraction
as mixed number.
9
113
89
1
b. 11 8 1
0
0
10
10
Copyright © Holt McDougal. All rights reserved.
8
13
2. Write mixed numbers as
improper fractions.
113 89
10
Write difference of numerators
over denominator.
24
2
2
1
0
5
Subtract. Then write fraction
as mixed number.
Chapter 5 • Pre-Algebra Notetaking Guide
91
Simplifying Variable Expressions
Example 4
4a
21
Write sum of numerators
over denominator.
10a
21
a. Add.
Simplify.
9
5b
4
5b
b. 4
5b
To subtract , add
Write sum of numerators
over denominator.
Add.
Simplify.
.
Checkpoint Find the sum or difference.
2
11
4
11
3. 3 5 2a
25
8a
25
5. 92
Chapter 5 • Pre-Algebra Notetaking Guide
5
13
6
13
4. 4 3 17
3c
5
3c
6. Copyright © Holt McDougal. All rights reserved.
Simplifying Variable Expressions
Example 4
4a
10a
4a 10a
a. 21
21
21
Write sum of numerators
over denominator.
14a
2
1
Add.
a
2
Simplify.
3
9
9
4
4
b. 5b 5b
5b
5b
4
4
To subtract , add 5b .
5b
9 4
5b
Write sum of numerators
over denominator.
5
Add.
5b
1
b
Simplify.
Checkpoint Find the sum or difference.
2
11
4
11
5
13
3. 3 5 6
13
4. 4 3 11
13
6
11
7
8 2a
25
17
3c
8a
25
5. 2a
5
92
Chapter 5 • Pre-Algebra Notetaking Guide
5
3c
6. 4
c
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5.3
Adding and Subtracting
Unlike Fractions
Goal: Add and subtract unlike fractions.
Adding and Subtracting Fractions
Example 1
7
15
1
5
1
5
7
15
a. 2
3
Write using LCD.
Write sum of numerators
over denominator.
Add.
Simplify.
3
4
b. Write fractions using LCD.
Write difference of numerators
over denominator.
Subtract.
Write fraction as a mixed number.
Checkpoint Find the sum or difference.
3
7
5
21
1. Copyright © Holt McDougal. All rights reserved.
1
4
3
10
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93
5.3
Adding and Subtracting
Unlike Fractions
Goal: Add and subtract unlike fractions.
Adding and Subtracting Fractions
Example 1
1
7
7
3
a. 1
5
5
15
15
73
15
Write sum of numerators
over denominator.
0
1
Add.
2
Simplify.
15
3
8
9
b. 12 1
2
3
4
2
1
5
Write using LCD.
3
Write fractions using LCD.
8 9
1
2
Write difference of numerators
over denominator.
17
Subtract.
12
5
11
2
Write fraction as a mixed number.
Checkpoint Find the sum or difference.
3
7
1
4
5
21
1. 2
3
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3
10
2. 1
20
Chapter 5 • Pre-Algebra Notetaking Guide
93
Adding Mixed Numbers
Example 2
1
6
3
10
5 2 Write mixed numbers as
improper fractions.
Write fractions using
LCD.
Write sum of numerators
over denominator.
Add. Then write fraction
as a mixed number.
Subtracting Mixed Numbers
Example 3
1
5
1
2
You are hiking a 12 -mile trail. You have already hiked 6 miles.
How many more miles do you have to hike before reaching the end
of the trail?
Solution
Your total hiking distance is
. You have already hiked
. To find the remaining distance, subtract.
1
5
1
2
12 6 Write mixed numbers as
improper fractions.
Write fractions using
LCD.
Write difference of
numerators over
denominator.
Subtract. Then write fraction
as a mixed number.
Answer: You need to hike
94
Chapter 5 • Pre-Algebra Notetaking Guide
miles.
Copyright © Holt McDougal. All rights reserved.
Adding Mixed Numbers
Example 2
10 31
3
1
5 2 6 6
310
60 1203 138
60 Write mixed numbers as
improper fractions.
Write fractions using
LCD.
310 (138)
Write sum of numerators
over denominator.
448
7
60 7 1
5
Add. Then write fraction
as a mixed number.
60
Subtracting Mixed Numbers
Example 3
1
5
1
2
You are hiking a 12 -mile trail. You have already hiked 6 miles.
How many more miles do you have to hike before reaching the end
of the trail?
