Download Lecture Notes - Midterm Exam Review - Pioneer Student

Midterm Exam Review – Richard Goldman
MAT116 Algebra I
Basic Mathematics for College Students – 3e
1.1
An Introduction to Whole Numbers
Set
Group of numbers
Subset
Set of numbers within another set
Natural Numbers: 1, 2, 3
Whole Numbers:
0, 1, 2, 3
Integers:
-2,-1, 0, 1, 2, 3
Braces { }
Parenthesis ( )
Brackets [ ]
Standard Notation
Expanded
Notation
Inequalities > <
Rounding
Used to define a set
Used to group a mathematical operation
1.2
Addends
Sum
Commutative
Property of
Addition
Associative
Property of
Addition
Carrying
Perimeter
Using a
Calculator to Add
1.3
Subtracting
Whole Numbers
Double Digit
Subtraction
Borrowing
Using a
Calculator
Mixed Addition
and Subtraction
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40,123
Four ten thousands + zero thousands + one hundred + two tens + three ones
> is Greater than, < is Less than
If 5 or greater – round up
If 4 or less – no change
Adding Whole Numbers
Numbers Added
Total of Addends
Numbers may be added in any order
Numbers may grouped in any manner
Demonstrate
The sum of the length of all side of an object
Demonstrate
Subtracting Whole Numbers
Minuend – top number
Subtrahend – bottom number
Difference – result of subtraction
Needed when subtracting from a smaller number in any particular column
5 – 3 + 6 = 8 (do subtraction first)
5+6–3=8
(combining integers is covered Section 2.3)
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1.4
Multiplying
Whole Numbers
Multiplication
Symbols
Properties
Multiply by
powers of 10
Multiplying
longhand
Area of a
rectangle
1.5
Division
Notation
Multiplying Whole Numbers
Factors – The numbers multiplied
Product – Result of multiplication
(Repeated Addition)
X

( ) ( ) – Number in parenthesis next to each other without other symbol
(4) (5) or 4 (5) or (5) 4
Commutative Property of Multiplication – any order gives same result
Associative Property of Multiplication – any grouping gives same result
10 add 1 zero
100 add 2 zeros
1,000 add 3 zeros
Single digit
Multiple digit
A=LW
Dividing Whole Numbers
Repeated subtraction (How many 4’s can be subtracted from 12?)
Dividend ÷ Divisor = Quotient
12 ÷
4
= 3 (Division Symbol)
12
/
12
4
3
4 ) 12
Zero & Division
Short Division
Long Division
Remainder
Check your work
1.6
Factor of Whole
Numbers
Odd & Even
Prime Number
Composite
Numbers
Prime Factors of
a Number
Finding Prime
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4
=
3
(Fraction Bar)
= 3
(Fraction Bar)
(Long Division)
Zero divided by any number is zero
Any number divide by zero is Undefined
One digit devisor
Two or more digit devisor
Number left over after dividing
Quotient ● Divisor + Remainder = Dividend
3
●
4
+
0
=
12
Prime Factors and Exponents
Whole numbers that are multiplied together
All the factors of 12 are 1, 2, 3, 4, 6, & 12
Even = Divisible by 2
Odd = Not divisible by 2
Whole number > 1 and not devisable (except by 1 and itself)
Not prime numbers
Prime numbers multiplied together to give the number

Tree Method – Keep dividing until you end up with all primes
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
Division Method – Divide by lowest prime the divide that result by lowest prim,
etc.
