Peterson 8th Section 1

```Section 1
Common Fractions and
Decimal Fractions
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Unit 1
Introduction to Common Fractions and Mixed
Numbers
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Unit 1
• Objectives
o Express fractions in lowest terms
o Express fractions as equivalent fractions
o Express mixed numbers as improper fractions
o Express improper fractions as mixed numbers
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Fractions
• Fraction: a value that shows the number of equal parts
taken of a whole quantity or unit
o Denominator: the number of equal parts the whole is divided
into
o Numerator: the number of equal parts that are taken
Numerator
• Format:
Denominator
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Types of Fractions
•
•
•
•
•
Common: numerator and denominator are whole numbers
Proper: numerator is smaller than the denominator
Improper: numerator is larger than or equal to the denominator
Mixed number: composed of a whole number and a fraction
Complex fraction: one or both terms of the fraction are fractions or mixed
numbers
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Equivalent Fractions
• Fractions that have the same value, but different forms
o Obtained by multiplying or dividing both the numerator and denominator by the
same number
• Useful when comparing two fractions
• Often necessary when adding or subtracting fractions (when denominators
must be the same)
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Lowest Terms
• A fraction in which the numerator and denominator do not contain a common
factor
• Reduce to lowest terms by dividing both the numerator and denominator by
each of their “common factors”
30 30  2 15 15  3 5
=
=
=
=
48 48  2 24 24  3 8
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Mixed Numbers and Improper Fractions (1 of 2)
• To convert a mixed number to an improper fraction:
(whole number &times; denominator)+numerator
denominator
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Example for Mixed Numbers and Improper
Fractions (1 of 2)
4
3 4  8 + 3 35
=
=
8
8
8
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Mixed Numbers and Improper Fractions (2 of 2)
• Improper fractions should be expressed as mixed numbers
• To convert an improper fraction to a mixed number:
o divide the numerator by the denominator; express the remainder as a fraction
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Example for Mixed Numbers and Improper
Fractions (2 of 2)
35
3
=4
8
8
4
8 35 R 3
Since 35 &divide; 8 = 4, with a remainder of 3. The fractional part of the mixed number
has the remainder as the numerator, and the divisor as the denominator
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Unit 2
Addition of Common Fractions and Mixed
Numbers
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Unit 2
• Objectives
o Determine lowest common denominators
o Express fractions as equivalent fractions having lowest common denominators
o Add fractions and mixed numbers
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Lowest Common Denominators (LCD)
• Before adding or subtracting fractions, they must have the same
denominators
• The LCD is the smallest denominator which is evenly divisible by the
denominators of each of the fractions being added
• Any common denominator can be used, but using the LCD simplifies the
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•
•
•
•
Convert the original fractions to equivalent fractions that have the LCD
Add the numerators of the fractions
Leave the denominator unchanged
Convert the result to best form (mixed number or lowest terms)
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•
•
•
•
Combine the whole number and fraction
Express the answer in lowest terms
o If the fractional portion is improper, convert it to a mixed number and simplify
further
o If the fractional portion is not in lowest terms, reduce it
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1
2
3
4
4 +2 = 4 +2
2
3
6
6
3+4
=6
6
7
=6
6
1
=7
6
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Unit 3
Subtraction of Common Fractions and Mixed
Numbers
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Unit 3
• Objectives
o Subtract fractions
o Subtract mixed numbers
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Subtracting Fractions
• Steps
o Convert to LCD form
o Subtract the numerators
o Leave the denominator unchanged
o Express answer in lowest terms
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Example for Subtracting Fractions
4 3
8
3
−
=
−
5 10 10 10
8−3
=
10
5
=
10
1
=
2
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Subtracting Mixed Numbers
• Mixed Numbers
o Subtract the fractions (borrow if necessary—see the next slide)
o Subtract the whole numbers
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Borrowing
• When subtracting DECIMAL numbers, we borrow “ten” from the next larger
decimal place
• When subtracting MIXED numbers, we sometimes need to borrow one unit
from the whole number
o The effect is to decrease the whole number by “one” and to add the denominator