Solution
1
Your total hiking distance is 12 5 miles . You have already hiked
1
6 miles . To find the remaining distance, subtract.
2
61
1
1
13
12 6 5 2
5
2
122
65
1
0 10
122 65
10
57
7
5
1
0
10
Write mixed numbers as
improper fractions.
Write fractions using
LCD.
Write difference of
numerators over
denominator.
Subtract. Then write fraction
as a mixed number.
Answer: You need to hike 5 7 miles.
10
94
Chapter 5 • Pre-Algebra Notetaking Guide
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Checkpoint Find the sum or difference.
5
6
4
9
1
3
3. 4 2 3
7
4. 2 3 Simplifying an Expression
Example 4
a
4
a
8
Simplify the expression .
a
a
a
p
4
8
4
a
8
a
a
Write using LCD.
8
4
Multiply.
Write difference of numerators
over denominator.
Subtract.
Checkpoint Find the sum or difference.
b
3
b
8
5. Copyright © Holt McDougal. All rights reserved.
c
5
c
7
6. Chapter 5 • Pre-Algebra Notetaking Guide
95
Checkpoint Find the sum or difference.
5
6
4
9
1
3
3. 4 2 3
7
4. 2 3 16
21
5
18
5 7 Simplifying an Expression
Example 4
a
4
a
8
Simplify the expression .
2
a
a
a
p 2
4
8
4
a
8
a
a
Write using LCD.
8
4
2a
8 Multiply.
2a a
Write difference of numerators
over denominator.
a
Subtract.
8
8
Checkpoint Find the sum or difference.
b
3
b
8
c
5
5. 11b
24
Copyright © Holt McDougal. All rights reserved.
c
7
6. 2c
35
Chapter 5 • Pre-Algebra Notetaking Guide
95
5.4
Multiplying Fractions
Goal: Multiply fractions and mixed numbers.
Multiplying Fractions
Words The product of two or more fractions is equal to the product
of the numerators over the product of the denominators.
3
5
4
7
Numbers p a
b
c
d
, where b 0 and d 0
Algebra p Multiplying Fractions
Example 1
5
3
5
p p
12
20
12
Assign negative sign to
numerator.
Use rule for multiplying
fractions.
Divide out common factors.
Multiply.
Checkpoint Find the product.
7
16
5
14
1. p 96
Chapter 5 • Pre-Algebra Notetaking Guide
2
15
5
18
2. p Copyright © Holt McDougal. All rights reserved.
5.4
Multiplying Fractions
Goal: Multiply fractions and mixed numbers.
Multiplying Fractions
Words The product of two or more fractions is equal to the product
of the numerators over the product of the denominators.
3
5
4
7
3p4
12
5p7
35
Numbers p a
b
ac
c
d
, where b 0 and d 0
Algebra p Multiplying Fractions
Example 1
bd
5
3
5
p p
12
20
12
203 5 p (3)
12 p 20
Assign negative sign to
numerator.
Use rule for multiplying
fractions.
1
1
p (
5
3)
p2
12
0
4
Divide out common factors.
4
1
1
16 1
6
Multiply.
Checkpoint Find the product.
7
16
5
14
2
15
1. p 5
32
96
Chapter 5 • Pre-Algebra Notetaking Guide
5
18
2. p 1
27
Copyright © Holt McDougal. All rights reserved.
Multiplying a Mixed Number and an Integer
Example 2
1
2
Water Use The showerhead in your home uses 2 gallons of
water per minute. If you take a 7-minute shower, how many gallons
of water do you use?
Solution
Number of
Gallons per
Gallons
p
minutes
minute
used
p
Substitute values.
Write numbers as improper
fractions.
p
Use rule for multiplying fractions.
Multiply.
Write fraction as a mixed number.
Answer: You use
Multiplying Mixed Numbers
Example 3
1
5
1
6
3 p 4 Copyright © Holt McDougal. All rights reserved.
gallons of water.
p
Write mixed numbers as improper
fractions.
Use rule for multiplying fractions.
Divide out common factors.
Multiply.
Write fraction as a mixed number.
Chapter 5 • Pre-Algebra Notetaking Guide
97
Multiplying a Mixed Number and an Integer
Example 2
1
2
Water Use The showerhead in your home uses 2 gallons of
water per minute. If you take a 7-minute shower, how many gallons
of water do you use?
