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Transcript
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 234820020 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) Page 1 of 7 Richard Goldman 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=LW 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 234820020 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 Page 2 of 7 Richard Goldman 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 234820020 Page 3 of 7 Richard Goldman 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 234820020 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 Page 4 of 7 Richard Goldman 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 234820020 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. Page 5 of 7 Richard Goldman 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 234820020 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 Page 6 of 7 Richard Goldman 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 5c 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 234820020 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 Page 7 of 7 Richard Goldman