Part 1 - Pre-Algebra Summary Page 1 of 22 1/19/12

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1 Part 1 - Pre-Algebra Summary Page 1 of 1/19/1 Table of Contents 1. Numbers NAMES FOR NUMBERS PLACE VALUES INEQUALITIES ROUNDING DIVISIBILITY TESTS PROPERTIES OF REAL NUMBERS PROPERTIES OF EQUALITY ORDER OF OPERATIONS Real Numbers OPERATIONS Fractions DEFINITIONS LEAST COMMON DENOMINATORS OPERATIONS Decimals OPERATIONS Conversions PERCENT TO DECIMAL TO FRACTIONS Algebra DEFINITIONS OPERATIONS OF ALGEBRA SOLVING EQUATIONS WITH 1 VARIABLE SOLVING FOR A SPECIFIED VARIABLE EXPRESSIONS VS. EQUATIONS Word Problems VOCABULARY FORMULAS STEPS FOR SOLVING RATIOS & PROPORTIONS GEOMETRY... Copyright Sally C. Zimmermann. All rights reserved.

2 Part 1 - Pre-Algebra Summary Page of 1/19/1 1. Numbers 1.1. Names for Numbers Real Numbers (points on a number line) Rationals (can be written as a quotient of integers) 3 4 Irrationals (can t be written as a quotient of integers) π = = = ± 1 3% = = 7 10 Integers, 3,, 1, 0, 1,, 3, Whole Numbers 0, 1,, 3, Natural Numbers 1,, 3 Copyright Sally C. Zimmermann. All rights reserved.

3 Part 1 - Pre-Algebra Summary Page 3 of 1/19/1 1.. Place Values AND HUNDRED- MILLIONS TEN-MILLIONS MILLIONS HUNDRED- THOUSANDS TEN- THOUSANDS THOUSANDS HUNDREDS TENS ONES TENTHS HUNDREDTHS THOUSANDTHS TEN- THOUSANDTHS ,000, ,000, ,000 10, Decimal Point 1 10, Place values go up by powers of can be expressed as ( 100) + (3 10) + (4 1) Commas In numbers with more than 4 digits, commas separate off each group of 3 digits, starting from the right. These groups are read off together. Reading numbers 13,406, = One hundred twenty-three million, four hundred six thousand, nine and twenty-three thousandths Copyright Sally C. Zimmermann. All rights reserved.

4 Part 1 - Pre-Algebra Summary Page 4 of 1/19/1 < Less than > Greater than Less than or equal to Greater than or equal to = Equal Not equal to 1.3. Inequalities Mouth eats larger number Represent numbers on a number line. The number to the right is always greater Convert numbers so everything is the same e.g. all decimals, all fractions with common denominators, etc. > 7 > 6 > 6 < 7 > 7 < 6 > 6 > 7 > 5/3? <.01 > 4/? 10/4 8/4 > 10/ Rounding 1. Locate the digit to the right of the given place value. If the digit is 5, add 1 to the digit in the given place value 3. If the digit is < 5, the digit in the given place value remains 4. Zero/drop remaining digits Round to the nearest hundreds: > 10 = 100 > = 1300 Round to the nearest hundredths: > = 5.13 > = 5.1 Copyright Sally C. Zimmermann. All rights reserved.