Exponent
Repeated Multiplication – 32 = 3 ● 3 = 9
1.7
Order of Operation & Mean
Order
Inner most groups first with this order
1. Exponents
2. Multiplication & Division
3. Addition and Subtraction
Solve numerator and denominator before dividing
Please
Excuse
My
Dear
Aunt
Sally
Parentheses Exponents
Multiplication Division
Addition
Subtraction
Mean
Same as Average
Divide the sum by the number of numbers
2.1
An Introduction to Integers
Negative
Left of zero on number line
Numbers
-3 is read as “Negative three”
Absolute Value
The distance between zero and any number on the number line
Always positive
Written as |5|
|5| is read as: “The absolute value of five”
Opposites
Each number has an opposite
The symbol for opposite is ( - )
The opposite of -5 is 5 Written as -(-5)
Double Negative Two negative equal a positive
-(-4) = +4
More Inequalities ≠
Not equal to
≥
Greater than or equal to
≤
Less than or equal to
Examples:
6≠5
6≥5
5≥5
2.2 & 2.3
Combining Integers
Show the following on a number line
+ Moves to right
- Moves to left
3+3=6
3- 3=0
-3 + 3 = 0
-3 - 3 = -6
Rule: Take smallest from largest; use sign of largest
2.4
Multiplication with Negative Numbers
Sign of Product
Even Number of negative factors = even product – Example: -2 x -2 = 4
Odd number of negative factors = negative product – Example: -2 x 2 = -4
Power of negative Even number exponents of a negative base evaluate to positive numbers
numbers
-32 = -3 x -3 = 9
Odd number exponents of a negative base evaluate to negative numbers
-33 = -3 x -3 x -3 = -27
Factors
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2.5
Sign of Quotient
Division with 0
2.6
Evaluating
Absolute Value
Expressions
Estimation
3.2
Multiplying
3.3
Reciprocal
Dividing
Fractions
3.4
Fractions with
Same
Denominators
LCD
Add or Subtract
Fractions with
Different
Denominators
Finding LCD
Convert Fractions
to LCD
Comparing
Fractions
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Dividing Integers
The Quotient of two like signs is positive.
The Quotient of two unlike signs is negative
-6 / -3 = 2 We know this is true because 2 X -3 = -6
0 divided by any number is 0
Any number divided by 0 is undefined
Order of Operations and Estimation
Evaluate any expressions within an Absolute Value as you would any grouped
expression.
Round off
Does the answer seem reasonable
Multiplying Fractions
Multiply Numerator times Numerator and
Multiply Denominator times Denominator
(2 is the same as 2/1)
Dividing Fractions
An inverted fraction
Multiply the first fraction by the reciprocal of the second
Adding and Subtracting Fractions
Add or Subtract the Numerators and use the common Denominator
Lowest Common Denominator
1. Find LCD
2. Convert fraction to LCD
3. Add or subtract
1. Find Prime Factors of each Denominator
2. Create a set of numbers that includes the Prime Factors of both Denominators
3. Find Product of this set (This is the LCD)
Example: 3/8 + 1/10
Prime Factors of 8 = 2, 2, 2
Prime Factors of 10 =
2, 5
Resulting Factors
2, 2, 2, 5
Product = 40 (the LCD)
4. Divide the Denominator of the first fraction into the LCD to find the multiple
5. Multiple both the Numerator and the Denominator of the first fraction by the multiple
6. Divide the Denominator of the second fraction into the LCD to find the multiple
7. Multiple both the Numerator and the Denominator of the second fraction by the
multiple
To compare fractions they must first be converted to LCD
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3.5
Mixed Number
Add, Subtract,
Multiply, or
Divide Mixed
Numbers
3.5
Convert Mixed
number to
improper Fraction
Convert Improper
Fraction to Mixed
number
Graphing
Fractions
Multiply and
Divide Mixed
Fractions
3.6
Adding and
Subtracting
Mixed Numbers
3.7
Order of
Operation
Complex
Fractions
4.1
Decimal
Decimal Places
Leading 0
Trailing 0
Reading
Comparing
Rounding
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Multiplying and Dividing Mixed Numbers
A whole number and a fraction – 3½ is mixed number = 3 + ½
1. Convert to whole number to a fraction and the add it to the fraction
2. Perform the designated operation (+, -, x, ÷)
Multiplying and Dividing Mixed Numbers
3¾ - Add 3 to ¾ = 3/1 + ¾ = 12/4 + ¾ = 15/4
(Multiply whole number by denominator then add numerator to it.)
15/4 – Divide numerator by denominator - the result is the whole number – Put the
remainder over the denominator to form a fraction
15/4 = 3 R3 = 3 ¾
Demonstrate on number line (Put 1 1/2, -2 1/4, etc. on a number line)
Convert to improper fractions then process as usual
Adding and Subtracting Mixed Numbers
Find LCD of fractions then add or subtract (carry and borrow as needed)
Order of Operation and Complex Fractions
Groups ( Parenthesis, Complex Fractions, Absolute Values)
Powers
Multiply & Divide (Left to Right)
Add & Subtract (Left to Right)
(Fraction over Fraction – write out horizontal and process)
An Introduction to Decimals
1/10 = 0.1
0.1
1 Tenth
0.01
1 Hundredth
0.001
1 Thousandth
0.0001
1 Ten Thousandth
0.00001
1 Hundred Thousandth
0.000001
1 Millionth
0.0000001
1 Ten Millionth
0.1 = .1 (Leading zero is a general convention used to help eliminate error)
0.10 = 0.1 (Trailing zero is an engineering/scientific convention use to display accuracy)
123.45 Is read as One hundred, twenty-three and 45 hundredths
0.45 Is read as Forty-five hundredths
(also – point Four, Five)
The number with the biggest different number in the column closest to decimal is the
largest value.