Solution
Number of
Gallons per
Gallons
p
minutes
minute
used
2 1 p 7
2
Substitute values.
5 p 7
Write numbers as improper
fractions.
5p7
Use rule for multiplying fractions.
5
3
Multiply.
171
2
Write fraction as a mixed number.
2
1
2p1
2
Answer: You use 171 gallons of water.
2
Multiplying Mixed Numbers
Example 3
1
1
16
25
3 p 4 5 p 6
5
6
8
5
1
6 p 25
p6
5
1
Use rule for multiplying fractions.
Divide out common factors.
3
40
Multiply.
3
1
13 3
Copyright © Holt McDougal. All rights reserved.
Write mixed numbers as improper
fractions.
Write fraction as a mixed number.
Chapter 5 • Pre-Algebra Notetaking Guide
97
Checkpoint Find the product.
2
9
3
4
3. 5 p 6
Example 4
1
3
4. 2 p 5 Simplifying Expressions
Simplify the expression.
Use rule for multiplying
fractions. Divide out
common factor.
m
10
a. p 4
7
n4 9n2
b. p 12 10
Multiply.
Use rule for multiplying
fractions. Divide out
common factor.
Product of powers property
Add exponents.
Checkpoint Simplify the expression.
2x
5
3x 2
8
5. p 98
Chapter 5 • Pre-Algebra Notetaking Guide
4y3
15
5y 6
16
6. p Copyright © Holt McDougal. All rights reserved.
Checkpoint Find the product.
2
9
3
4
3. 5 p 6
1
3
2
3
14 31
Example 4
1
3
4. 2 p 5 Simplifying Expressions
Simplify the expression.
5
m p ()
10
m
10
a. p 4
7
p7
4
Use rule for multiplying
fractions. Divide out
common factor.
2
5m
5m 14
14
Multiply.
3
n4 p 9n2
n4 9n2
b. p 12 10
2 p 10
1
Use rule for multiplying
fractions. Divide out
common factor.
4
3n4 2
Product of powers property
3n6
Add exponents.
40
40
Checkpoint Simplify the expression.
2x
5
4y3
15
3x 2
8
3x 3
20
98
Chapter 5 • Pre-Algebra Notetaking Guide
5y 6
16
6. p 5. p y9
12
Copyright © Holt McDougal. All rights reserved.
Focus On
Measurement
Converting Temperatures
Use after Lesson 5.4
Goal: Convert temperatures between degrees Celsius and degrees
Fahrenheit.
Temperature Conversions
Fahrenheit to Celsius
The Celsius scale is
also known as the
centigrade scale
because there are
100 degrees
between the freezing
point of water (0°C)
and the boiling point
of water (100°C).
To convert degrees Fahrenheit to degrees Celsius, use the formula
C
.
Celsius to Fahrenheit
To convert degrees Celsius to degrees Fahrenheit, use the formula
F
.
Converting a Temperature to Degrees Celsius
Example 1
Cooking The internal temperature of a baked chicken is 167° F.
Convert this temperature to degrees Celsius.
Solution
C 5 (
9
5
(
9
5
9
(
Write formula for degrees Celsius.
)
)
Substitute
)
for
.
.
Use rule for multiplying fractions.
Divide out common factor.
5 p (13 5)
9 p1
.
Answer: In degrees Celsius, the internal temperature of the
baked chicken is
Copyright © Holt McDougal. All rights reserved.
.
Chapter 5 • Pre-Algebra Notetaking Guide
99
Focus On
Measurement
Converting Temperatures
Use after Lesson 5.4
Goal: Convert temperatures between degrees Celsius and degrees
Fahrenheit.
Temperature Conversions
Fahrenheit to Celsius
The Celsius scale is
also known as the
centigrade scale
because there are
100 degrees
between the freezing
point of water (0°C)
and the boiling point
of water (100°C).
To convert degrees Fahrenheit to degrees Celsius, use the formula
C 5 (F 32) .
9
Celsius to Fahrenheit
To convert degrees Celsius to degrees Fahrenheit, use the formula
9
F C 32 .
5
Converting a Temperature to Degrees Celsius
Example 1
Cooking The internal temperature of a baked chicken is 167° F.
Convert this temperature to degrees Celsius.