5 Part 1 - Pre-Algebra Summary Page 5 of 1/19/ Divisibility Tests If last digit is 0,,4,6, or 8, 30, 50, 68, If sum of digits is divisible by 3 13 is divisible by 3 since = 6 (and 6 is divisible by 3) 864 is divisible by 4 since 64 is divisible by 4 4 If number created by the last digits is divisible by 4 5 If last digit is 0 or 5 5, 10, 15, 0, 5, 30, 35, If divisible by & 3 5 is divisible by 6 since it is divisible by & 3 9 If sum of digits is divisible by 9 61 is divisible by 9 since = 9 (and 9 is divisible by 9) 10 If last digit is 0 10, 0, 30, 40, 50, 5550 For Addition 1.6. Properties of Real Numbers For Subtraction For Multiplication For Division Commutative a + b = b + a a b b a ab = ba a/b b/a Associative (a+b)+c = a+(b+c) (a b) c a (b c) (ab)c = a(bc) (a b) c a (b c) Identity 0+a = a & a+0 = a a 0 = a a 1=a & 1 a=a a 1 = a Inverse a + ( a) = 0 & ( a) + a = 0 a a = 0 1/a a = 1 & a 1/a =1 if a 0 a a = 1 if a 0 Distributive Property a(b + c) = ab + ac a(b c) = ab ac a(b + c) = ab ac a(b c) = ab + ac Addition Property of Equality 1.7. Properties of Equality If a = b then a + c = b + c Multiplication Property of Equality If a = b then ac = bc Multiplication Property of 0 0 a = 0 and a 0 = 0 Copyright Sally C. Zimmermann. All rights reserved.

6 Part 1 - Pre-Algebra Summary Page 6 of 1/19/ Order of Operations 1 SIMPLIFY ENCLOSURE SYMBOLS: > Absolute value, parentheses ( ), or brackets [ ] [ 3( + 1) ] If multiple enclosure symbols, do innermost 1 st = If fraction, pretend it has ( ) around its ( 3(3) ) numerator and ( ) around its denominator CALCULATE EXPONENTS (Left to Right) 3 PERFORM MULTIPLICATION & DIVISION (Left to Right) 4 PERFORM ADDITION & SUBTRACTION (Left to Right) = ( 9) 18 = 18 = 1 > Evaluate x for x = 5 (means the base is 5, not 5! It's very helpful to put the value of the variable in parentheses) x = ( 5) = ( 5)( 5) = 5 > ( 5) = ( 5)( 5) = (5) = > 5 = 5 = 5 > 5 = 5 = 5 > = = = 5 + ( 0) + 6 = = 9 5 SIMPLIFY FRACTIONS 5 > = > = = > = = OR 3 9 3/ Copyright Sally C. Zimmermann. All rights reserved.

7 Part 1 - Pre-Algebra Summary Page 7 of 1/19/1. Real Numbers Absolute Value x Addition + Subtraction _ Multiplication Division (Divisors zero) Exponential Notation Double Negative.1. Operations The DISTANCE (which is always positive) of a number from zero on the number line If the signs of the numbers are the SAME, ADD absolute values If the signs of the numbers are DIFFERENT, SUBTRACT absolute values The answer has the sign of the number with the largest absolute value Change subtraction to ADDITION OF THE OPPOSITE NUMBER ADD numbers AS ABOVE MULTIPLY the numbers DETERMINE THE SIGN OF THE ANSWER If the number of negative signs is EVEN, the answer is POSITIVE If the number of negative signs is ODD, the answer is NEGATIVE DIVIDE the numbers DETERMINE THE SIGN of the answer by using the MULTIPLICATION SIGN RULES AS ABOVE A exponent is a shorthand way to show how many times a number (the base) is multiplied by itself An exponent applies only to the base Any number to the zero power is 1 The opposite of a negative is POSITIVE > > = = 0 = 0 > > > > 3 + ( ) = 1 addend addend > + 3 = 1 > 3 + ( ) = 5 > 3 + ( ) = 1 > 3 = 3 + ( ) = 1 = = > = ( ) = 4 minuend subtrahend difference > ( 3) = + 3 > 3 = 3 + ( ) > 3 ( ) = 3+ > 3 = 3 = (3)() = 6 factor factor > ( 3)( ) = 6 product > 3 = 3 ( ) = (3)( ) = 6 > ( 3)( )( ) = 1 > ( 1) ( ) = 6 dividend divisor quotient > ( 1)/ = 6 > 1 = 6 sum 0 > 3 = = 3 3 = 3 3 = 9 > 5 = 5 5 > ( 5) = 5 5 > = (base is ) > ( ) = (base is ) > ( x) = 1( 1 x) = x Copyright Sally C. Zimmermann. All rights reserved.

