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Chapter 3: Rational and Real Numbers

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Chapter 3: Rational
and Real Numbers
Regular Math
Section 3.1: Rational Numbers
пЃ¬
A rational number is any number that
can be written as a fraction.
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Relatively prime numbers have no
common factors other than 1.
Example 1: Simplifying
Fractions
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Simplify.
6
9
21
25
пЂ­ 24
32
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Try these on your
own…
5
10
16
80
пЂ­18
29
Example 2: Writing Decimals as
Fractions
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пЃ¬
Write each decimal as a fraction in simplest form.
•
0.5
•
-2.37
•
0.8716
Try these on your own…
•
•
•
-0.8
•
-4/5
5.37
•
5 37/100
0.622
•
311/500
Example 3: Writing Fractions as
Decimals
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Write each fraction
as a decimal.
•
•
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Try these on your
own…
5/4
• 1.25
•
11/9
•
7/20
1/6
• 0.16666
• 1.22222
• 0.35
Section 3.2: Adding and
Subtracting Rational Numbers
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Example 1:
In the 2001 World Championships 100-meter
dash, it took Maurice Green 0.132 seconds to
react to the starter pistol. His total race time,
including this reaction time, was 9.82
seconds. How long did it take him to run the
actual 100 meters?
Try this one on your own…
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In August 2001 at the World University
Games in Beijing, China, Jimyria Hicks
ran the 200-meter dash in 24.08
seconds. Her best time at the U.S.
Senior National Meet in June of the
same year was 23.35 seconds. How
much faster did she run in June?
• She ran 0.73 seconds faster in June.
Example 2: Using a Number Line
to Add Rational Numbers
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-0.4 + 1.3
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Try these on your
own…
• 0.3 +(-1.2)
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-5/8 + (-7/8)
• -0.9
• 1/5 + 2/5
• 3/5
Example 3: Adding and Subtracting
Fractions with Like Denominators
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Add or Subtract.
• 6/11 + 9/11
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Try these on your
own…
• -2/9 – 5/9
• -7/9
• -3/8 – 5/8
• 6/7 + (-3/7)
• 3/7
Example 4: Evaluating Expressions
with Rational Numbers
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Try these on your own…
• 12.1 – x for x = -0.1
• 12.2
• 7/10 + m for m = 3 1/10
•3
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4/5
Evaluate each expression for the given
value of the variable.
• 23.8 + x for x = -41.3
• -1/8 + t for t = 2
5/8
Section 3.3: Multiplying Rational
Numbers
Example 1: Multiplying a
Fraction and an Integer
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Multiply. Write each
answer in simplest
form.
• 6 (2/3)
• -4 (2
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Try these on your
own…
• -8(6/7)
• -6 6/7
• 2(5
3/5)
1/3)
• 10
2/3
Example 2: Multiplying Fractions
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Multiply. Write each
answer in simplest
form. пЂ­1 пѓ¦ пЂ­ 3 пѓ¶
пѓ§ пѓ·
2пѓЁ 5 пѓё
5 пѓ¦ 12 пѓ¶
пѓ§пЂ­ пѓ·
12 пѓЁ 5 пѓё
2пѓ¦ 7 пѓ¶
6 пѓ§ пѓ·
3 пѓЁ 20 пѓё
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Try these on your
own…
1пѓ¦6пѓ¶
пѓ§ пѓ·
8пѓЁ7пѓё
2пѓ¦9пѓ¶
пЂ­ пѓ§ пѓ·
3пѓЁ 2пѓё
3пѓ¦1пѓ¶
4 пѓ§ пѓ·
7пѓЁ2пѓё
Example 3: Multiplying Decimals
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Multiply.
• -2.5(-8)
• -0.07(4.6)
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Try these on your own…
• 2(-0.51)
• -1.02
• (-0.4)(-3.75)
• 1.5
Example 4: Evaluating Expressions
with Rational Numbers
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Evaluate -5 1/2t for
each value of t.
• t = -2/3
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Try these one on
your own…
• Evaluate -3 1/8x for
each value of x.
