Quick study academic SAT tips math 600dpi

WORLD'S #1 ACADEMIC OUTLINE SAT TIPS: m ~ ~z lI1lJC()c1llc:iIl~ ~~ ~~~S:f:}~~~ ~ This guide reflects the changes in the March 2005 SAT ­ T -­ he Scholastic Aptitude Test (SAT): One of the measures used by colleges and universities to determine who gets in These pages will give you guidance and examples to familiarize you with the test, and offer helpful hints to achieve the best results J:~~ ~;.~~~~~~S English language skills • Critical Reading (Understanding and interpreting written materials) • Writing (Ability to use basic standard writing skills) Mathematical (Basic number skills) Time's Up! •two See QuickStudy«' guide "SAT Tips: Verbal" for more detailed information • Critical Reading • A section of short reading passages followed by comprehension questions • A long (400-900 words) reading passage testing the same • Sentence completions: Incomplete sentences with multiple-choice answers • Writing • Multiple-choice questions based on a short written passage that test knowledge of basics (grammar, spelling, diction, agreements, etc.) • A brief persuasive essay to test writing skills • Regular math questions, covering basic arithmetic, geometry, and algebra through H.S Algebra II • Grid-ins: Ask you to solve a problem and enter the answer itself in the grid Hovv long is it? Getting Your Best Score Remember this is only another test Designed to see what you know • Geared to your age and learning level -if you have studied, at least most of the time, you should pretty well %$ mmutea • 35mtnuteeaectione wrIdD8: Us9 and seateIlCe/parasraph coulCdalD: 25 mID Where the SAT diHers from other tests is in the time it takes, how the test is set up, and how you answer the questions The Time Fador • The test is hours 45 minutes long; with the exception of the writing questions (35 minutes) no section lasts mo re than 25 minutes • In the Math portion of the SAT you cannot skip ahead to another section • You cannot return to a section once the time for that section is over ,lest I Factor MIIt8rI.I you will b given for the t:eet:: A sealed test booklet, Inclucling the ue and space for writIn theeaay A pencil An aMWW rid .t • The answer grid sheet contains colunms of numbers corresponding to the test questions next to colunms containing open ovals with the letters A - E beneath them • Blacken in the appropriate oval on the line corresponding to the question It is easy to accidentally blacken the oval across from, say, question 45 with the answer to question 44 - At best, you may get one unfairly marked wrong answer - At worst, this may lead to all subsequent answers being wrong! / Do not spend too much time on any one question; if stumped, move on and time allowing come back to that question later / Do not be "thrown" if one section is difficult / Do not try to "make up time" in one section if you lost time in the one before it Should you guess1 - I point is given for every correct answer while 1/4 of a point is deducted for a wrong answer - No points are lost for an answer left blank, but no points are gained, either - If you have no idea of the correct answer, and all of the choices seem just as valid, don't guess - You may discover you not know the right answer, but you can eliminate all the wrong answers , • • • • ••••••••••••••••••••••• ** Arithmetic : : • Integers: Whole numbers including I negatives: { • -3 -2 -1 O 2.3 ) I • Consecutive integers: Arranged in I sequence: (12 13 14 15) or (-5 -4 -3) or I generally (n n + n + ) • Even numbers: Integers divisible by 2: : I I I I I { -4 -2 O ) • • Set: A collection of elements; relations between sets include: Intersection: Union: (n): All elements that belong to both sets (U): All elements that belong to either set or both I • Odd numbers: Integers not divisible by 2: I { -3 -1 1.3.5.7 ) I I • Prime number: Divisible by only and itself: 2.3.5.7.11.13.17.19 • Composite number: An integer that is not prime; every integer can be factored into a product of primes in only one way • Digits: The numerals O 1.2.3.4.5.6 and • Arithmetic sequence: A constant number is added to produce the next term; the following sequence adds between consecutive terms: 11 15 19.23 • Geometric sequence: A constant number is multiplied to produce the next term; in the sequence 2.4.8 16 24.48 the next term is obtained by multiplying by • Rational numbers: Can be expressed as a ratio of integers mIn; in decimal form such a number either terminates or repeats; Ex: 3/8 = 0.375 2/3= 0.66666 • 5!g= 0.55555 • 12h = 12 • Irrational numbers: Cannot be expressed as a ratio of integers; in decimal form irrational numbers neither terminate nor reJ?eat; Ex: j3 = 1.7320508 /2 = 1.414213 1t =3.14159 • Percent refers to hundredths: 35% = 0.35 = 35/100 = 7/20; Ex: What is 40% of 80? 