Solution
C 5 ( F 32 )
9
5
( 167 32 )
9
5
( 135 )
9
15
5 p (13 5)
1
9 p1
75
Write formula for degrees Celsius.
Substitute 167 for F .
Subtract .
Use rule for multiplying fractions.
Divide out common factor.
Multiply .
Answer: In degrees Celsius, the internal temperature of the
baked chicken is 75°C .
Copyright © Holt McDougal. All rights reserved.
Chapter 5 • Pre-Algebra Notetaking Guide
99
Checkpoint Convert the temperature to degrees Celsius.
1. 95° F
2. 5° F
Converting a Temperature to Degrees Fahrenheit
Example 2
Walking Mrs. Hochstetler likes to go walking when the temperature
is around 25°C. Convert this temperature to degrees Fahrenheit.
Solution
F 9
5
9
5
(
Write formula for degrees Fahrenheit.
)
9 p 25
5 p1
Substitute
for
.
Use rule for multiplying fractions.
Divide out common factor.
Answer: In degrees Fahrenheit, Mrs. Hochstetler likes to go
walking at
.
Checkpoint Convert the temperature to degrees Fahrenheit.
3. 68° C
100
Chapter 5 • Pre-Algebra Notetaking Guide
4. 37° C
Copyright © Holt McDougal. All rights reserved.
Checkpoint Convert the temperature to degrees Celsius.
1. 95° F
2. 5° F
35° C
15° C
Converting a Temperature to Degrees Fahrenheit
Example 2
Walking Mrs. Hochstetler likes to go walking when the temperature
is around 25°C. Convert this temperature to degrees Fahrenheit.
Solution
F 9 C 32
5
9
Write formula for degrees Fahrenheit.
( 25 ) 32
5
Substitute 25 for C .
9 p 25
32
5p
Use rule for multiplying fractions.
Divide out common factor.
45 32
Multiply.
77
Add.
5
1
1
Answer: In degrees Fahrenheit, Mrs. Hochstetler likes to go
walking at 77° F .
Checkpoint Convert the temperature to degrees Fahrenheit.
3. 68° C
4. 37° C
154.4° F
100
Chapter 5 • Pre-Algebra Notetaking Guide
98.6° F
Copyright © Holt McDougal. All rights reserved.
5.5
Dividing Fractions
Goal: Divide fractions and mixed numbers.
Vocabulary
Reciprocals:
Using Reciprocals to Divide
Words To divide by any nonzero number, multiply by its reciprocal.
2
9
3
7
2
9
Numbers p
a
b
c
d
a
b
Algebra p
, where b 0, c 0, and d 0
Dividing a Fraction by a Fraction
Example 1
3
7
6
11
p
Use rule for multiplying fractions.
Divide out common factor.
Multiply by reciprocal.
Multiply.
Check: To check, multiply the quotient by the divisor to see if you
get the dividend:
6
11
p Copyright © Holt McDougal. All rights reserved.
.
Chapter 5 • Pre-Algebra Notetaking Guide
101
5.5
Dividing Fractions
Goal: Divide fractions and mixed numbers.
Vocabulary
Reciprocals:
Two nonzero numbers whose product is 1 are
reciprocals.
Using Reciprocals to Divide
Words To divide by any nonzero number, multiply by its reciprocal.
2
3
2
14
7
Numbers p 3 2
7
9
7
9
a
a
c
d
ad
Algebra p c b
, where b 0, c 0, and d 0
c
b
b
d
Example 1
Dividing a Fraction by a Fraction
3
6
3
11
7 p 6
7
11
Multiply by reciprocal.
1
3 p 11
7p6
Use rule for multiplying fractions.
Divide out common factor.
2
11
11
14 14
Multiply.
Check: To check, multiply the quotient by the divisor to see if you
get the dividend:
6
11
3
p 14
7
11
Copyright © Holt McDougal. All rights reserved.
Solution checks .
Chapter 5 • Pre-Algebra Notetaking Guide
101
Dividing a Mixed Number by a Mixed Number
Example 2
1
2
3
4
2 3 Write mixed numbers as
improper fractions.
p
Multiply by reciprocal.
Use rule for multiplying
fractions. Divide out
common factors.
Multiply.
Checkpoint Find the quotient.
8
21
9
14
5
24
2. 3
5
4. 3 5 7
10
3. 4 1 102
5
12
1. Chapter 5 • Pre-Algebra Notetaking Guide
1
4
1
2
Copyright © Holt McDougal. All rights reserved.
