Download Sec. 4.1 Introduction to Fractions and Mixed Numbers

Sec. 4.1 Introduction to Fractions and Mixed Numbers
Learning Objectives:
1. Identify the numerator and the denominator of a fraction.
2. Write a fraction to represent parts of figures or real-life data.
3. Graph Fractions on a Number Line.
4. Review division properties for 0 and 1.
5. Write mixed numbers as improper fractions.
6. Write improper fractions as mixed numbers or whole numbers.
7. Key Vocabulary: fractions, numerator, denominator, proper fraction, improper fraction, mixed
number.
1. Identify numerator and denominator of a fraction
Definition:
Fraction—is an expression of the form
Example 1. Identify the numerator and the denominator of a fraction.
10
Numerator:________________________ Denominator:_____________________________
7
-------------------------------------------------------------------------------------------------------------------------------2. Write a fraction to represent parts of figures or real-life data
1.
Example 2. Write a fraction to represent the shaded part of each figure.
1.
2.
Answer:________________________
Answer:________________________
-------------------------------------------------------------------------------------------------------------------------------Example 3. Writing Fraction from a Real-Life Data
Of the 207 students taking Basic Mathematics, 143 are freshman. What fraction of the class is
freshman?
Answer:________________________
-------------------------------------------------------------------------------------------------------------------------------3. Graph Fractions on a Number Line
Example 4. Graph each fraction on a number line.
1.
5
8
1
2.
7
6
-------------------------------------------------------------------------------------------------------------------------------4. Review division properties for 0 and 1
Division Properties:
1.
a
=
a
2.
Let a be any whole number, a ≠ 0 , then
a
=
1
3.
0
=
a
4.
a
=
0
Example 5. Simplify.
1.
0
3
2.
−5
5
3.
2
0
Answer:__________________
Answer:__________________
Answer:__________________
-------------------------------------------------------------------------------------------------------------------------------5. Write mixed numbers as improper fractions
Definitions:
1. Proper Fraction—is a fraction whose numerator is less than its denominator.
Example:
2.
Improper Fraction— is a fraction whose numerator is grater than or equal to its
denominator.
Example:
3.
Mixed Number— is an expression of the form a
b
where a, b and c are any real numbers;
c
c ≠0.
Example:
Example 6. Identify each number as a proper fraction, improper fraction, or mixed number.
3
6
1.
2.
5
6
Answer:__________________
Answer:__________________
3.
15
13
4.
Answer:__________________
2
1
3
Answer:__________________
2
Example 7. Represent the shaded part of each figure group as both an improper fraction and a
mixed number.
Answer:__________________
-------------------------------------------------------------------------------------------------------------------------------Steps for Writing a Mixed Number as an Improper Fraction
1.
2.
3.
Multiply the denominator of the fraction by the whole number.
Add the numeration of the fraction to the product from step 1.
Write the sum from step 2 as the numerator of the improper fraction over the original
denominator.
Example 8. Write each mixed number as an improper fraction.
5
7
8
Answer:__________________
-------------------------------------------------------------------------------------------------------------------------------6. Write Improper Fractions as Mixed numbers or Whole Numbers
Steps for Writing Improper Fractions as Mixed numbers or Whole Numbers
1.
2.
Divide the denominator into the numerator.
The whole number part of the mixed number is the quotient. The fraction part of the mixed
number is the remainder over the original denominator.
Re mainder
original deno min ator
Example 9. Write each mixed number as an improper fraction or a whole number.
Mixed Number = quotient
1.
51
7
Answer:________________________
2.
48
4
Answer:________________________
3
Sec. 4.2 Factors and Simplest Form
Learning Objectives:
1. Write a number as a product of prime numbers.
2. Write a fraction in simplest form.
3. Determine whether two fractions are equivalent.
4. Solve problems by writing fractions in simplest form.
5. Key Vocabulary: factor, prime factorization, prime numbers, composite number, simplest form,
lowest terms.
1. Write a number as a product of prime numbers
Definitions:
Prime number—is a natural number greater than 1 that has exactly two different factors, 1and
itself.
Example of prime numbers:
Composite number—is any natural number, other than 1, that is not prime.
Note:
The natural number 1 is neither prime nor composite.