to the numerator
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Borrowing (example) (1 of 2)
• In the following example,
• So,
1
12
cannot be subtracted from
16
16
16
are borrowed from the whole number
16
1
3
1
12
3
−1 = 3
−1
16
4
16 16
17 12
= 2 −1
16 16
5
=1
16
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Borrowing (example) (2 of 2)
• Effect: decrease whole number by 1; and add the denominator to the
numerator before subtracting
1
3
1
12
3
−1 = 3
−1
16
4
16 16
17 12
= 2 −1
16 16
5
=1
16
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Unit 4
Multiplication of Common Fractions and Mixed
Numbers
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Unit 4
• Objectives
o Multiply fractions
o Multiply mixed numbers
o Divide by common factors (cancellation)
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Multiplying Fractions (1 of 2)
•
•
•
•
Multiply the numerators
Multiply the denominators
Reduce the resulting fraction
NOTE: Finding a common denominator is NOT necessary for multiplication
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Multiplying Fractions (2 of 2)
• Examples
o Without cancellation
5 4
20 1

=
=
8 15 120 6
o Cancelling common factors
51 41 1

=
82 153 6
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Multiplying Mixed Numbers
• Convert mixed numbers to improper fractions; multiply as fractions; convert
the answer to a mixed number (shown without using cancellation)
2
1 5 9 45
1 2 =  =
3
4 3 4 12
45
9
=3
12
12
3
=3
4
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Unit 5
Division of Common Fractions and Mixed Numbers
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Unit 5
• Objectives
o Divide fractions
o Divide mixed numbers
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Dividing Fractions
• Division by a number is the same as multiplying by the reciprocal of the
number
32  4 = 32 
1 32
=
=8
4 4
• The same concept applies to fractions
1
4 128
32  = 32  =
= 128
4
1
1
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Dividing Mixed Numbers
• Convert to improper fractions, as you did for multiplication; then, divide and
reduce to lowest terms
3
3
1 27 9 27 4 108
2 =
 =
 =
8
4 8 4 8 9 72
36
=1
72
1
=1
2
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Unit 6
Combined Operations of Common Fractions and
Mixed Numbers
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Unit 6
• Objectives
o Solve problems that involve combined operations of fractions and mixed numbers
o Solve complex fractions
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Combined Operations
• A helpful memory aid: PEMDAS, or “Please Excuse My Dear Aunt Sally”
o Parentheses (grouping symbols)
o Exponents (powers and roots)
o Multiplication
o Division
o Subtraction
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Parentheses
• Parentheses: Perform operations within parentheses first
o The fraction line, brackets, or other grouping symbols should be treated the same
as parentheses
o If grouping symbols are “nested” work outward, starting with the innermost group
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Other Operations
• Exponents: Perform operations involving powers or roots next
• Multiplication and Division: Perform these operations in the order in which
they appear, from left to right
• Addition and Subtraction: Perform these operations last, in order from left to
right
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• If you find it helpful, draw boundaries at the plus and minus signs
• Work between those boundaries before you perform the addition or
subtraction
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Complex Fractions (1 of 2)
• Complex fractions include numerators or denominators that are, themselves,
fractions or mixed numbers
• Treat the fraction line as a grouping symbol, like parentheses
2
 1
 12 + 2 3 
=


 21 
3 

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Complex Fractions (2 of 2)
• Remember that the fraction line is a grouping
symbol, and perform operations accordingly
1
2
1 +2
2
3 =
1
2
3
2  1
1
1
 1
12 + 2 3    2 3  = 4 6  2 3

 

25 7
=

6 3
25 3
=

6 7
75
=
42
11
=1
14
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Unit 7
Computing with a Calculator: Fractions and Mixed
Numbers
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Unit 7
• Objectives
o Perform individual operations of addition, subtraction, multiplication, and division
with fractions using a calculator
o Perform combinations of operations with fractions using a calculator
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Scientific Calculator
• Scientific calculators usually have a key labeled A b/c, or something similar
• This key is used as a “separator” between the numerator and the
denominator of a fraction, and between the whole number and numerator,
and again between the numerator and denominator, of a mixed number
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Machinist Calc Pro2 (1 of 3)
• To enter a fraction, such as
Solution: Press
3
:
4
to clear the calculator. Then press
You should see the calculator screen in Figure 7-1.
