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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.
пЃ¬
Relatively prime numbers have no
common factors other than 1.
Example 1: Simplifying
Fractions
пЃ¬
Simplify.
6
9
21
25
пЂ­ 24
32
пЃ¬
Try these on your
ownвЂ¦
5
10
16
80
пЂ­18
29
Example 2: Writing Decimals as
Fractions
пЃ¬
пЃ¬
Write each decimal as a fraction in simplest form.
вЂў
0.5
вЂў
-2.37
вЂў
0.8716
вЂў
вЂў
вЂў
-0.8
вЂў
-4/5
5.37
вЂў
5 37/100
0.622
вЂў
311/500
Example 3: Writing Fractions as
Decimals
пЃ¬
Write each fraction
as a decimal.
вЂў
вЂў
пЃ¬
Try these on your
ownвЂ¦
5/4
вЂў 1.25
вЂў
11/9
вЂў
7/20
1/6
вЂў 0.16666
вЂў 1.22222
вЂў 0.35
Subtracting Rational Numbers
пЃ¬
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 ownвЂ¦
пЃ¬
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?
вЂў She ran 0.73 seconds faster in June.
Example 2: Using a Number Line
пЃ¬
-0.4 + 1.3
пЃ¬
Try these on your
ownвЂ¦
вЂў 0.3 +(-1.2)
пЃ¬
-5/8 + (-7/8)
вЂў -0.9
вЂў 1/5 + 2/5
вЂў 3/5
Fractions with Like Denominators
пЃ¬
вЂў 6/11 + 9/11
пЃ¬
Try these on your
ownвЂ¦
вЂў -2/9 вЂ“ 5/9
вЂў -7/9
вЂў -3/8 вЂ“ 5/8
вЂў 6/7 + (-3/7)
вЂў 3/7
Example 4: Evaluating Expressions
with Rational Numbers
пЃ¬
вЂў 12.1 вЂ“ x for x = -0.1
вЂў 12.2
вЂў 7/10 + m for m = 3 1/10
вЂў3
пЃ¬
4/5
Evaluate each expression for the given
value of the variable.
вЂў 23.8 + x for x = -41.3
вЂў -1/8 + t for t = 2
5/8
Section 3.3: Multiplying Rational
Numbers
Example 1: Multiplying a
Fraction and an Integer
пЃ¬
Multiply. Write each
form.
вЂў 6 (2/3)
вЂў -4 (2
пЃ¬
Try these on your
ownвЂ¦
вЂў -8(6/7)
вЂў -6 6/7
вЂў 2(5
3/5)
1/3)
вЂў 10
2/3
Example 2: Multiplying Fractions
пЃ¬
Multiply. Write each
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
пЃ¬
Try these on your
ownвЂ¦
1пѓ¦6пѓ¶
пѓ§ пѓ·
8пѓЁ7пѓё
2пѓ¦9пѓ¶
пЂ­ пѓ§ пѓ·
3пѓЁ 2пѓё
3пѓ¦1пѓ¶
4 пѓ§ пѓ·
7пѓЁ2пѓё
Example 3: Multiplying Decimals
пЃ¬
Multiply.
вЂў -2.5(-8)
вЂў -0.07(4.6)
пЃ¬
вЂў 2(-0.51)
вЂў -1.02
вЂў (-0.4)(-3.75)
вЂў 1.5
Example 4: Evaluating Expressions
with Rational Numbers
пЃ¬
Evaluate -5 1/2t for
each value of t.
вЂў t = -2/3
пЃ¬
Try these one on
вЂў Evaluate -3 1/8x for
each value of x.
вЂў t=8
вЂў x=5
вЂў
-15 5/8
вЂў x = 2/7
вЂў
-25/28
Section 3.4: Dividing Rational
Numbers
пЃ¬
A number and its reciprocal have a
product of 1.
Example 1: Dividing Fractions
пЃ¬
Try these on your
ownвЂ¦
5 1
п‚ё
11 2
3
2 п‚ё2
8
пЃ¬
Divide. Write each
forms.
7 2
п‚ё
12 3
1
3 п‚ё4
4
Example 2: Dividing Decimals
пЃ¬
Divide.
вЂў 2.92 / 0.4
вЂў 7.3
пЃ¬
Try this one on your ownвЂ¦
вЂў 0.384 / 0.24
вЂў 1.6
Example 3: Evaluating Expressions
with Fractions and Decimals
пЃ¬
Evaluate each expression for the given
value of the variable.
вЂў 7.2/n for n = 0.24
вЂў M / (3/8) for M = 7 1/2
пЃ¬
Evaluate each expression for the given
value of the variable.
вЂў 5.25/n for n = 0.15
вЂў 35
вЂў K / (4/5) for K = 5
вЂў6
1/4
Example 4: Problem Solving
пЃ¬
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 ownвЂ¦
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?