Example 1. Identify each number as prime, composite or neither.
1.
47
Answer:________________________________
2.
51
Answer:________________________________
3.
1
Answer:________________________________
-------------------------------------------------------------------------------------------------------------------------------Prime Factorization of a number—is the factorization in which all the factors are prime
numbers.
Divisibility Tests: A whole number is divisible by
• 2 if the last digit is even that is 0, 2, 4, 6, or 8.
Example. 536 is divisible by 2 since the last digit is a 6.
• 3 if the sum of the digits is divisible by 3.
Example. 912 is divisible by 3 since 9+1+2 = 12 is divisible by 3.
• 4 if its last two digits are divisible by 4.
Example. 744 is divisible by 4 since 44 is divisible by 4.
• 5 if the last digit is 0 or 5.
Example. 915 is divisible by 5 since the last digit is a 5.
• 6 if it’s divisible by 2 and 3.
• 9 if the sum of its digits is divisible by 9.
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Example 2.
Write the prime factorization of 480. Use exponents with any repeated factors.
Answer:______________________________________
-------------------------------------------------------------------------------------------------------------------------------2. Write a fraction in simplest form
Simplest Form of a Fraction—A fraction is written in simplest form or lowest terms when the
numerator and the denominator have no common factors other than 1.
Writing a Fraction in Simplest Form—to write a fraction in simplest form is to write the prime
factorization of the numerator and the denominator and then divide both by all common factors.
Example 3. Write in simplest form.
1.
138
42
2.
39x
−
51
Answer:______________________________________
Answer:______________________________________
---------------------------------------------------------------------------------------------------------------------------------3. Determine whether two fractions are equivalent
Two ways to Determine Whether Two Fractions Are Equivalent
a×c
1. Simplifying:
=
b×c
2.
Cross Product Rule:
a c
=
b d
then
Example 4. Determine whether the pair of fractions is equivalent by simplifying.
12
21
and
20
35
Answer:_________________________________
5
Example 5. Determine whether the pair of fractions is equivalent by cross products.
30
15
and
46
24
Answer:________________________________
---------------------------------------------------------------------------------------------------------------------------------4. Solve problems by writing fractions in simplest form
Example 6. Solve. Write each fraction in simplest form.
1. Alicia was scheduled to work 6 hours at the tanning salon. What fraction of Alicia’s shift is
represented by 4 hours?
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------2. There are 140 students in a freshman lecture class. If 16 students are absent, what fraction of the
students are absent?
Answer:_________________________________
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Sec. 4.3 Multiplying and Dividing Fractions
Learning Objectives:
1. Multiply fractions.
2. Evaluate exponential expressions with fractional bases.
3. Divide fractions.
4. Multiply and divide given fractional replacement values.
5. Solve applications that require multiplication of fractions.
6. Key Vocabulary: reciprocal, ”of”.
1. Multiply fractions
Multiplying Fractions if a, b, c and d represent positive whole numbers, then
a c
⋅ =
b d
Example 1.
1.
Multiply. Write each answer in simplest form.
2 2
⋅
5 7
Answer:_________________________________
2.
7 9 8
⋅ ⋅
3 14 15
Answer:_________________________________
3.
a3 b
⋅
b2 a 2
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------2. Evaluate exponential expressions with fractional bases
Example 2. Evaluate.
1.
 1
− 
 3
4
Answer:_________________________________
2
2.
2 1
  ⋅
7 4
Answer:_________________________________
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3. Divide fractions
Dividing Fractions: if a, b, c and d represent positive whole numbers, then
a b
÷ =
c d
Example 3. Divide and simplify. Write each answer in simplest form.
1.
7 3
÷
8 4
Answer:_________________________________
2.
5 10
− ÷
9 9
3.
3y
÷ 5y3
4
Answer:_________________________________
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------4. Multiply and divide given fractional replacement values
Example 4. Given the following replacement values, evaluate x ÷ y when x =
5
5
and y = −
7
9
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------5. Solve applications that require multiplication of fractions
Example 5. Solve. Write each answer in simplest form.
3
1. Find of − 63
7
Answer:_________________________________
2.
A bike trail is 27 miles long. Michelle bikes
2
of the trail. How many miles did Michelle bike?