Figure 7-1
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Machinist Calc Pro2 (2 of 3)
5
8
• To enter a mixed number, such as 7 :
METHOD 1: Press
and you get the image in Figure 7-8.
Figure 7-8
METHOD 2: Press
in Figure 7-8.
and you again should see the image
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Fractional Resolution
• The default setting on the Machinist Calc Pro2 is to display fractions rounded
to the nearest 164
• This setting can be changed by the user
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Unit 8
Computing with a Spreadsheet: Fractions and
Mixed Numbers
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Unit 8
• Objectives
o Enter fractions and mixed numbers in a spreadsheet
o Perform individual operations of addition, subtraction, multiplication, division,
powers, and roots with fractions using a spreadsheet
o Perform combinations of operations with fractions using a spreadsheet
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Displaying Fractional Values
• Number formatting applied to a cell determines the display of decimal numbers
o Choose “Fraction,” rather than “Number,” if fractional values are preferred
o Use a “/” to separate numerator and denominator
o For mixed numbers, use “+” to separate the whole number and fractional parts
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Key Points for Displaying Fractional Values
• An equal sign is needed at the beginning of a formula
• For mixed numbers, parentheses are often needed to ensure that combined
operations are performed correctly
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Combined Operations for Displaying Fractional Values
• The order of operations will be followed
• Parentheses must be used to group terms where needed, and when finding
powers or roots of expressions
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Unit 9
Introduction to Decimal Fractions
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Unit 9
• Objectives
o Locate decimal fractions on a number line
o Express common fractions having denominators of powers of ten as equivalent
decimal fractions
o Write decimal numbers in word form
o Write numbers expressed in word form as decimal fractions
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Decimal Fractions
• Decimal fractions: fractions that have denominators that are powers of 10
• The denominator is indicated by the number of digits to the right of the
decimal point
• See Figure 9-3
1
= 0.1
10
31
= 0.31
100
809
= 0.809
1000
47
= 0.0047
10000
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• Read the decimal point as “and”
• Read the decimal fraction as a whole number, adding the name of the last
decimal place
• 4321.123 is read as: “Four thousand, three hundred twenty-one, and one
hundred twenty-three thousandths”
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Writing Common Fractions as Decimal Fractions
• To write a common fraction in decimal form, if the original denominator is a
power of 10:
o The number of decimal places is the same as the number of zeroes in the
original denominator
12
= 0.12
100
12
= 0.0012
10000
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Unit 10
Rounding Decimal Fractions and Equivalent
Decimal and Common Fractions
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Unit 10
• Objectives
o Round decimal fractions to any required number of decimal places
o Express common fractions as decimal fractions
o Express decimal fractions as common fractions
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Precision of Decimals
• Computations involving decimals often result in more decimal places than
required
• The degree of precision required depends on how we plan to use the
resulting number
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Examples of Precision
• A calculated money amount might be \$12.5375, which would have to
rounded to the nearest cent
• A calculation might yield a length of 2.476535 inches. We might round that to
2.477 inches if measuring to the nearest thousandth of an inch.