Round from the right just like with whole numbers.
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4.2
Adding &
Subtracting
Decimals
4.3
Multiplying
Decimals
Multiplying by
powers of 10’s
Order of
Operation
4.4
Dividing
Decimals
Rounding
Dividing by
powers of 10’s
Order of
Operation
4.5
Converting
Fractions into
Decimals
Repeating
Decimal
Quotients
Rounding
Repeating
Decimals
Working
Problems with
Fractions and
Decimals
4.6
Square Root
Radical Symbol
√
Negatives
Perfect Squares
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Adding and Subtracting Decimals
1. Line up Decimal points
2. Add trailing zeros to match longest decimal
3. Add or subtract as you would whole numbers
Multiplying Decimals
1. Multiply as you would whole numbers
2. Count total decimal places in both factors and add to product (add leading 0’s if
necessary)
Move the decimal place to the right for each 0 in the power.
10 x 123.456 = 1,234.56
100 X 123.456 = 12,345.6
Same
Dividing Decimals
1. Move the decimal place in the devisor to make it a whole number
2. Move the decimal place in the dividend the same number places
3. Divide until there is no remainder or the desired precision is reached (add trailing
zeros to the dividend as necessary)
Carry out division one digit past desired accuracy and then round back
Move the decimal place to the Left for each 0 in the power.
123.456 ÷ 10 = 12.3456
123.456 ÷ 100 = 1.23456
Same
Fractions and Decimals
Carry out the indicated division
½ = 1 ÷ 2 = 0.5
_
10 ÷ 3 = 3.33333333333333 or 3.3… or 3.3 (overbar)
Carry out to one digit past desired precision and then round back.
1/3 ≈ 0.3333 (approximately equal to)
Either convert to all fractions or all decimals.
Examples:
0.24 = 24/100
0 .6
3 5 .0
3/5 = 0.6
Square Root
The number that must be squared to produce the number
The answer is really two numbers - both a negative and a positive number
We normally only deal with the positive square root
Means square root
Number under radical is the radicand
√36 = 6, -√36 = -6, √-36 is not a real number (anything squared will always be
positive)
Whole number roots
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Pythagorean
Theorem
(optional)
a2 + b2 = c2
Demo: 3, 4, 5 Rt. Triangle:
a2  b2  c2
32  4 2  c 2
9  16  c 2
25  c 2
25  c 2
5.1
Percent
Convert % to
Fraction
Convert % to
Decimal
Convert Decimal
to %
Convert Fraction
to %
5.2
Equations
Percentage
Speed=Distance
/Time
5.3
Total Price
Sales Tax
Withholding Tax
5c
Percent, Decimals, and Fractions
1/100
Drop % sign and put over 100
Drop % sign and divide by 100 (move decimal point 2 places to the left)
Add % sign and multiply by 100 (move decimal point 2 places to the right)
Convert fraction to decimal (divide the fraction) and then multiply by 100 (move decimal
point 2 places to the right)
Solving Percent Problems
A statement that two quantities are equal
May perform the same operation to each side of an equation with out changing the
statement (it will still be equal)
Percent = (Amount / Base) x 100
3/5 x 100 = 60%
Amount = (Percent x Base) / 100
Base = (Amount / Percent) x 100
Speed = Miles/Hr.
10 miles / 1 Hr = 10 M/Hr
Speed x Hr = Miles
10 M/Hr x 1 Hr = 10 Miles
Hr = Miles/ Speed
1 Hr = 10 Miles/ 10 M/Hr
Applications of Percent
Total Price = Purchase Price + Sales Tax
Sales Tax = Percent x Purchase Price
(Amount = Percent x Base)
Percent = Amount / Base
Percent = $11.04 / $240.00 = .046 = 4.6%
Commissions
5.4
Principal
Interest Rate
Time
Simple Interest
Compound
Interest
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Interest
Amount borrowed (or loaned)
Percent used to calculate interest
Time borrowed
Interest = Principal x Rate x Time
Interest paid on accumulated interest plus principal
nt
 r
A  P 1  
 n
A = Amount, P= Principal, r = rate, n=numb of compounds/yr, t=time in years
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