Dividing a Mixed Number by a Mixed Number
Example 2
1
3
145 5
2 3 2 2
4
145 5
2 p
1
2
5 p (
)
4
p 15
2
1
Write mixed numbers as
improper fractions.
Multiply by reciprocal.
Use rule for multiplying
fractions. Divide out
common factors.
3
2
2
3 3
Multiply.
Checkpoint Find the quotient.
8
21
9
14
5
12
1. 5
24
2. 16
27
3
5
1
4
7
10
3. 4 1 12
17
Chapter 5 • Pre-Algebra Notetaking Guide
1
2
4. 3 5 2 102
2
13
22
Copyright © Holt McDougal. All rights reserved.
Example 3
Dividing a Whole Number by a Mixed Number
1
5
Dogs You have two dogs that eat about 1 pounds of dog food per
day. How many whole days will a 5-pound bag of dog food last?
Solution
Divide to find how long the bag of dog food will last.
Number of pounds
Number of
Number
eaten per day
pounds in bag
of days
Substitute values.
Write numbers as improper fractions.
p
Multiply by reciprocal.
Use rule for multiplying fractions.
Multiply.
Write fraction as a mixed number.
Answer: A 5-pound bag of dog food will last
Copyright © Holt McDougal. All rights reserved.
.
Chapter 5 • Pre-Algebra Notetaking Guide
103
Example 3
Dividing a Whole Number by a Mixed Number
1
5
Dogs You have two dogs that eat about 1 pounds of dog food per
day. How many whole days will a 5-pound bag of dog food last?
Solution
Divide to find how long the bag of dog food will last.
Number of pounds
Number of
Number
eaten per day
pounds in bag
of days
5 1 1
5
Substitute values.
5 6
Write numbers as improper fractions.
5 p 5
1
Multiply by reciprocal.
5p5
Use rule for multiplying fractions.
5
2
Multiply.
4 1
6
Write fraction as a mixed number.
1
5
6
1p6
6
Answer: A 5-pound bag of dog food will last 4 whole days .
Copyright © Holt McDougal. All rights reserved.
Chapter 5 • Pre-Algebra Notetaking Guide
103
5.6
Using Multiplicative Inverses
to Solve Equations
Goal: Use multiplicative inverses to solve equations.
Vocabulary
Multiplicative inverse:
Multiplicative Inverse Property
Words The product of a number and its multiplicative inverse is 1.
3
5
5
3
Numbers p 1
a
b
b
a
Algebra p 1, where a 0, b 0
Example 1
Solving a One-Step Equation
3
x 15
5
Original equation
Multiply each side by multiplicative
3
x 5
(15)
x
(15)
x
inverse of
.
Multiplicative inverse property
Multiply.
Answer: The solution is
.
Checkpoint Solve the equation. Check your solution.
6
11
1. x 18
104
Chapter 5 • Pre-Algebra Notetaking Guide
7
13
2. x 28
Copyright © Holt McDougal. All rights reserved.
5.6
Using Multiplicative Inverses
to Solve Equations
Goal: Use multiplicative inverses to solve equations.
Vocabulary
Multiplicative inverse:
The multiplicative inverse of a nonzero
number is the number’s reciprocal.
Multiplicative Inverse Property
Words The product of a number and its multiplicative inverse is 1.
3
5
5
3
Numbers p 1
a
b
b
a
Algebra p 1, where a 0, b 0
Example 1
Solving a One-Step Equation
3
x 15
5
5
3
3
x 5 (15)
5
3
Original equation
Multiply each side by multiplicative
inverse of 3 .
5
1 x 5 (15)
3
x 25
Multiplicative inverse property
Multiply.
Answer: The solution is 25 .
Checkpoint Solve the equation. Check your solution.
6
11
7
13
1. x 18
2. x 28
33
104
Chapter 5 • Pre-Algebra Notetaking Guide
52
Copyright © Holt McDougal. All rights reserved.
Solving a Two-Step Equation
Example 2
3
4
7
12
1
2
Original equation
1
2
Subtract
x 7
12
3
4
x 7
12
x Write fractions using LCD.
7
12
x 7
12
Subtract.
x x
Example 3
from each side.
Multiply each side by
multiplicative inverse of
.
Multiply.