3
Answer:_________________________________
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Sec. 4.4 Adding and Subtracting Like Fractions and Least Common Denominator
Learning Objectives:
1. Add or subtract like fractions.
2. Add or subtract given fractional replacement values.
3. Solve problems by adding or subtracting like fractions.
4. Find the least common denominator of a list of fractions.
5. Write equivalent fractions.
6. Key Vocabulary: like fractions, unlike fractions, (LCD) least common denominator, (LCM) least
common multiple, and equivalent fractions.
1. Add or subtract like fractions
Definitions:
1. Like Fractions-are fractions with the same denominators.
2. Unlike Fractions-are fractions with different denominators.
Rules: If a, b and c represent nonzero whole number, then
1.
a c
+ =
b b
Example 1.
1.
2.
a c
− =
b b
Add or subtract and simplify. Write each answer in simplest form.
5
3
4
+
+
32 32 32
Answer:_________________________________
2.
9 1 3  5
+ − + − 
12 12 12  12 
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------2. Add or subtract given fractional replacement values
3
1
Example 2. Evaluate x + y if x = and y = −
5
5
Answer:_________________________________
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3. Solve problems by adding or subtracting like fractions
Example 3. Solve.
1.
Find the perimeter of a triangle with sides:
6
9
5
inch,
inch, and
inch.
25
25
25
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------4. Find the least common denominator(LCD) of a list of fractions
The least common denominator (LCD) of a list of fractions—is the smallest positive number
divisible by all the denominators in the list. (The least common denominator is also the least
common multiple (LCM) of the denominators.)
Steps for Finding the LCD of a List of Numbers Using Prime Factorization.
1. Write the prime factorization of each number using exponent.
2. For each different prime factor in Step 1, circle the greatest number or time that factor occurs
or the highest exponent in any one factorization.
3. The LCM is the product of the circles factors.
Example 4. Find the LCD of
7 1
13
,
and
using Prime Factorization.
20 24
45
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------5. Write equivalent fractions
Example 5. Write the fraction as an equivalent fraction with the given denominator.
12
=
7 105
Answer:_________________________________
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Sec. 4.5 Adding and Subtracting Unlike Fractions
Learning Objectives:
1. Add or subtract unlike fractions.
2. Write fractions in order.
3. Evaluate expressions given fractional replacement values.
4. Solve problems by adding or subtracting unlike fractions.
5. Key Vocabulary: least common denominator (LCD).
1. Add or subtract unlike fractions
Steps to add or subtract Unlike Fractions:
1. Find the LCM of the denominators of the fractions. This number is the Least Common Denominator
(LCD).
2. Write each fraction as an equivalent fraction whose denominator is the LCD.
3. Add or subtraction the numerators of the like fractions.
4. Simplify the result in simplest from if possible.
Example 1. Add or subtract and simplify if needed.
1.
13 3 11
+ +
14 7 28
Answer:_________________________________
2.
2 1 2
− −
3 4 5
Answer:_________________________________
3.
5−
y
4
Answer:_________________________________
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2. Write fractions in order
Example 2. Insert < or > to form a true sentence.
1.
2
1
________________
3
9
2.
5
4
− ______________ −
6
5
---------------------------------------------------------------------------------------------------------------------------------3. Evaluate expressions given fractional replacement values
Example 3. Evaluate each expression if x =
1.
1
3
and y = −
4
5
x+ y
Answer:_________________________________
2.
x⋅ y
Answer:_________________________________
3.
x− y
Answer:_________________________________
4.
x÷ y
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------4. Solve problems by adding or subtracting unlike fractions
Example 4. Solve.
1.
Sharon is making matching holiday outfits for her three children. Each outfit required
7
yards.
8
How many yards of material will be needed to make the three outfits?
Answer:_________________________________
12
Sec. 4.6 Complex Fractions, Order of Operations, and Mixed Numbers
Learning Objectives:
1. Simplify complex fractions.
2. Review the order of operations.
3. Evaluate expressions given replacement values.
4. Key Vocabulary: complex fraction.
1. Simplify complex fractions
Definition:
Complex Fraction—is a fraction whose numerator or denominator or both numerator and
denominator contain fractions.
