chapter 3: rational and real numbers regular math
TRANSCRIPT
Chapter 3: Rational and Real Numbers
Regular Math
Section 3.1: Rational Numbers
A rational number is any number that can be written as a fraction.
Relatively prime numbers have no common factors other than 1.
Example 1: Simplifying Fractions
Simplify. Try these on your own…
32
24
25
21
9
6
29
18
80
16
10
5
Example 2: Writing Decimals as Fractions 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
Write each fraction as a decimal.• 5/4
• 1.25
• 1/6
• 0.16666
Try these on your own…
• 11/9
• 1.22222
• 7/20
• 0.35
Section 3.2: Adding and Subtracting Rational Numbers
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…
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
-0.4 + 1.3
-5/8 + (-7/8)
Try these on your own…• 0.3 +(-1.2)
• -0.9
• 1/5 + 2/5• 3/5
Example 3: Adding and Subtracting Fractions with Like Denominators
Add or Subtract.• 6/11 + 9/11
• -3/8 – 5/8
Try these on your own…• -2/9 – 5/9
• -7/9
• 6/7 + (-3/7)• 3/7
Example 4: Evaluating Expressions with Rational Numbers
Try these on your own…• 12.1 – x for x = -0.1
• 12.2
• 7/10 + m for m = 3 1/10• 3 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
Multiply. Write each answer in simplest form.• 6 (2/3)
• -4 (2 3/5)
Try these on your own…• -8(6/7)
• -6 6/7
• 2(5 1/3)• 10 2/3
Example 2: Multiplying Fractions
Multiply. Write each answer in simplest form.
Try these on your own…
20
7
3
26
5
12
12
5
5
3
2
1
2
1
7
34
2
9
3
2
7
6
8
1
Example 3: Multiplying Decimals
Multiply.• -2.5(-8)
• -0.07(4.6)
Try these on your own…• 2(-0.51)
• -1.02
• (-0.4)(-3.75)• 1.5
Example 4: Evaluating Expressions with Rational Numbers
Evaluate -5 1/2t for each value of t.• t = -2/3
• t = 8
Try these one on your own…• Evaluate -3 1/8x for
each value of x.
• x = 5
• -15 5/8
• x = 2/7
• -25/28
Section 3.4: Dividing Rational Numbers
A number and its reciprocal have a product of 1.
Example 1: Dividing Fractions
Try these on your own…
Divide. Write each answer in simplest forms.
44
13
3
2
12
7
28
32
2
1
11
5
Example 2: Dividing Decimals
Divide.• 2.92 / 0.4
• 7.3
Try this one on your own…• 0.384 / 0.24
• 1.6
Example 3: Evaluating Expressions with Fractions and Decimals
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…
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
You pour 2/3 cup of sports drink into a glass. The serving size is 6 ounces, or ¾ 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…
A cookie recipe calls for ½ cup of oats. You have ¾ 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
Add or subtract.• 2/3 + 1/5
• 3 2/5 + (-3 ½)
Try these on your own…• 1/8 + 2/7
• 23/56
• 1 1/6 + 5/8• 1 19/24
Example 2: Evaluating Expressions with Rational Numbers
Evaluate n – 11/16 for n = -1/3.
Try this one on your own…• Evaluate t – 4/5 for t = 5/6.
• 1/30
Example 3: Consumer Application
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 ½ yards long. How many yards will be left on the bolt?
Try this one on your own…
Two dancers are making necklaces from ribbon for their costumes. They need pieces measuring 13 ¾ inches and 12 7/8 inches How much ribbon will be left over after the pieces are cut from 36-inch length?• There will be 9 3/8 inches left.
Section 3.6: Solving Equations with Rational Numbers
Example 1: Solving Equations with Decimals• Solve.
• y – 12.5 = 17
• -2.7p = 10.8
• t/7.5 = 4
Try these on your own…
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
Solve.• x + 1/5 = -2/5
• x – ¼ = 3/8
• 3/5(w) = 3/16
Try these on your own…• n + 2/7 = -3/7
• n = -5/7
• y – 1/6 = 2/3• y = 5/6
• 5/6(x) = 5/8• x = 3/4
Example 3: Solving Word Problems Using Equations Try this one on your own…
• Mr. Rios wants to prepare a casserole that requires 2 ½ cups of milk. If he makes the casserole, he will have only ¾ cup of milk left for his breakfast cereal. How much milk does Mr. Rios have?• Mr. Rios has 3 ¼ cups of milk.
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
Solving Inequalities with Decimals
Try these on your own…
305.7
5.05.0
t
x
118.3
8.04.0
y
x
Example 2: Solving Inequalities with Fractions
Solve.
Try these on your own…
103
13
12
1
y
x
94
12
23
2
n
x
Example 3: Problem Solving Application 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 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… 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 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
The principal square root is the non-negative square root.
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…
Find the two square roots of each number.• 49
• + or - 7
• 100• + or - 10
• 225• + or - 15
Example 2: Computer Application
The square computer icon (pg. 147) contains 676 pixels. How many pixels tall is the icon?
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
Evaluate each expression.
Try these on your own…
7169
5162
9521
7363
Section 3.9: Finding Square Roots
Estimating Square Roots of Numbers…
Try these on your own…
150
30
90
55
Example 2: Problem Solving Application
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…
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
Use a calculator to find . Round to the nearest tenth.
Try this one on your own…• Use a calculator to
find . Round to the nearest tenth.• 22.4
300
500
Section 3.10: The Real Numbers Irrational numbers can only be written as
decimals that do not terminate or repeat.
The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
The Density Property of real numbers states that between any two real numbers is another real number.
Example 1: Classifying Real Numbers
Write all the names that apply to each number.
Try these on your own…
3
9
85.56
3
216
75.12
5
Example 2: Determine the Classification of All Numbers
Try these on your own…
State if the number is rational, irrational, or not a real number.
17
4
1
0
3
10
9
4
9
3
0
15
Example 3: Applying the Density Property of Real Numbers
Find a real number between 2 1/3 and 2 2/3.
Try this one on your own…• Find a real number between 3 2/5 and 3 3/5.
• 3 ½