M3: Chapter 5 Notes
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Academic
Pre-Algebra
Chapter 5 Notes
Rational Numbers
and
Equations
Name_________________ Pd._____
M3: Chapter 5 Notes
Sections 5.1-5.6 Vocabulary List
Section 4.3:




Equivalent fractions
Simplest form
Greatest common factor
Relatively prime
Section 5.1:
 Rational number
 Terminating decimal
 Repeating decimal
Section 5.2:
 Like fractions
Section 5.3:
 Unlike fractions
 Least common multiple
 Least common denominator
Section 5.5:
 Reciprocals
Section 5.6:
 Multiplicative inverse
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M3: Chapter 5 Notes
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Section 4.3: Equivalent Fractions
Learning Goal: We will write equivalent fractions and write fractions in simplest
form.
Fractions:
a
b
Vocabulary:
 Equivalent Fractions –
M3: Chapter 5 Notes
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Example 1: Writing Equivalent Fractions
Write two fractions that are equivalent to
15
.
18
ON YOUR OWN:
 Greatest Common Factor – the greatest whole number that is a
common factor of two or more nonzero whole numbers
 Relatively Prime – two or more nonzero whole numbers whose
greatest common factor is 1
 Simplest Form – a fraction whose numerator and denominator are
relatively prime
Example 2: Writing Fractions in Simplest Form
Write the fraction in simplest form.
12
25
b.
a.
30
40
M3: Chapter 5 Notes
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Example 3: Simplifying a fraction
Julia served 42 customers during the breakfast shift at a diner.
Twenty-eight of the customers ordered eggs. Write the fraction in
simplest form, of customers she served who ordered eggs.
ON YOUR OWN:
SIMPLIFYING VARIABLE EXPRESSIONS:
1.
2.
Example 4: Simplifying a Variable Expression
Write the fraction in simplest form.
10 xy
16bc 3
a.
b.
15 y 2
24b 2 c
M3: Chapter 5 Notes
ON YOUR OWN:
a
a.
abc
Page 6 of 25
2mn
b.
6m
24 x 2 y
c.
8 xy
3ab 2
d.
12 ac
M3: Chapter 5 Notes
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Section 5.1: Rational Numbers
Learning Goal: We will write fractions as decimals and decimals as fractions.
Vocabulary:
 Rational number –
Example 1: Identifying Rational Numbers
Show that the number is rational by writing it as a quotient of two
integers.
a.
8
b.
 14
c.
3
5
4
d.
1
9
2
 Terminating decimal –
 Repeating decimal –
Example 2: Writing Fractions as Decimals
3
a. Write as a decimal.
8
5
b. Write
11 as a decimal.
M3: Chapter 5 Notes
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ON YOUR OWN:
Write the fraction or mixed number as a decimal.
a.
3
10
b.

2
3
c.
1
9
20
d.
29
80
Example 3: Using Decimals to Compare Fractions
Of the 20 students on the girls’ swim team, 9 are seniors. Of the 24
students on the boys’ swim team, 10 are seniors. On which team is the
fraction of students who are seniors greater?
ON YOUR OWN:
Of the 50 mammal species found in Canyonlands, National Park, 20
species belong to the order Rodentia. Of the 54 mammal species found
in Badlands National Park, 24 belong to Rodentia. In which park is the
fraction of mammal species belonging to Rodentia greater?
M3: Chapter 5 Notes
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WRITING DECIMALS AS FRACTIONS:
***To write a terminating decimal as a fraction or mixed number, use
the place of the ______________________________ to determine
the ___________________________ of the fraction.
Example 4: Writing Terminating Decimals as Fractions
a.
0.7
b.
 3.05
c.
0.4
d.
0.324
What happens when you write a repeating decimal as a fraction?
Proof:
Hint:
M3: Chapter 5 Notes
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Example 5: Writing a Repeating Decimal as a Fraction
a. Write
fraction.
0.93 as a
b. Write
fraction.
0.78 as a
Example 6: Ordering Rational Numbers
Order the numbers from least to greatest.
5
5
13


 0.2 , 4.31 ,  3 , 2 , 3
a.
4,
5 4
9


 2.3 , 2
b. 0.7 ,  1 ,
4, 3 ,
c. Write
fraction.
2.6 as a
M3: Chapter 5 Notes
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Section 5.2: Adding and Subtracting Like Fractions
Learning Goal: We will add and subtract like fractions.
Adding and Subtracting Like Fractions
To add or subtract fractions with the same denominator, write the sum
or difference of the numerators over the denominator.
Numbers
Algebra
4 1 5
 
9 9 9
9 2 7
 
11 11 11
a b ab
 
,c  0
c c
c
a b a b
 
,c  0
c c
c
Example 1: Adding Like Fractions
a.
7 1

9 9
77
9
c. Find the sum of
and
.
100
100
b.

2 5

7 7
M3: Chapter 5 Notes
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Example 2: Subtracting Like Fractions
a.