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Rounding
• A decimal number can be shortened by “rounding” it to a specific number of
decimal places
• Rounding is based only on the digit immediately to the right of the last
required decimal place
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Rounding Down
• If the next digit is less than 5, drop all digits to the right of the last required
decimal place
• 12.39645 would round to:
o 12.396 (if rounded to three decimal places)
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Rounding Up
• If the next digit is 5 or greater, increase the digit in the last required decimal
place by 1
• 12.39645 would round to:
o 12.3965 (if rounded to four decimal places)
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Expressing Common Fractions as Decimal
Fractions
• To write a common fraction in decimal form, if the original denominator is
NOT a power of 10:
o Divide the numerator by the denominator, carrying the quotient to one place
beyond the required number of places without rounding
o Then, round the result
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Expressing Decimal Fractions as Common
Fractions
• Write the number after the decimal point as the numerator
• Write the denominator as 1 followed by as many zeroes as there are decimal
places in the original decimal fraction
• Reduce to lowest terms
12.465 = 12
465
93
= 12
1000
200
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Unit 11
Addition and Subtraction of Decimal Fractions
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Unit 11
• Objectives
o Add combinations of decimals, mixed decimals, and whole numbers
o Subtract decimal fractions
o Subtract combinations of decimals, mixed decimals, and whole numbers
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• Arrange the numbers so that the decimal points are aligned directly under
each other
• Add or subtract as with whole numbers
• Place the decimal point in the result directly beneath the other decimal points
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Unit 12
Multiplication of Decimal Fractions
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Unit 12
• Objectives
o Multiply decimal fractions
o Multiply combinations of decimals, mixed decimals, and whole numbers
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Multiplying Decimal Numbers
• Multiply as with whole numbers
• Counting from the right side, the product will have as many decimal places
as in both of the numbers being multiplied
 2.15
0.05
 0.12
9.04075
0.0060
4.205
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Unit 13
Division of Decimal Fractions
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Unit 13
• Objectives
o Divide decimal fractions
o Divide decimal fractions with whole numbers
o Divide decimal fractions with mixed decimals
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Dividing Decimal Numbers
• If the divisor is NOT a whole number:
o Move the decimal point of the divisor to the right, so the divisor is a whole
number
o Move the decimal point of the dividend to the right, the same number of decimal
places
o Divide as with whole numbers
o Using zeroes as necessary, place the decimal point in the quotient directly above
the decimal point in the dividend
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Example for Dividing Decimal Numbers
• If dividing: 4.71 53.6970
Move the decimal points, so that the divisor is a whole number; position
the decimal point in the answer
11.40
471 5369.70
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Unit 14
Powers
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Unit 14
• Objectives
o Raise numbers to indicated powers
o Solve problems involving combinations of powers and other operations
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Definitions
•
•
•
•
Factors: numbers that are being multiplied
Power: the product of two or more equal factors
Exponent: shows how many times a number is taken as a factor
Formula: expresses a mathematical relationship using symbols
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Powers
o 122 is read as “twelve to the second power,” or as “twelve squared”
o 123 is “twelve to the third power,” or “twelve cubed”
o 45 is read as “four to the fifth power”
• Powers are commonly found in geometric problems—see the following
examples
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Examples for Powers
• Area of a square: A = s 2
• Volume of a cube: V = s 3
• NOTE: Powers are applied to the units of measure as well as the numbers.
So, areas might be expressed as square feet; volumes as cubic feet.
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Powers Involving Fractions
• Apply powers to the appropriate factor or “group”
2
2
9
3
3 3 3
 4  =  4    4  = 2 = 16
 
    4
32 33 9
=
=
4
4
4
3
3
3
=
=
2
4  4 16
4
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Using Powers
• An exponent shows how many times a number is taken as a factor
• Perform repeated multiplication, based on the value of the exponent
• Scientific calculators have special keys for entering exponents and
parentheses
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Unit 15
Roots
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Unit 15
• Objectives
o Extract whole number roots
o Calculate the root of any positive number
o Solve problems that involve combinations of roots with other basic arithmetic
operations
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Terminology
• A root of a number is a value that is taken two or more times as an equal
factor of the number
• The radical sign is used to indicate the operation of finding a root of a
number
• The index indicates the number of times the root is to be taken as a factor. If
the index is 2, it is not written next to the radical sign.
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Examples for Terminology (1 of 2)
4
81 = 9
check: 9  9 = 81
81 = 3
check: 3  3  3  3 = 81
• Find the length of the side of a square whose area is 36:
A = s2
36 = s 2
36 = s 2 = s
s=6
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Examples for Terminology (2 of 2)
• Find the length of the side of a cube whose volume is 729:
V = s3
s = 3V
If V = 729, s = 3 729 = 9
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Fractions Containing Roots
• If the radical sign contains only the numerator (or only the denominator), find
the root of the number that is in the radical sign; then simplify the fraction
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Roots of Entire Fractions
• If the radical sign contains both the numerator and the denominator,
EITHER:
o find the root of each number; then simplify the fraction, OR,
o find the root of the fraction as a whole
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Examples for Roots of Entire Fractions
36
36
=
= 12
3
9
36 6 2
= =
9
9 3
36
36 6
=
= =2
9
3
9
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Roots of Expressions (1 of 2)
• The radical sign is a grouping symbol, similar to parentheses
• Perform the operations within the symbol first; then find the root of the result
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Roots of Expressions (2 of 2)
12 + 37 = 49 = 7
20  (2 + 3) = 20  5 = 100 = 10
20  2 + 3 = 40 + 3 = 43  6.56
NOTE: The last answer is not exact. It was rounded to two places.