• t=8
• x=5
•
-15 5/8
• x = 2/7
•
-25/28
Section 3.4: Dividing Rational
Numbers
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A number and its reciprocal have a
product of 1.
Example 1: Dividing Fractions
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Try these on your
own…
5 1
п‚ё
11 2
3
2 п‚ё2
8
пЃ¬
Divide. Write each
answer in simplest
forms.
7 2
п‚ё
12 3
1
3 п‚ё4
4
Example 2: Dividing Decimals
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Divide.
• 2.92 / 0.4
• 7.3
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Try this one on your own…
• 0.384 / 0.24
• 1.6
Example 3: Evaluating Expressions
with Fractions and Decimals
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Evaluate each expression for the given
value of the variable.
• 7.2/n for n = 0.24
• M / (3/8) for M = 7 1/2
Try these on your own…
пЃ¬
Evaluate each expression for the given
value of the variable.
• 5.25/n for n = 0.15
• 35
• K / (4/5) for K = 5
•6
1/4
Example 4: Problem Solving
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You pour 2/3 cup of
sports drink into a
glass. The serving
size is 6 ounces, or
Вѕ cup. How many
servings will you
consume? How
many calories will
you consume?
Calories
50
Total Fat 0g
0%
Sodium 110mg
5%
Potassium 30mg
1%
Total Carbs 0g
5%
Sugar 14g
5%
Protein 0g
0%
Try this one on your own…
пЃ¬
A cookie recipe calls for ВЅ cup of oats.
You have Вѕ cup of oats. How many
batches of the cookies can you bake?
• You can bake 1 ½ batches of the cookies.
Section 3.5: Adding and Subtracting
with Unlike Denominators
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Add or subtract.
• 2/3 + 1/5
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Try these on your
own…
• 1/8 + 2/7
• 23/56
• 3 2/5 + (-3 ½)
• 1 1/6 + 5/8
•1
19/24
Example 2: Evaluating Expressions
with Rational Numbers
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Evaluate n – 11/16 for n = -1/3.
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Try this one on your own…
• Evaluate t – 4/5 for t = 5/6.
• 1/30
Example 3: Consumer
Application
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A folkloric dance skirt pattern calls for 2
2/5 yards of 45-inch-wide material to
make the ruffle and 9 1/3 yards to make
the skirt. The material for the skirt and
ruffle will be cut from a bolt that is 15 ВЅ
yards long. How many yards will be left
on the bolt?
Try this one on your own…
пЃ¬
Two dancers are making necklaces from
ribbon for their costumes. They need
pieces measuring 13 Вѕ inches and 12
7/8 inches How much ribbon will be left
over after the pieces are cut from 36inch length?
• There will be 9 3/8 inches left.
Section 3.6: Solving Equations
with Rational Numbers
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Example 1: Solving Equations with
Decimals
• Solve.
•
y – 12.5 = 17
• -2.7p = 10.8
• t/7.5 = 4
Try these on your own…
пЃ¬
Solve.
• M + 4.6 = 9
• M = 4.4
• 8.2p = -32.8
• p = -4
• x/1.2 = 15
• x = 18
Example 2: Solving Equations
with Fractions
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Solve.
• x + 1/5 = -2/5
пЃ¬
Try these on your
own…
• n + 2/7 = -3/7
• n = -5/7
• x – ¼ = 3/8
• y – 1/6 = 2/3
• y = 5/6
• 5/6(x) = 5/8
• 3/5(w) = 3/16
• x = 3/4
Example 3: Solving Word
Problems Using Equations
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Try this one on your own…
•
Mr. Rios wants to prepare a casserole that requires 2
ВЅ cups of milk. If he makes the casserole, he will have
only Вѕ cup of milk left for his breakfast cereal. How
much milk does Mr. Rios have?
• Mr. Rios has 3 ¼ cups of milk.
пЃ¬
In 1668 the Hope diamond was reduced
from its original weight by 45 1/6 carats
to a diamond weighing 67 1/8 carat. How
many carats was the original diamond?
Section 3.7: Solving Inequalities
with Rational Numbers
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Solving Inequalities
with Decimals
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Try these on your
own…
0.5x п‚і 0.5
0.4 x п‚Ј 0.8
t пЂ­ 7.5 пЂѕ 30
y пЂ­ 3.8 пЂј 11
Example 2: Solving Inequalities
with Fractions
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Solve.