Property: For any real number a, b, c or d where b ≠ 0, c ≠ 0, d ≠ 0,
a
b = a ÷ c = a ⋅ d = ad
c b d b c bc
d
Example 1. Simplify each complex fraction.
1.
1
6
2
3
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------1 1
+
6
2
2.
1 3
+
3 4
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------2. Review the order of operations
Example 2. Use the order of operations to simplify each expression.
 2 3  2 3 
1.  +  − 
 7 14  7 14 
Answer:_________________________________
13
2
2.
1 2 1
+  −
2 3 3
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------3. Evaluate expressions given replacement values
1
3
7
Example 3. Evaluate each expression if x = − , y = and z =
2
5
10
1. x 2 + 2 y
Answer:_________________________________
2.
x+ y
z
Answer:_________________________________
14
Sec. 4.7 Operations On Mixed Numbers
Learning Objectives:
1. Graph positive and negative fractions and mixed numbers.
2. Multiply or divide mixed or whole numbers.
3. Add or subtract mixed numbers.
4. Solve problems containing mixed numbers.
5. Perform operations on negative mixed numbers.
1. Graph positive and negative fractions and mixed numbers
Example 1. Graph each list of numbers on a number line.
1
3 1
− 3, − 3 , − 1, , 4
2
4 3
---------------------------------------------------------------------------------------------------------------------------------2. Multiply or divide mixed or whole numbers
Multiplying Fractions and Mixed numbers or Whole Numbers—is to write any mixed or whole
numbers as fractions and then multiply as usual.
Dividing Fractions and Mixed Numbers or Whole Numbers—is to write the mixed or whole
number as a fraction and then divide as usual.
Example 2. Multiply. Write each answer in simplest form.
1.
3 2
7 ⋅2
4 3
Answer:_________________________________
2.
7
×5
10
Answer:_________________________________
3.
1
4 ×0
8
Answer:_________________________________
15
4.
3 3
2 ÷1
5 5
Answer:_________________________________
5.
8
2 ÷ 13
9
Answer:_________________________________
6.
1
6 ÷0
3
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------3. Add or subtract mixed numbers
Steps to Add or Subtract Mixed Numbers:
To add or subtract mixed numbers, add or subtract the fraction parts and then add or subtract the whole
number parts.
Example 3. Add or subtract and simplify if needed.
1.
1
1
9 + 18
2
3
Answer:_________________________________
2.
5
2
14 − 9
6
3
Answer:_________________________________
16
3.
5 1
3
2 −1 + 5
8
6
4
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------4. Solve problems containing mixed numbers
1
5
Example 4. Ray played hockey for 3 hours on Monday and played hockey on Wednesday for 4
2
9
hours. What was the total amount of time Ray played hockey?
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------3
Example 5. John cuts a board 13 feet long from one 20 feet long. How long is the remaining piece?
7
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------5. Perform operations on negative mixed numbers
Example 6. Perform the indicated operation.
7
1
1. − 5 ÷ 5
8
4
Answer:_________________________________
17
2.
1  1
− 8 ⋅ −1 
3  5
Answer:_________________________________
3.
1  5
10 +  − 5 
9  9
Answer:_________________________________
4.
3  18 
19 −  − 5 
5  20 
Answer:_________________________________
18
Sec. 4.8 Solving Equations Containing Fractions
Learning Objectives:
1. Solve Equations Containing Fractions.
2. Solve Equations by Multiplying by the LCD.
3. Review Adding and Subtracting Fractions.
1. Solve Equations Containing Fractions
Example 1. Solve each equation. Check your proposed solution.
3
11
1. 8 x − − 7 x =
5
5
Answer:_________________________________
2.
− 8x =
16
25
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------2. Solve Equations by Multiplying by the LCD
Example 2. Solve each equation.
1.
x
− x = −5
7
Answer:_________________________________
2.
y y 3
= +
2 5 2
Answer:_________________________________
19
3.
y
y
−2 = −4
5
3
Answer:_________________________________
---------------------------------------------------------------------------------------------------------------------------------3. Review Adding and Subtracting Fractions
Example 3. Add or subtract as indicated.
1.
n 4
+
2 7
Answer:_________________________________
2.
5c c
−
8 4
Answer:_________________________________
20