80x 0.40 =32 • Percent increase or decrease: = increase or decrease original quantity; Ex: An item usually priced at $400 is given a sale price of $340; what is the percent decrease? Decrease = $400 - $340 = $60 percent decrease = $60 1$400 = 0.15 = 15% • Average speed: Total distance /total time; not simply average the speeds! Ex: Jennifer travels for hours at 50 mph then for hours at 60 mph; what is her average speed? Total distance = x 50 + x 60 = 200 + 180 = 380 miles total time = + = hours average speed = 380 miles I hours = 54 2hmph I • If one choice can be made in m ways and another can be made in n ways I then there are mn ways to make both I • This principle extends to situations where several sequential choices are to I be made; I - Ex: A diner offers a lunch special in which customers can choose a soup I a sandwich and a beverage; if they offer kinds of soup sandwiches I and beverages how many different specials can be composed? Since ~ there are ways to choose a soup ways to choose a sandwich and ways I to choose a beverage, there are a total of2 x x = 70 combinations _ -­ The number of ways in which x objects can be arranged in order is x! ("factorial") • This is the product of all the integers between x and I: xl =x(x-l)(x-2) (I); Ex: 51 = x x x x = 120 - Ex: Six chess players compete in a tournament; how many rankings are possible for the order in which they finish? Since there are players who could come in first, then who could come in second, then who could come in third, and so on, the total number ofpossible rankings is 6! = x x x x2x1=720 • If only x objects out of a larger group of size n are being arranged, the number of arrangements is nll(n - x) ! - Ex: If, out often runners in a race, awards are to be given for 1st • 2nd and 3nt place, how many ways can they be aSsigned? There are 10 runners who could win 1st place, and then who could win 2"d, and then who could win 3"", so the total number ofarrangements is 10 x x =1O! I (1 - 3)f =720 • When counting the number of choices of x things out of a group of n without regard to order the number of choices is n!/(n - x)!xl since the xl arrangements of each group chosen are counted as one - Ex: Three people must be chosen from among the five in the accounting department to attend a meeting; how many choices are possible? There are Sf! (5 - 3)f 3! = Sf I 2f 3! =5 x x x x 12 x x x x J = 10 ways to c1wose the three - Or, call the five employees A B C D and E • There are five choices for the first then four fo r the second then three for the third employee to attend the m eeting • But this takes into account the order which doesn't matter here: there are six sequences for every possible group • That is, ABC, ACB BAe BCA, CAB, and CBA all consist of the same employees • So the x x that we would obtain should be divided by the x x ways to arrange the members yielding 60/6 = 10 groups Systems of Eq,uations (x.y) a =xa.ya Xa'Xb=X(a+b) (xa) b = X a • b X a/b= b EX: S4/J = Jj84 = eM =24 = if x"l 0, 16 /9 = 3, not ± x_ a = l/x a xO=l It is useful 10 bear in mind that an equivalent equation - that is, one with the same solution set - can be obtained by addinq, subtracting, multiplying, or dividing the entirety of both sides by the same /Xa Unless noted otherwise, ["denotes the positive (principal) root That is, a ~ =x(a - b) (y /x) a = ya/xa Factoring I Remove a common factor: 3x2 + 12x = 3x(x + 4) I Perfect square: 4x + 12x + = (2x + 3) (2x +3) = (2x + 3)2 I Difference of squares: x - 25 = (x + 5) (x - 5) I Quadratic: 3x + 13x -IO = (3x- 2)(x + 5) expression • Ex: If 5x + 3y = 11 and 3x - 6y = 30, solve for x and y - By substitution: Taking the second equation, we can add 6y to both sides to produce 3x = 6y + 30 • Dividing both sides