8 3. Fractions Fractions Proper fraction Improper fraction Part 1 - Pre-Algebra Summary Page 8 of 1/19/ Definitions Numerator/denominator o Numerator < Denominator o Numerator > Denominator > /7 > 7/ Mixed number Integer + fraction > 3 ½ Factor A whole number that divides into another number Factors of 18: 1,, 3, 6, 9, & 18 Prime Number A whole number > 1 whose only factors are 1 > >, 3, 5, 7, 11, 13 and itself Composite Number A whole number > 1 that is not prime > 4, 6, 8, 9, 10 Prime Factorization A composite number written as a product of > 18 = 3 3 prime numbers Factor Tree A method for determining prime factorization 18 of a number 3 6 Lowest Terms 3 1 or 1 Equivalent 1 5 = 10 Numerator and denominator have NO COMMON FACTORS other than 1. Reduce fractions by cancelling common factors. If the numerator and/or denominator has addition or subtraction, it is not in factored form. No cancelling of factors Numbers that represent the same point on a number line Multiplying a number by any fraction equal to 1 does not change the value of the number (see Multiplication Identity) > = = x + 8y > NO!! x > = x 18 = = = Copyright Sally C. Zimmermann. All rights reserved.

9 Part 1 - Pre-Algebra Summary Page 9 of 1/19/1 Least Common Multiple (LCM) (Smallest number that the given numbers will divide into) Least Common Denominator (LCD) 3.. Least Common Denominators 1 METHOD 1 (USE THE LARGEST NUMBER) 1. Start with the largest number. Do the other numbers divide into it?. If yes, you re done! 3. If no, double the largest number. Do the other numbers divide into it? 4. If yes, you re done! 5. If no, triple the largest number. Do the other numbers divide into it? 6. Keep going until you re done METHOD (PRIME FACTORIZATION) 1. Determine the prime factorization of each number. The LCM will have every prime factor that appears in each number. Each prime factor will appear the number of times as it appears in the number which has the most of that factor. 3 METHOD 3 (L METHOD) 1. Find a number that divides into at least two of the numbers. Perform the division 3. Repeat steps 1 & until there are no more numbers that divide into at least two of the numbers 4. Multiple the leftmost and bottommost numbers together The smallest positive number divisible by all the denominators. The LCD is also the LCM of the denominators. Ex. Find the LCM of 4,6,9 9, no 18, no 7, no 36, YES Ex. Find the LCM of 4,6, LCM = 3 3 = 36 Ex. Find the LCM of 4,6, LCM = = Ex.,, LCD = 36 Copyright Sally C. Zimmermann. All rights reserved.

10 Part 1 - Pre-Algebra Summary Page 10 of 1/19/1 Addition/ Subtraction + Multiplication Division Converting an Improper Fraction to a Mixed Number 3.3. Operations Convert mixed numbers to improper fractions & solve as fractions Denominator 0; Always write final answer in lowest terms If the DENOMINATORS are the SAME: 1 1 > = COMBINE NUMERATORS Write answer over common denominator If the DENOMINATORS are DIFFERENT: 1 1 Write an equivalent expression using the > LCD = 4 4 LEAST COMMON DENOMINATOR ADD/SUBTRACT (see If the = = DENOMINATORS are the SAME ) = 4 FACTOR numerators and denominators 5 / 5/ 5 5 Write problem as one big fraction > = = 5 6 5/ / 3 3 CANCEL common factors MULTIPLY top top & bottom bottom RECIPROCATE (flip) the divisor MULTIPLY the fractions (See Multiplication) 1. Numerator Denominator. Whole-number part of the quotient is the whole-number part of the mixed number. Remainder is the fractional part. Divisor 6 5 > 5 = = > = ( 7) = 3 remainder 1 1 = 3 Converting a Mixed Number to an Improper Fraction 1. If the mixed number is negative, ignore the sign in step and add the sign back in step 3. (Denominator whole-number part) + numerator Answer to step 3. Original denominator 1 > = 7 7 = Copyright Sally C. Zimmermann. All rights reserved.

11 Part 1 - Pre-Algebra Summary Page 11 of 1/19/1 4. Decimals Addition/ Subtraction + Multiplication Division To Multiply by Powers of 10 (shortcut) To Divide by Powers of 10 (shortcut) 4.1. Operations LINE UP the decimals Pad with 0 s Perform operation as though they were whole numbers Remember, the decimal point come straight down into the answer Multiply the decimals as though they were whole numbers Take the results and position the decimal point so the number of decimal places is equal to the SUM OF THE NUMBER OF DECIMAL PLACES IN THE ORIGINAL PROBLEM If the DIVISOR contains a DECIMAL POINT MOVE THE DECIMAL POINT TO THE RIGHT so that the divisor is a whole number MOVE THE DECIMAL POINT IN THE DIVIDEND THE SAME NUMBER OF DECIMAL PLACES TO THE RIGHT Divide the decimals as though they were whole numbers The decimal point in the answer should BE STRAIGHT ABOVE THE DECIMAL POINT IN THE DIVIDEND Move the DECIMAL POINT TO THE RIGHT the same number of places as there are zeros in the power of 10 Move to the right because the number should get bigger (Add zeros if needed) Move the DECIMAL POINT TO THE LEFT the same number of places as there are zeros in the power of 10 Move to the left because the number should get smaller (Add zeros if needed) > = > = = 10. =. 1 > > = = > 67.6 / 1000 = = Copyright Sally C. Zimmermann. All rights reserved.

12 5. Conversions From Percent* 13% Part 1 - Pre-Algebra Summary Page 1 of 1/19/ Percent to Decimal to Fractions To Percent* To Decimal To Fraction Drop the % sign & divide Write the % value over 100. by 100. (move the decimal Always reduce. 13/100 point digits to the left) If there is a decimal point, multiply numerator & 1.3 denominator by a power of 10 to eliminate it If there is a fraction part, write the percent value as an improper fraction % = = = From Decimal 1.34 From Fraction 1 6 Multiply by 100 (move the decimal point digits to the right) & attach the % sign 13.4% Method 1 - To express as a mixed number multiply by = = 16 % 3 Method - To express as a decimal convert to decimal & multiply by = 16.7% Perform long division Write number part. Put decimal part over place value of right most digit. Always reduce = % means per hundred Copyright Sally C. Zimmermann. All rights reserved.

13 Part 1 - Pre-Algebra Summary Page 13 of 1/19/1 6. Algebra 6.1. Definitions Constant A NUMBER > 5 Variable A LETTER which represents a number > x Coefficient A NUMBER associated with variable(s) > 3x Term A COMBINATION of coefficients and variable(s), or a > 3x constant Like Terms Each variable (including the exponent) of the terms is exactly the same, but they don t have to be in the same order Linear Expression One or more terms put together by a + or The variable is to the first power Linear Equation Has an equals sign The variable is to the first power > 3x & x > 3x & x > 3xy & xy > 3x + 3 > 3x+ 3 = 1 Copyright Sally C. Zimmermann. All rights reserved.

14 Part 1 - Pre-Algebra Summary Page 14 of 1/19/1 Addition/ Subtraction + (Combining Like Terms) Multiplication/ Division 6.. Operations of Algebra Only COEFFICIENTS of LIKE TERMS are > 4x 3x = x combined > 3xy xy + 3 = 5xy + 3 Underline terms (or use box & circle ) as they are combined > xyz + 3xy Can't be combined > y + y Can't be combined COEFFICIENTS AND VARIABLES of ALL TERMS are combined > (4 x)( 3 x) = 1x > ( 3 xy)( xy)(3) = 18x y > (xyz)( 3 xy) = 6 > ( y )( y) = y 3 x y z Copyright Sally C. Zimmermann. All rights reserved.

15 Part 1 - Pre-Algebra Summary Page 15 of 1/19/ Solving Equations with 1 Variable 1ELIMINATE FRACTIONS Multiply both sides of the equation by the LCD x + 1 Ex. Solve x = (x + 1) 3 x = x + 1 3x = 0 REMOVE ANY GROUPING SYMBOLS SUCH AS PARENTHESES Use Distributive Property 3SIMPLIFY EACH SIDE Combine like terms 4GET VARIABLE TERM ON ONE SIDE & CONSTANT TERM ON OTHER SIDE Use Addition Property of Equality (moves the WHOLE term) 5ELIMINATE THE COEFFICIENT OF THE VARIABLE Use Multiplication Property of Equality (eliminates PART of a term) 6CHECK ANSWER Substitute answer for the variable in the original equation x + 1 3x = 0 x + 1 3x = 0 x + 1 = 0 x = 0 1 x = 1 ( 1) ( x) = ( 1) ( 1) x = 1 (1) + 1 (1) = = 0 Note: Occasionally, when solving an equation, the variable cancels out : If the resulting equation is true (e.g. 5 = 5), then all real numbers are solutions. If the resulting equation is false (e.g. 5 = 4), then there are no solutions. Copyright Sally C. Zimmermann. All rights reserved.

16 Part 1 - Pre-Algebra Summary Page 16 of 1/19/1 1CIRCLE THE SPECIFIED VARIABLE TREAT THE SPECIFIED VARIABLE AS THE ONLY VARIABLE IN THE EQUATION & USE THE STEPS FOR SOLVING LINEAR EQUATIONS 1. Eliminate fractions. Remove parenthesis 3. Simplify each side 4. Get variable term on one side & constant term on other side (use addition/subtraction) 5. Eliminate the coefficient of the variable (use multiplication/division) 6. Check answer 6.4. Solving for a Specified Variable 5 Ex. Solve T F = T C+3 9 for TC, where T C is the temperature is Celsius and T is the temperature in Fahrenheit F 5 T F-3= T C ( TF -3) = T ( TF -3) = TC 5 C Copyright Sally C. Zimmermann. All rights reserved.

17 Part 1 - Pre-Algebra Summary Page 17 of 1/19/1 Definition Equivalent Expand Opposite of factoring rewrite without parenthesis Factor Opposite of expanding rewrite as a product of smaller expressions Cancel Only within a fraction in factored form Simplify/ Evaluate/ Add/Subtract/ Multiply/Divide An equalivent expression with a smaller number of parts Evaluate for a number Substitute the given number (put it in parenthesis) & simplify Solve 6.5. Expressions vs. Equations Expressions One or more terms put together by a + or Ex. x + x + 1 Evaluate both for x = & get same answer Ex. x + x + 1 = x + 3x x + 1 Ex. ( x + 1) = x + Ex. x + = ( x + 1) ( x + ) Ex. + 3 = x Cancelling common factors (top & bottom) & collecting like terms / ( x + ) Ex x + 6 x + 8 = + = ~denominators stay when adding fractions Ex. Evaluate x for x = base is ( ), not! = ( ) ( ) = 4 Equations Expression=Expression + + = Same solution Ex. x x 1 0 x + x + Ex. 1 0 x + = x + = 4 0 Often used in solving equations Often used in solving equations Often used in solving equations Often used in solving equations ~denominators are eliminated when simplifying equations Used to check answers Ex. Evaluate 0 = x + for x = 0 = ( ) + 0 = 0 so is a solution Find all possible values of the variable(s) Ex. x + = 0 x = Copyright Sally C. Zimmermann. All rights reserved.

18 Part 1 - Pre-Algebra Summary Page 18 of 1/19/1 7. Word Problems Addition Subtraction Multiplication Division General Percent Word Problems 7.1. Vocabulary The sum of a and b The total of a, b, and c 8 more than a a increased by 3 a subtracted from b The difference of a and b 8 less than a a decreased by 3 1/ of a The product of a and b twice a a times 3 The quotient of a and b 8 into a a divided by 3 Variable words Multiplication words Equals words 50% of 60 is what 50% of what is 30 What % of 60 is 30 a + b a + b + c a + 8 a + 3 b a a b a 8 a 3 (1/)a a b a 3a a b a/8 a/3 what, how much, a number of is, was, would be 50% 60 = a 50% a = 30 (a%) 60 = 30 Copyright Sally C. Zimmermann. All rights reserved.

19 Part 1 - Pre-Algebra Summary Page 19 of 1/19/1 Commission Sales Tax Total Price Amount of Discount Sale Price Simple Interest Percent increase/decrease 7.. Formulas Commission = commission rate sales amount Sales tax = sales tax rate purchase price Total price = purchase price + Sales tax Amount of discount = discount rate original price Sale price = original price Amount of discount Simple interest = principal interest rate time amount of increase/decrease Percent increase/decrease = amount 100 original Copyright Sally C. Zimmermann. All rights reserved.

20 Part 1 - Pre-Algebra Summary Page 0 of 1/19/1 1 UNDERSTAND THE PROBLEM As you use information, cross it out or underline it. Remember units! NAME WHAT x IS Start your LET statement x can only be one thing When in doubt, choose the smaller thing 3 DEFINE EVERYTHING ELSE IN TERMS OF x 4 WRITE THE EQUATION 5 SOLVE THE EQUATION 6 ANSWER THE QUESTION Answer must include units! 7.3. Steps for Solving Kevin s age is 3 years more than twice Jane s age. The sum of their ages is 39. How old are Kevin and Jane? Let x = Jane s age (years) x + 3 = Kevin s age 7 CHECK = 39 Kevin's age + Jane's age = 39 x x = 39 3x + 3 = 39 ( 3 ) + 3x + 3 = 39 + ( 3) 1 3x 36 1 = x = 1 Jane s age = 1 years Kevin s age = (1) + 3 = 7 years Copyright Sally C. Zimmermann. All rights reserved.

21 Part 1 - Pre-Algebra Summary Page 1 of 1/19/1 Rate Ratio Proportion Cross Product (shortcut) 7.4. Ratios & Proportions The quotient of two quantities Used to compare different kinds of rates A statement that TWO RATIOS or RATES are EQUAL On a map 50 miles is represented by 5 inches. 10 miles would be represented by how many inches? Method for solving for x in a proportion Multiply diagonally across a proportion If the cross products are equal, the proportion is true. If the cross products are not equal, the proportion is false 1 person gallons, 100 people 50 miles 10 (miles) 50 (miles) = x (inches) 5 (inches) x = 10 5 = = 50x 5 50 = x 50 5 = x x Copyright Sally C. Zimmermann. All rights reserved.

22 Part 1 - Pre-Algebra Summary Page of 1/19/1 Terminology Formulas Perimeter/ Circumference Area Surface Area Volume Square s Rectangle l Parallelogram Trapezoid w 7.5. Geometry Measures the length around the outside of the figure (RIM); the answer is in the same units as the sides Measures the size of the enclosed region of the figure; the answer is in square units Measures the outside area of a 3 dimensional figure; the answer is in square units Measures the enclosed region of a 3 dimensional figure; the answer is in cubic units PERIMETER: P = 4s AREA: A = s (s = side) PERIMETER: P = l + w AREA: A = lw (l = length, w = width) DEFINITION: a four-sided figure with two pairs of parallel sides. PERIMETER: P = h + b AREA: A = hb (h = height, b = base) DEFINITION: a four-sided figure with one pair of parallel sides PERIMETER: P = b 1 + b + other two sides AREA: A = ((b 1 + b ) / )h (h = height, b = base) Rectangular Solid SURFACE AREA: A =hw + lw + lh VOLUME: V = lwh (l = length, w = width, h = height) Circle CIRCUMFERENCE: C = π r d AREA: A = π r r = radius, d = diameter, r = 1 d r Sphere SURFACE AREA: A = π d 4 3 VOLUME: V = 3 π r ( ) Triangle a h b c PERIMETER: P = a + b + c 1 AREA: A = bh Copyright Sally C. Zimmermann. All rights reserved.