вЂў You can bake 1 ВЅ batches of the cookies.
with Unlike Denominators
пЃ¬
вЂў 2/3 + 1/5
пЃ¬
Try these on your
ownвЂ¦
вЂў 1/8 + 2/7
вЂў 23/56
вЂў 3 2/5 + (-3 ВЅ)
вЂў 1 1/6 + 5/8
вЂў1
19/24
Example 2: Evaluating Expressions
with Rational Numbers
пЃ¬
Evaluate n вЂ“ 11/16 for n = -1/3.
пЃ¬
Try this one on your ownвЂ¦
вЂў Evaluate t вЂ“ 4/5 for t = 5/6.
вЂў 1/30
Example 3: Consumer
Application
пЃ¬
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 ownвЂ¦
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?
вЂў There will be 9 3/8 inches left.
Section 3.6: Solving Equations
with Rational Numbers
пЃ¬
Example 1: Solving Equations with
Decimals
вЂў Solve.
вЂў
y вЂ“ 12.5 = 17
вЂў -2.7p = 10.8
вЂў t/7.5 = 4
пЃ¬
Solve.
вЂў M + 4.6 = 9
вЂў M = 4.4
вЂў 8.2p = -32.8
вЂў p = -4
вЂў x/1.2 = 15
вЂў x = 18
Example 2: Solving Equations
with Fractions
пЃ¬
Solve.
вЂў x + 1/5 = -2/5
пЃ¬
Try these on your
ownвЂ¦
вЂў n + 2/7 = -3/7
вЂў n = -5/7
вЂў x вЂ“ Вј = 3/8
вЂў y вЂ“ 1/6 = 2/3
вЂў y = 5/6
вЂў 5/6(x) = 5/8
вЂў 3/5(w) = 3/16
вЂў x = 3/4
Example 3: Solving Word
Problems Using Equations
пЃ¬
Try this one on your ownвЂ¦
вЂў
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?
вЂў Mr. Rios has 3 Вј 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
пЃ¬
Solving Inequalities
with Decimals
пЃ¬
Try these on your
ownвЂ¦
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
пЃ¬
Solve.
1
пЂј1
2
xпЂ«
пЂ­3
пЃ¬
1
y п‚і 10
3
xпЂ«
2
пЂѕ 2
3
пЂ­2
1
nп‚Ј9
4
Example 3: Problem Solving
Application
пЃ¬
With first-class mail, there is an extra cost
in any of these cases:
вЂў The length is greater than 11 ВЅ inches.
вЂў The height is greater than 6 1/8 inches.
вЂў The thickness is greater than Вј inch.
вЂў 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 ownвЂ¦
пЃ¬
With first-class mail, there is an extra cost in any
of these cases:
вЂў
вЂў
вЂў
вЂў
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.
вЂў 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
пЃ¬
The principal square root is the nonnegative square root.
пЃ¬
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
вЂў Find the two square roots of each number.
вЂў 64
вЂў1
вЂў 121
пЃ¬
Find the two square roots of each
number.
вЂў 49
вЂў + or - 7
вЂў 100
вЂў + or - 10
вЂў 225
вЂў + or - 15
Example 2: Computer
Application
пЃ¬
The square
computer icon (pg.
147) contains 676
pixels. How many
pixels tall is the
icon?
пЃ¬
Try this one on your
ownвЂ¦
вЂў A square window
has an area of 169
square inches. How
wide is the window?
вЂў The window is 13
inches wide.
Example 3: Evaluating Expressions
Involving Square Roots
пЃ¬
Evaluate each expression.
2 16 пЂ« 5
9 пЂ« 16 пЂ« 7
пЃ¬
3 36 пЂ« 7
21пЂ­ 5 пЂ« 9
Section 3.9: Finding Square
Roots
пЃ¬
Estimating Square
Roots of NumbersвЂ¦
30
пЂ­ 150
пЃ¬
Try these on your
ownвЂ¦
55
пЂ­ 90
Example 2: Problem Solving
Application
пЃ¬
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 ownвЂ¦
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.
вЂў The length of each side of the table is about
вЂў
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
пЃ¬
Use a calculator to
300
find
. Round
to the nearest tenth.
пЃ¬
Try this one on your
ownвЂ¦
вЂў Use a calculator to
find 500 . Round to
the nearest tenth.
вЂў 22.4
Section 3.10: The Real Numbers
пЃ¬
Irrational numbers can only be written as
decimals that do not terminate or repeat.
пЃ¬
The set of real numbers consists of the set
of rational numbers and the set of irrational
numbers.
пЃ¬
The Density Property of real numbers
states that between any two real numbers
is another real number.
Example 1: Classifying Real
Numbers
пЃ¬
Write all the names
that apply to each
number.
пЃ¬
Try these on your
ownвЂ¦
5
3
пЂ­ 56.85
9
3
пЂ­ 12.75
16
2
Example 2: Determine the
Classification of All Numbers
пЃ¬
Try these on your
ownвЂ¦
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
пЃ¬
Find a real number between 2 1/3 and 2
2/3.
пЃ¬
Try this one on your ownвЂ¦
вЂў Find a real number between 3 2/5 and 3 3/5.
вЂў3ВЅ
```
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