by produces X= 2y+ 10 • By subsituting 2y + 10 for x in the first equation, we obtain 5(2y + 10) +3y= 11 • Distributing the yields lOy + 50 + 3y = II; combining like terms gives 13y + 50 = 11 • Subtract 50 from both sides: 13y = -39; then y = -3 • Now we can substitute -3 for y in either equation to find x = 4, so the solution is y = -3, x = • By combination: Multiply both sides of the first equation by to produce 10x+ 6y= 22 • Then add the two equations together, combining all like terms: (lOx + 6}1 + (3x- 6}1 = 22 + 30, so 13x = 52, or x = 4; then substitution yields y = -3 SolVing Eq,uations • The equation can also be solved by factoring: if 3x - 7x + = 0, then (3x- I)(x- 2) = - Since the product will be zero only if one of the factors is equal to zero, 3x - = or x- = - Solving the first produces x = 113, the second, x = 2; the solution is x = or x = 1/3 - Check x = 2: 3(2)2 - 7(2) = 3(4) -14 = 12 -14 =-2 - Checkx= 1/3: 3(1/3)2-7(1/3) = 1/3 -713 = -% =-2 • Rational equations: Contain variables in the denominators of rational expressions; solve them by mUltiplying to eliminate the denominators • Ex: Solve for x: 3/x- 1/2= 7/4 - First, add liz to each side to produce 3/x= % Multiplying by x produces = (9/4)x Then dividing by % yields x = 3(4/9) =4/3 - Check: 3/(4/3) - _ = 3(314) - _ = % - 24 = 7/4 • Ex: Solve for w: 3/(w + 2) -lIw = 1I(4w); multiply each term by the appropriate expression to obtain the common denominator of 4w(w + 2) - That is, [3/(w + 2)][4w14w) - [lIw)[4(w + 2) I 4(w + 2)) = [11 (4w))[(w + 2) I (w + 2») so 12w I 4w( w + 2) - 4(w + 2) I • The solution set of an equation consists of all values of the variable that make the equation true • Equations can be solved by performing operations to transform into equivalent equations the whole expressions on both sides of the equation; preserves the solution set • Work toward isolating the variable • When multiplying or dividing, be careful to perform the operation on every term of each side • Always check your solution(s) against the original equation • Linear equations have degree (the highest exponent of a variable) I; they can be solved by simple arithmatic operations • Ex: Solve for z: 3z+ = 12; subtracting from both sides produces 3z= - Dividing both sides by produces the solution z = 7/3 - Check:3(7/3)+5=7+5=12 • Ex: Solve for y: - 5/y= 14 - Subtracting from both sides, we have -Sly = 10 - Multiplying by y, -5 = lOy; then y = _Sly) = - 1/2 - Check: - 51 (-liz) = + 5(2) = + 10 = 14 • Quadratic equations: Have a squared term; that is, they can be expressed Ax2 + Bx+ C= 0, where A, B, and Care constant coefficients • They can be solved by checking that they are set equal to zero, and then either factoring the quadratic expression or applying the quadratic formula: given a quadratic equation set equal to zero, 4w(w+ 2) x= -B±jB2-4AC 2A • Ex: Solve for x: 3x -7x=-2 - First, set the equation equal to zero: 3x2 -7x + = - Then appl the uadratic formula, with A = 3, B = -7, and C =2, x = (7 ± 49 - 4x3x2) 16 = (7 ± j2s) 16 = (7 ± 5) 16 = 12/6 or 216 SO =(w+ 2) 14w(w+ 2) - Now that all terms have a common denominator, that denominator doesn't matter, except that it must be nonzero - That is, 4w(w+ 2) 0, so 4w and (w+ 2) 0, so W" and W" -2 (These values must be excluded from the solution set) - Or, we can mUltiply the entire equation by 4w(w + 2) which must be nonzero to give: 12w- 4(w + 2) =w + 2, so 12w- 4w- = w+ 2, 8w- = w+ 2, 7w = 10 so W= 1% - Check: 3/(lOh+ 2) -lI(lOh) = 1/(4 x IOh) 31(24h) - 7/10= 7/40 7/8 - 7/10 = 7/40 35/40 - 28/40 = 7/40 the solution is x = or x = 1/3 • When solving inequalities, you can perform the arithmetic operations as with equations, but remember to change the direction o/the inequality when mUltiplying or dividing by a negative number • Always check your answer against the original inequality with some relevant x values • Ex: Solve for x: - x 7; -x -2, so x ~ - Check: x = fails, x = works, x = works • The solution to a system of inequalities is the intersection of the solutions of the inequalities • Ex: Solve for x: 2x + 12 and - 3x 13 - x + 2, and -3x - X and x ~ -2; so the solution is -2 x - Check: x =-3 fails the second inequality, x = - satisfies both, x = satisfies both, x = fails the first Algebra continued: Functions • A function - denoted I(x) - is a relation in which each element of the domain is matched with only one element of the range • The domain consists of all numbers x for which fix) is a real number • Ex: For I(x) = 3/(x + 3), the domain is x 3, since division by is undefined; for fix) = j x ­ 4, the domain is x ~ 4, since the square root of a negative is imaginary Probability • Its value is always between and 1; if the probability of an event is O the event is impossible; if probability is the event is certain • If all outcomes are equally likely the probability of an event is the ratio: Numbu of ways tM evellt COlI occur Total numbu of possible outColfUS • Ex: There are 14 girls and 11 boys in a kindergarten class; one is chosen at random; what is the probability of choosing a girl? 14 t.: 11 - ~~ = 0.62 9" ses III An is a mefI8lIte used to describe data, usualIyMferrJDg to the ,Ic'ywedc The mean of a set of II numbers is defined 81 the sum oftbe numbers divided by II • Ex: The mean of 550, 820, and 830 is: SSO + 'f + §30 _ lye» - 600 Absolute Value On the SAT, the word "average" refers to the artthmetic mean, except for probIBms illuoluingaverage speed • The absolute value of x, denoted lxi, is its distance from zero • That is,lxl = x, if x~ 0; -x, if x< • Equations involving absolute value can be solved by taking the positive and the negative of the expression inside the absolute value • Ex: Solve for x: 13x- 41 = 17 - (3x-4) =17 or-(3x-4) =-3x+ 4=,17 - 3x= 21 or-3x= 13 - X= or X= -13/3 • Inequalities involving absolute value can be similarly solved, but the direction of the inequality must be switched when taking the negative • A ">" inequality results in two disjoint solutions, and a " 113: the only integer greater than 1/3 but less than 115 is 3.350; Some n% of a quantity q is (n/100)q, so 28% of 300 = (28/100)300 = 84; this result is 24% of x, that is, (24/100) x = 84 • Dividing both sides by 241100 (or 24) produces X= 350 4.2112 or 10.5; since "yand x vary inversely" means that there is a constant b such that y = b I x, multiplying by x produces xy = b; with y= 7, X= 12, xy= 84 = b • For x = 8, Y= 84/8 = 2112 or 10.5 144; the area of a circle is 1tr2 = 361t, so r2 = 36, and r = 6; since the circle is inscribed in the square, the sides of the square must have length 2r= 12, producing an area of 12 x 12 = 144 6.26 Let C M, and S represent the number sold by Carmen, Melvin, and Sam; "Carmen sold twice as many candy bars as Sam" means that C = 2S, and "Melvin sold five fewer than Carmen" means that M = C- = 2S - • The total number of candy bars sold is 105, so C + M + S = 105; substituting produces (2S) + (2S - 5) + S = 125, so 5S = 130, producing S = 26 • Using this value for S yields C =2(26) =52, and M =2(26) - = 47 • The result can be checked: C + M + S = 52 + 47 + 26 = 125 So Sam sold the least number of bars: 26 ~ mproveyour Z rC~ko ~ut t"; ;it.r ~: score! 00 0 0 00, QuickStuclye guides: A ~-g=,,", ,-=-= -­ c: "'=.:.: :.= - - - - - - - ­ ~ ~ Commonly Misspelled and Confused Words LL CREDITS Author: Stephen Kizlik PRICE: u.s $5.95 CAN $8.95 20878 NOTE TO STUDENT Use this QUICICSTUDY ~ guide as a resource to help you Improve your test scores, but not as a substitute for dHigent study, homework and class attendance : Math Review o 2005 Ba~haI1•• Inc :o SAT Verbal All rlghts re&erved No part of this publication may be reproduced or transmitled In any form or by any means , electronic or mechanical, i~tuding phOtocopy, recording or any informati(ln stora ge and retrleval system, without wrtnen permiSSion from the publisher •• • .• • .• 0306 Customer Hotline # 1.800.230.9522 We welcome your feedback so we can maintain and exceed your expectations hun.dr~ gs qUICKS i fuay.com titles at ISBN - 13 : 78 - 15722287 - ISBN- D: 157222878 - 911~ lll,lli~~II ~ ~ll1Jl1l11 1 1il111rIlillll ... this QUICICSTUDY ~ guide as a resource to help you Improve your test scores, but not as a substitute for dHigent study, homework and class attendance : Math Review o 2005 Ba~haI1•• Inc :o SAT Verbal... and x ~ -2; so the solution is -2 x - Check: x =-3 fails the second inequality, x = - satisfies both, x = satisfies both, x = fails the first Algebra continued: Functions • A function - denoted... the intercept b /