1
пЂј1
2
xпЂ«
пЂ­3
пЃ¬
1
y п‚і 10
3
Try these on your own…
xпЂ«
2
пЂѕ 2
3
пЂ­2
1
nп‚Ј9
4
Example 3: Problem Solving
Application
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With first-class mail, there is an extra cost
in any of these cases:
• The length is greater than 11 ½ inches.
• The height is greater than 6 1/8 inches.
• The thickness is greater than ¼ inch.
• The length divided by the height is less than 1.3
or greater than 2.5
пЃ¬
The height of an envelope is 4.5 inches.
What are the minimum and maximum
lengths to avoid an extra charge?
Try this one on your own…
пЃ¬
With first-class mail, there is an extra cost in any
of these cases:
•
•
•
•
пЃ¬
The length is greater than 11 ВЅ inches.
The height is greater than 6 1/8 inches.
The thickness is greater than Вј inch.
The length divided by the height is less than 1.3 or
greater than 2.5
The height of an envelope is 3.8 inches. What
are the minimum and maximum lengths to avoid
an extra charge.
• The length of the envelope must be between 4.94 inches
and 9.5 inches to avoid extra charges.
Section 3.8: Squares and Square
Roots
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The principal square root is the nonnegative square root.
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A perfect square is a number that has
integers as its square roots.
Example 1: Finding the Positive and
Negative Square Roots of a Number
• Find the two square roots of each number.
• 64
•1
• 121
Try these on your own…
пЃ¬
Find the two square roots of each
number.
• 49
• + or - 7
• 100
• + or - 10
• 225
• + or - 15
Example 2: Computer
Application
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The square
computer icon (pg.
147) contains 676
pixels. How many
pixels tall is the
icon?
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Try this one on your
own…
• A square window
has an area of 169
square inches. How
wide is the window?
• The window is 13
inches wide.
Example 3: Evaluating Expressions
Involving Square Roots
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Evaluate each expression.
2 16 пЂ« 5
9 пЂ« 16 пЂ« 7
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Try these on your own…
3 36 пЂ« 7
21пЂ­ 5 пЂ« 9
Section 3.9: Finding Square
Roots
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Estimating Square
Roots of Numbers…
30
пЂ­ 150
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Try these on your
own…
55
пЂ­ 90
Example 2: Problem Solving
Application
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You want to install a square skylight that
has an area of 300 square inches.
Calculate the length of each side and the
length of trim you will need, to the
nearest tenth of an inch.
Try this one on your own…
пЃ¬
You want to sew a fringe on a square
tablecloth with an area of 500 square
inches. Calculate the length of each side
of the tablecloth and the length of fringe
you will need to the nearest tenth of an
inch.
• The length of each side of the table is about
•
22.4 inches.
You will need about 89.6 inches of fringe.
Example 3: Using a Calculator to
Estimate the Value of a Square Root
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Use a calculator to
300
find
. Round
to the nearest tenth.
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Try this one on your
own…
• Use a calculator to
find 500 . Round to
the nearest tenth.
• 22.4
Section 3.10: The Real Numbers
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Irrational numbers can only be written as
decimals that do not terminate or repeat.
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The set of real numbers consists of the set
of rational numbers and the set of irrational
numbers.
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The Density Property of real numbers
states that between any two real numbers
is another real number.
Example 1: Classifying Real
Numbers
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Write all the names
that apply to each
number.
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Try these on your
own…
5
3
пЂ­ 56.85
9
3
пЂ­ 12.75
16
2
Example 2: Determine the
Classification of All Numbers
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Try these on your
own…
15
0
3
пЃ¬
State if the number
is rational, irrational,
or not a real 10
number.
3
0
пЂ­9
4
9
1
4
пЂ­ 17
Example 3: Applying the Density
Property of Real Numbers
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Find a real number between 2 1/3 and 2
2/3.
пЃ¬
Try this one on your own…
• Find a real number between 3 2/5 and 3 3/5.
•3½
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