Writing and Solving a Two-Step Equation
Tree Growth The height of a certain Norway Spruce is 10 feet.
1
2
If the tree’s height grows 2 feet per year, find how long it will
take the tree to reach a height of 25 feet.
Solution
Number of
New
Growth
Current
p
years
height
rate
height
1
2
10 2 x 25
Write equation.
1
2
Subtract
10 2 x 25 Simplify. Write mixed number
as improper fraction.
x
x
x
(
)
Multiply each side by
multiplicative inverse of
.
Multiply.
Answer: The tree will be 25 feet tall after
Copyright © Holt McDougal. All rights reserved.
from each side.
years.
Chapter 5 • Pre-Algebra Notetaking Guide
105
Example 2
Solving a Two-Step Equation
3
4
7
12
1
2
x Original equation
7
3
1
3
3
x 4 4
12
4
3
Subtract 4 from each side.
2
7
3
2
x 4 4
12
7
Write fractions using LCD.
1
x 4
12
12
7
12
x 7
7
12
Subtract.
1
4
x 3
7
Example 3
Multiply each side by
7
multiplicative inverse of 1
.
2
Multiply.
Writing and Solving a Two-Step Equation
Tree Growth The height of a certain Norway Spruce is 10 feet.
1
2
If the tree’s height grows 2 feet per year, find how long it will
take the tree to reach a height of 25 feet.
Solution
Number of
New
Growth
Current
p
years
height
rate
height
1
2
10 2 x 25
Write equation.
1
2
Subtract 10 from each side.
10 2 x 10 25 10
Simplify. Write mixed number
as improper fraction.
5
x 15
2
2
5
5
2
2
x 5
x 6
(
15
)
Multiply each side by
5
multiplicative inverse of 2 .
Multiply.
Answer: The tree will be 25 feet tall after 6 years.
Copyright © Holt McDougal. All rights reserved.
Chapter 5 • Pre-Algebra Notetaking Guide
105
5.7
Equations and Inequalities
with Rational Numbers
Goal: Use the LCD to solve equations and inequalities.
Solving an Equation by Clearing Fractions
Example 1
1
2
3
x 4
5
10
Original equation
14 x 130 25
Multiply each side by
LCD of fractions.
25
Use distributive
property.
Simplify.
Subtract
each side.
Simplify.
from
Divide each side
by
.
x
Simplify.
Checkpoint Solve the equation by first clearing the fractions.
1
3
5
6
7
9
1. x 106
Chapter 5 • Pre-Algebra Notetaking Guide
3
10
7
15
2
3
2. x Copyright © Holt McDougal. All rights reserved.
5.7
Equations and Inequalities
with Rational Numbers
Goal: Use the LCD to solve equations and inequalities.
Solving an Equation by Clearing Fractions
Example 1
1
2
3
x 4
5
10
Original equation
Multiply each side by
LCD of fractions.
25 Use distributive
property.
1
2
3
20 x 20 4
5
10
20
1
x
4
20
3
10
20 5x 6 8
Simplify.
5x 6 6 8 6
5x 2
5x
2
5
5
Subtract 6 from
each side.
Simplify.
Divide each side
by 5 .
2
x 5
Simplify.
Checkpoint Solve the equation by first clearing the fractions.
1
3
5
6
7
9
3
10
1. x 1
6
106
Chapter 5 • Pre-Algebra Notetaking Guide
7
15
2
3
2. x 11
14
Copyright © Holt McDougal. All rights reserved.
Example 2
Solving an Equation by Clearing Decimals
Solve the equation 2.75 6.15 0.4m.
Because the greatest number of decimal places in any of the
terms with decimals is
, multiply each side of the equation
by
, or
.
2.75 6.15 0.4m
(2.75) (6.15 0.4m)
m
Copyright © Holt McDougal. All rights reserved.
Original equation
Multiply each side
by
.
Use distributive property.
Simplify.
Subtract
each side.
from
Simplify.
Divide each side
by
.
Simplify.
Chapter 5 • Pre-Algebra Notetaking Guide
107
Example 2
Solving an Equation by Clearing Decimals
Solve the equation 2.75 6.15 0.4m.
Because the greatest number of decimal places in any of the
terms with decimals is 2 , multiply each side of the equation
by 10 2 , or 100 .
2.75 6.15 0.4m
100 (2.75) 100 (6.15 0.4m)
Use distributive property.