4 2

7 7
b.
1  3
  
10  10 
ON YOUR OWN:
Find the sum or difference.
a.
3 2

8 8
b.
1 5
 
6 6
c.
2 7

15 15
d.
1  7
  
12  12 
***To add or subtract mixed numbers, you can first write the mixed
numbers as _____________________________.
Example 3: Adding and Subtracting Mixed Numbers
a.
2
11
7
6
12
12
b.
8
9
2
 10
15
15
M3: Chapter 5 Notes
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ON YOUR OWN:
a.
3
3
2 1
4
4
b.
2
1
6 3
3
3
c.
1
3
4 2
5
5
Example 4: Simplifying Variable Expressions
3k 9k

a.
28 28
4
9

b.
7y 7y
3a 5a

c.
20 20
8  2

 
d.
3b  3b 
d.
2
3
3 6
7
7
M3: Chapter 5 Notes
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Section 5.3: Adding and Subtracting Unlike Fractions
Learning Goal: We will add and subtract unlike fractions.
 Least common multiple (LCM) – the least number that is a common
multiple of two or more numbers
 Least common denominator (LCD) – the least common multiple of
the denominators of two or more fractions
Example 1: Adding and Subtracting Fractions
a. 
3 1

8 2
ON YOUR OWN:
b. 
1  2
  
4  3
M3: Chapter 5 Notes
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Example 2: Adding Mixed Numbers
4
5
2 
6
a. 3  1
b.  4    1 
9 12
5  11 
ON YOUR OWN:
Find the sum or difference.
Example 3: Subtracting Mixed Numbers
Tyrone volunteered to work 7 ½ hours at a weekend fundraiser. On
2
Saturday, he worked for 2 hours. How many hours will he be working
3
at the fundraiser on Sunday?
M3: Chapter 5 Notes
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ON YOUR OWN:
Texas blind salamanders have been found in lengths varying from
3
1
3 inches to 5 inches. Find the range of these lengths.
8
4
Example 4: Simplifying an Expression
ON YOUR OWN:
Simplify the expression.
d d

a.
4 12
b.
m 2m

5
3
c.
3x 7 x

8 12
M3: Chapter 5 Notes
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Section 5.4: Multiplying Fractions
Learning Goal: We will multiply fractions and mixed numbers.
Example 1: Multiplying Fractions
Find the product
3 11
 
5 12
ON YOUR OWN:
Find the product.
1.
2 7

3 8
2.
 5  3 
   
 12  10 
3.
3
  12 
4
M3: Chapter 5 Notes
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Example 2: Multiplying a Mixed Number and an Integer
ON YOUR OWN:
A recipe for one loaf of bread requires
1
3 cups of flour. Joe wants to
4
make 15 loaves. How much flour does he need?
Example 3: Multiplying Mixed Numbers
2 7
7 2
3 10
M3: Chapter 5 Notes
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ON YOUR OWN:
Example 4: Simplifying Expressions
2 x  3x 

 
b.
9  4 
c 14

a.
7 15
ON YOUR OWN:
Simplify the expression.
3y y

a.
4 9
b. 
3z 2 z

25 15
c. 
4v 7v

21 16
M3: Chapter 5 Notes
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Section 5.5: Dividing Fractions
Learning Goal: We will divide fractions and mixed numbers.
Vocabulary:
 Reciprocal –
M3: Chapter 5 Notes
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Example 1: Dividing a Fraction by a Fraction
Find the quotient 
2  5
  
3  6
Example 2: Dividing a Mixed Number by a Mixed Number
Find the quotient  6
ON YOUR OWN:
2 5
1 .
3 9
M3: Chapter 5 Notes
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Example 3: Dividing a Whole Number by a Mixed Number
ON YOUR OWN:
Carissa mixes 2 gallons (32 cups) of fruit punch for a cookout. If each
1
3
of the tumblers she plans to serve the punch in holds 2 cups, how
many tumblers can she fill?
M3: Chapter 5 Notes
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Section 5.6: Using Multiplicative Inverses to Solve Equations
Learning Goal: We will use multiplicative inverses to solve equations.
Vocabulary:
 Multiplicative inverse –
Example 1: Solving a One-Step Equation
a. Solve 
7
k  56 .
8
b. Solve
2
x  12 .
9
M3: Chapter 5 Notes
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Example 2: Solving a Two-Step Equation
a. Solve
13 3
1
 g .
16 8
2
ON YOUR OWN:
b. Solve
4
x  7  31 .
9
M3: Chapter 5 Notes
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Example 3: Writing and Solving a Two-Step Equation
There are currently 1680 students at Fairview Middle School. So far
1
2
this school year, an average of 3 new students have enrolled at the
school each week. The school has a maximum capacity of 1750
students. If this growth rate continues, in how many weeks will the
school reach its maximum student capacity?
ON YOUR OWN:
The length of the United States flag is 1
9
times the width of the
10
flag. A particular U.S. flag is 5 feet long. Write and solve an equation
to find the width of the flag.
EXTRA PRACTICE: