Multiply Decimals Multiply # s, Ignore Decimals, Count # of Decimals, Place in Product from right counting in to left Dividing Decimals Quotient (answer to prob), Dividend (the # being subdivided) & Divisor (# of equal parts dividend is to be subdivided into) Decimal always placed after ones in dividend, zeros can be added indefinitely to the right Terminating Decimal (an exact answer is achieved) Repeating Non-Terminating Decimal (a bar is used over the repeat to show, never use fractions in a decimal, don t round unless specifically asked or need to for a real-world application) Non-Repeating Non-Terminating Decimals are irrational numbers and can t be achieved by dividing one # by another Division by decimal: Move decimal out of divisor, moving it the same number of places (to the right) in the dividend Multiplication Property of Zero Anything times zero is zero: a 0 = 0 Sets of Numbers A group or collection of things (elements or members); In math numbers form sets Real Numbers (R) Natural Numbers (N): subset of real, whole, integers, rationals Whole Numbers (W): subset of real, integers, rationals Integers (Z): can be subdivided into positive (whole numbers) and negative integers; subset of reals & rationals Rational Numbers (Q): subset of reals; mutually exclusive of irrationals Irrational Numbers (I): subset of reals; mutually exclusive of rationals Subset is a set contained within another set Comparison of Numbers < is less than, > is greater than, is less than or equal to & is greater than or equal to How to tell apart: Old way is little eats big, new way is point out the small guy Order Property of Real # s: On # line left gets smaller & right gets bigger (neg. are all smaller than pos.) Decimals: Compare number by number, find the smaller, you ve found the smaller Fractions: Cross mult. up (denom to num) and the larger product is larger fraction Neg. # s: Larger the number looks without it s sign the smaller it is! Adding Fractions Must have an LCD to add Find LCD by prime factorization and unique primes to highest exponent (find product) Build higher terms by using Fundamental Theorem of Fractions (mult. old denom by constant to get LCD & mult. old num by same constant to get new num) All fractional answer are in lowest terms/reduce. Fundamental Thm of Fractions to divide out GCF Improper Fractions should be changed to mixed # s Integer Addition Subtraction is not allowed, change to addition by adding the opposite of the number following the subtraction symbol Like Signs when adding the numbers add and you keep the sign Unlike Signs when adding the big minus the small and bigger # s sign is sign of answer Absolute Value The distance from zero regardless of direction (the number w/out its sign) Absolute values DON T distribute ; they are grouping symbols do problem inside and then take abs. val Y. Butterworth Translation Problems Review 1
Order of Operations PEMDAS Parentheses, Exponents, Mult/Division (left to right order), Add/Subt. (left to right order) Parentheses is generic for grouping symbols which include parentheses, brackets, braces, absolute values, radicals, fractions bars Most common errors: add/subt. before mult. divide & mult. before dividing Adding/Subtracting Decimals Line up decimal places and add/subtract as normal, bringing down the decimal as it is crossed Multiplying Integers + + = +, = +, + = or + = Multiplying Fractions Cancel if possible (dividing out a common factor from num & denom) Mult. numerators & mult. denominators If improper change to mixed number Check for common factors, especially if you didn t try to cancel Exponents Represents repeated multiplication (base used as a factor number of times indicated by exponent) Fractions: numerator & denominator to exponent (if lowest terms to start will be in lowest terms in end) Decimals: see mult. decimals Grouping Symbols: Simplify inside 1 st then take single number to power One to any power is one: 1 n = 1 Negative number to even power (parentheses around the negative #) is always positive Neg. # to odd power (parentheses around neg. #) is always negative -a n (-a) n when n is even -a n is read as: The opposite of a to the n th power (-a) n is read as: A negative number used as a factor n times Anything to the zero power is one: a 0 = 1 Evaluation Put in the values given for the variables, using parentheses to replace the variables with the values Simplify using order of operations Distributive property should never be used in lieu of order of operations This is taught as a first step in a check for an equation, and it s use in many solution methods Properties of the Real Numbers Commutative Property of Addition and of Multiplication (move addend/factors around) Associate Property of Addition and of Multiplication (group addends/factors in different orders) Identity Property of Addition and of Multiplication (gives back the identity using identity element) Identity Element of Addition: ZERO Identity Element of Multiplication: ONE Inverse Property of Addition and of Multiplication (inverse is used to give back identity element) Distributive Property (Multiplication distributes over add/subt) Division by Zero: UNDEFINED Zero Divided by Anything: ZERO (division is multiplication by a recip so becomes zero times anything) Multiplication by Zero: ZERO Translation There is a whole separate sheet with all the translation nuances Y. Butterworth Translation Problems Review 2
Solving Equations Simplify: 1) Distribute 1 st 2) Clear fractions/decimals 3) Combine like terms Addition Property of Equality used to move things that are add/subt. from one another across the equal sign (can be used twice) Multiplication Property of Equality used to remove numeric coefficient of variable (last step used only once) Give answers as x = #, or as a solution set in roster form 3 Types of Equations: 1) Conditional 2) Identity 3) Contradictions 3 Types of Solutions from 3 Types of Eq.: 1) Single Solution 2) All Real Numbers 3) No Solution Solving an Equation for 1 Variable Follow the process for solving an equation, only focusing on the variable of interest Percentage Problems Percentage to decimal conversion: Move the decimal 2 places left (remember that decimal always comes after ones position) Decimal to Fraction Conversion: Read the decimal and write what you read or count the number of decimal places and put the number in the decimal over a factor of 10 with the number of decimal places that you just counted Set up as algebra problem: % of (whole) is (part) where percent as a decimal is multiplied by the whole and is equal to the part Set up as a proportion: is over of equals some part of one hundred Simple Interest: PRT = I % Increase/Decrease Problems: Original Price (op) is unknown and % is a known, and final result is known (price after increase or decrease) % of (op) is (increase/decrease) and then an equation results: op ± increase/decrease = price after which can be solved for op This is not all the concepts in Chapter 1, but this is an adequate review. I do not have time to cover every concept in the detail that I would like. I will leave it to you to review on your own and to look over my supplementary notes (Ch. 1 on my web page). Please do not put the review on the back burner, for it may become very important sometime in the very near future! Y. Butterworth Translation Problems Review 3
Addition Word Phrasing Symbols Sum The sum of 7 and 2 7 + 2 more than 5 more than 10 10 + 5 added to 6 added to 10 10 + 6 greater than 7 greater than 9 9 + 7 increased by 4 increased by 20 4 + 20 years older than 15 years older than John. John is 20. 20 + 15 total of The total of 6 and 28 6 + 28 plus 8 plus 281 8 + 281 Subtraction Word Phrasing Symbols difference of The difference of 5 and 2 The difference of 2 and 1 5 2 2 1 *years younger than Sam's age if he is 3 years younger than John. John is 7. 7 3 diminished by 15 diminished by 9 21 diminished by 15 15 9 21 15 *less than 17 less than 49 7 less than 17 49 17 17 7 decreased by 29 decreased by 15 15 decreased by 7 29 15 15 7 *subtract(ed) from Subtract 13 from 51 Subtract 51 from 103 51 13 103 51 take away 79 take away 61 79 61 subtract 54 subtract 2 54 2 less 16 less 4 16 4 * - Means that the numbers come in opposite order than they appear in the sentence. Multiplication Word Phrasing Symbols product The product of 6 and 5 6 5 times 24 times 7 24(7) twice Twice 24 2(24) multiplied by 8 multiplied by 15 8*15 at 9 items at $5 a piece ($5)9 "fractional part" of A quarter of 8 (¼)(8) or 8 / 4. "Amount" of "$" or Amount of money in 25 dimes ($0.1)(25) or (10)(25) " " (nickels, quarters, pennies, etc.) percent of 3 percent of 15 0.03(15) Division Word Phrasing Symbols divide Divide 81 by 9 81 9 quotient The quotient of 6 and 3 The quotient of 24 and 6 6 3 24 6 divided by 100 divided by 20 20 divided by 5 100 20 20 5 ratio of The ratio of 16 to 8 The ratio of 8 to 2 16 8 8 2 shared equally among 65 apples shared equally among 5 people 65 5 Note: Division can also be written in the following equivalent ways, i.e. x 6 = x/6 = 6 x = x 6 Y. Butterworth Translation Problems Review 4
Exponents Words Phrasing Algebraic Expression squared Some number squared x 2 square of The square of some number x 2 cubed Some number cubed x 3 cube of The cube of some number x 3 (raised) to the power of Some number (raised) to the power of 6 x 6 Equality Words Phrasing Algebraic Equation yields A number and 7 yields 17. Let x = #. x + 7 = 17 equals 7 and 9 equals 16 7 + 9 = 16 is The sum of 5 and 4 is 9. 5 + 4 = 9 will be 12 decreased by 4 will be 8. 12 4 = 8 was The quotient of 12 and 6 was 2. 12 6 = 2 Note: Any form of the word is can be used to mean equal. Parentheses Parentheses are indicated in four ways. The first is the use of a comma, such as: The product of 5, and 16 less than a number. The second is the use of two operators' phrases next to one another, such as: 17 decreased by the sum of 9 and 2. *Notice how decreased by is followed by the sum of and not a number, this indicates that we will be doing the sum first; hence a set of parentheses will be needed. Next, you may notice that the expected 'and' between the two numbers being operated on is after a prepositional phrase [A phrase that consists of a preposition (usually of in our case) and the noun it governs (usually number in our case) and acts like an adjective or adverb]. Such as: The sum of 9 times a number and the number. *Usually we would see the 'and' just after the number 9, but it does not appear until after the prepositional phrase 'of 9 times a number'. If you think of this in a logical manner, what you should see is that you have to have two numbers to operate on before you can complete the operation, which would require the use of parentheses to tell you to find a number first! Finally, you may notice a phrase containing another operator after the 'and' where you would expect a number. An example here might be: The difference of 51 and the product of 9 and a number. *The note about thinking in a logical manner applies here too! You must have two numbers to operate on! Y. Butterworth Ch. 1 Concepts Review Int. Alg. FTHL 5
Chapter 1 Pretest Circle the best answer. Support with work whenever possible. 1. (-6.4)(3)(0) = (5)(0)(5) a) True b) False 2. -12 < -13 a) True b) False 3. Simplify: (- 4 / 9 ) (- 3 / 4 ) a) 11 / 36 b) -1 7 / 36 c) - 1 / 9 d) 19 / 36 4. Simplify: 5 12 + 2 4 11 2 a) 18 b) 12 c) 32 d) 16 5. Simplify: 5(-0.2) (0.1)(2) a) -1.2 b) 0.8 c) -0.8 d) -0.3 6. Simplify: 11 + -54 6 2(3) a) 0 b) -4 c) 14 d) -7 7. Simplify: 4(- 4 / 5 ) 3 2 (2) + ( 2 / 3 ) 2 a) -22 8 / 9 b) 1 / 81 c) - 7 / 36 d) -20 34 / 45 8. Simplify: (-3 + 1) 2 + 7 9 24 6 + 2(4) a) - 1 / 2 b) - 1 / 6 c) 3 / 5 d) 1 / 2 9. Evaluate when x = 6, y = -1 and z = 0: x + 6y z 6x y + z a) 1 / 37 b) 0 c) 5 12 / 37 d) undefined Y. Butterworth Ch. 1 Concepts Review Int. Alg. FTHL 6
10. Name the property illustrated: 3(x + y) = 3(y + x) a) associative prop. of mult. b) communtative prop. of add. c) distributive property d) associative prop. of add. 11. Name the property illustrated: (a + b) 0 = 0 a) multiplicative identity prop. b) assoiciative prop. of add. c) multiplicative prop. of zero d) distributive prop. 12. Translate the statement using mathematical symbols. The quotient of seven and the sum of x and two is equal to four. a) 7 = 4 b) 7(x + 2) = 4 x + 2 c) 7 = 4 d) 7 = 4 x 2 x + 2 13. Translate the statement using mathematical symbols. The difference of 3 and twice x, multiplied by 4, is 12. a) 4(3 2x) = 12 b) 3 2x = 4(12) c) 3 4(2x) = 12 d) 4(3 + 2x) = 12 14. Translate the statement using mathematical symbols. Eight less than x is twice x. a) 8 x = 2x b) x 8 = x 2 c) x 8 = 2x d) 8 x = x 2 15. Translate the statement using mathematical symbols. Three times x squared subtracted from 5 is the product of 8 and x. a) 3x 2 5 = 8x b) 3(x 2 5) = 8x c) 5 3x 2 = 8x c) 3(5 x 2 ) = 8 / x 16. Solve: -4(2 3x) = -(4 + x) 3(x + 2) a) x = - 1 / 8 b) x = -1 c) x = 5 / 7 d) x = 1 Y. Butterworth Ch. 1 Concepts Review Int. Alg. FTHL 7
17. Solve: 3x 5 + 8(x 4) = 5(2x 7) a) x = 4 b) x = -26 c) x = -4 d) x = 0 18. Solve: 2x + 5 = 4x 3 3 8 a) x = -3 3 / 8 b) x = -7 3 / 4 c) x = -12 1 / 4 d) x = -1 3 / 4 19. Solve: 6 5m = 5 3(m + 1) 2m + 4 a) No Sol.or b) m = 5 / 6 c) m = 0 d) All Real # s 20. Solve for y: 5x + 11y = 6 a) y = 5x / 11 + 6 / 11 b) y = 11 / 6 5x / 6 c) y = 6 / 11 5x / 11 d) y = 11 / 5 6x / 5 21. Solve for y: 4 / x + 3 / y = 1 / z a) y = 4xz b) y = 3xz z 3x 4z x c) y = 3xz b) y = 3xz 4x z x 4z 22. Jose is deciding whether to accept a sales position at an electronics store. He is offered a salary of $1200 monthly plus a 8% commission on his sales. If his sales are $12,000, what is his pay? a) $2160 b) $960 c) $1056 d) $2560 Y. Butterworth Ch. 1 Concepts Review Int. Alg. FTHL 8