Simplify.
275 615 40m
275 615 615 40m 615
340 40m
340
40m
40
40
8.5 m
Copyright © Holt McDougal. All rights reserved.
Original equation
Multiply each side
by 100 .
Subtract 615 from
each side.
Simplify.
Divide each side
by 40 .
Simplify.
Chapter 5 • Pre-Algebra Notetaking Guide
107
Solving an Inequality with Fractions
Example 3
Geometry Describe the possible values
of x if the area of the rectangle is at least
24 square inches.
6
2
x
5
Solution
Length p Width
p
≥
Area
≥
Substitute.
≥
Use distributive property.
≥
Subtract
≥
≥
(
)
Answer: The possible values of x are
Chapter 5 • Pre-Algebra Notetaking Guide
from each side.
Simplify.
≥
108
2
Multiply each side by
multiplicative inverse of
.
Simplify.
.
Copyright © Holt McDougal. All rights reserved.
Solving an Inequality with Fractions
Example 3
Geometry Describe the possible values
of x if the area of the rectangle is at least
24 square inches.
6
2
x
5
Solution
Length p Width
25 x 2
12
x
5
12
x
5
≥
Area
p 6 ≥ 24
Substitute.
12 ≥ 24
Use distributive property.
12 12 ≥ 24 12
12
x ≥ 12
5
5
12
12
x
5
≥ 5
12
x ≥ 5
2
Subtract 12 from each side.
Simplify.
( 12 )
Multiply each side by
12
multiplicative inverse of 5 .
Simplify.
Answer: The possible values of x are 5 or more .
108
Chapter 5 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Example 4
Solving an Inequality by Clearing Fractions
1
6
5
6
5
12
m ≤ 16 m 152 ≤
≤
≤
≤
Original inequality
56
Multiply each side by
LCD of fractions.
Use distributive
property.
5
6
Simplify.
Add
≤
to each side.
Simplify.
Divide each side by
.
the
inequality symbol.
Simplify.
m
Checkpoint Solve the inequality by first clearing the fractions.
4
11
2
3
3. x 1 < Copyright © Holt McDougal. All rights reserved.
3
7
1
4
1
2
4. x < Chapter 5 • Pre-Algebra Notetaking Guide
109
Solving an Inequality by Clearing Fractions
Example 4
1
6
5
6
5
12
m ≤ Original inequality
Multiply each side by
LCD of fractions.
Use distributive
property.
1
5
5
12 m ≤ 12 6
6
12
12
1
6
m
12
5
12
5
6
≤ 12 2m 5 ≤ 10
Simplify.
2m 5 5 ≤ 10 5
Add 5 to each side.
2m ≤ 5
Simplify.
Divide each side by
2 . Reverse the
inequality symbol.
2m
5
≥ 2
2
m ≥
5
2
Simplify.
Checkpoint Solve the inequality by first clearing the fractions.
4
11
2
3
3
7
3. x 1 < 11
12
x < Copyright © Holt McDougal. All rights reserved.
1
4
1
2
4. x < 7
12
x < Chapter 5 • Pre-Algebra Notetaking Guide
109
5
Words to Review
Give an example of the vocabulary word.
Rational number
Terminating decimal
Repeating decimal
Inductive reasoning
Deductive reasoning
Counterexample
Reciprocal
Multiplicative inverse
Review your notes and Chapter 5 by using the Chapter Review on
pages 264–267 of your textbook.
110
Chapter 5 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
5
Words to Review
Give an example of the vocabulary word.
Rational number
Terminating decimal
3
4
3
0.75
4
Repeating decimal
Inductive reasoning
1
0.3
3
Examples: 2 4 6, 4 6 10,
6 8 14
Conjecture: The sum of two
even integers is even.
Deductive reasoning
Counterexample
Two even integers can be
represented by 2a and 2b, where
a and b are any integers. The
sum 2a + 2b can be written as
2(a + b). This sum is even
because it is a multiple of 2.
Therefore, the sum of two
integers is even.
Reciprocal
Conjecture: An integer multiplied
by a negative integer is negative.
Counterexample: 0 • –4 = 0
Multiplicative inverse
3
4
4
3
The reciprocal of is .
3 4
p 1
4 3
Review your notes and Chapter 5 by using the Chapter Review on
pages 264–267 of your textbook.
110
Chapter 5 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.