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Fractional Exponents
• Fractional exponents can be used to indicate roots
49 2 = 49 = 7
1
4 2 = 43 = 64 = 8
3
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Unit 16
Table of Decimal Equivalents and Combined
Operations of Decimal Fractions
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Unit 16
• Objectives
o Write decimal or fraction equivalents using a decimal equivalent table
o Determine nearer fraction equivalents of decimals by using a decimal equivalent
table
o Solve problems by applying the order of operations
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Fractions or Decimals
• Some engineering drawings might show dimensions expressed as fractions,
although more recent drawings would use decimal inch (or metric)
dimensions
• Decimal equivalent tables are useful for converting from fractions to decimal
forms … Table Page 97
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Nearer Fractional Equivalent
• If you need to convert a decimal measurement to fractional form, you might
need to find the “nearer fractional equivalent”
• This term means exactly what is says: “the fractional equivalent that most
closely matches the decimal fraction”
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Procedure
• Find the common fractions that are “just less than” and “just greater than” the
decimal fraction
• Find which of the two fractions is “nearer” by subtraction, and determining
the smaller difference
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Example for Procedure
• For the decimal fraction: 0.361
• The value is between:
23
3
and
64
8
23
• By subtracting, you will find that 0.361 is nearer to
64
23
• So, the nearer fractional equivalent is
64
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Combined Operations (Order of Operations)
• Parentheses are grouping symbols. Work within them first.
• Exponents: Powers and roots are done next.
• Multiplication and Division are performed in the order in which they occur,
from left to right.
• Addition and Subtraction are performed in the order in which they occur, from
left to right.
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Reminders
• DO NOT think that you need to do all multiplication before any division.
Do these operations from left to right, in sequence.
• The same rule applies to addition and subtraction.
• Sometimes, students find it helpful to draw boundaries at the plus and minus
signs, so they complete the operations between those signs before doing the
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Unit 17
Computing with a Calculator: Decimals
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Unit 17
• Objectives
o Perform individual operations of addition, subtraction, multiplication, division,
powers, and roots with decimals using a calculator
o Perform combinations of operations with decimals using a calculator
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Decimals
• The decimal point key is used to enter the decimal point in the appropriate
place in decimal numbers
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Powers–Scientific Calculator
• The “x-squared” key is used to raise a number to the second power
• The “caret” key ( an upward pointing arrow) is used to raise a number to any
power
• Some calculators have a &quot; y x &quot; key instead
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Machinist Calc Pro2 (3 of 3)
• To square a number:
Example To calculate 28.752 , enter 28.75
Solution
28.75
826.5625 Ans
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Roots on a Scientific Calculator
• Roots are usually alternate key functions, accessed using a “shift” or “2nd”
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Unit 18
May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Unit 18
• Objectives
o Perform individual operations of addition, subtraction, multiplication, division,
powers, and roots with decimals using a spreadsheet
o Perform combinations of operations with decimals using a spreadsheet
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Displaying Decimal Values
• Number formatting applied to a cell determines the display and rounding of
decimal numbers
o Choose “Number”, rather than “Fraction” if decimal values are preferred
o Specify the number of places displayed after the decimal point
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Key Points for Displaying Decimal Values
• An equal sign is needed at the beginning of a formula
• Powers can be found using the “^” symbol or POWER function
• Roots can be found using the SQRT function, or using fractional powers
not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Combined Operations for Displaying Decimal Values
• The order of operations will be followed
• Parentheses must be used to group terms where needed, and when finding
powers or roots of expressions