SAT Math Must-Know Facts & Formulas

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1 SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas pus specia numbers suc as 2, 3 and π Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses / Exponents / Mutipy / Divide / Add / Subtract) eac term is equa to te previous term pus d Sequence: t 1, t 1 + d, t 1 + 2d,... Exampe: d = 4 and t 1 = 3 gives te sequence 3, 7, 11, 15,... Geometric Sequences: eac term is equa to te previous term times r Sequence: t 1, t 1 r, t 1 r 2,... Exampe: r = 2 and t 1 = 3 gives te sequence 3, 6, 12, 24,... Factors: te factors of a number divide into tat number witout a remainder Exampe: te factors of 52 are 1, 2, 4, 13, 26, and 52 Mutipes: te mutipes of a number are divisibe by tat number witout a remainder Exampe: te positive mutipes of 20 are 20, 40, 60, 80,... Percents: use te foowing formua to find part, woe, or percent part = percent 100 woe Exampe: 75% of 300 is wat? Sove x = (75/100) 300 to get 225 Exampe: 45 is wat percent of 60? Sove 45 = (x/100) 60 to get 75% Exampe: 30 is 20% of wat? Sove 30 = (20/100) x to get 150 pg. 1

2 SAT Mat Must-Know Facts & Formuas Averages, Counting, Statistics, Probabiity average = sum of terms number of terms average speed = tota distance tota time Fundamenta Counting Principe: sum = average (number of terms) mode = vaue in te ist tat appears most often median = midde vaue in te ist (wic must be sorted) Exampe: median of {3, 10, 9, 27, 50} = 10 Exampe: median of {3, 9, 10, 27} = (9 + 10)/2 = 9.5 If an event can appen in N ways, and anoter, independent event can appen in M ways, ten bot events togeter can appen in N M ways. Probabiity: probabiity = number of desired outcomes number of tota outcomes Exampe: eac SAT mat mutipe coice question as five possibe answers, one of wic is te correct answer. If you guess te answer to a question competey at random, your probabiity of getting it rigt is 1/5 = 20%. Te probabiity of two different events A and B bot appening is P(A and B) = P(A) P(B), as ong as te events are independent (not mutuay excusive). Powers, Exponents, Roots x a x b = x a+b (x a ) b = x a b x 0 = 1 x a /x b = x a b (xy) a = x a y a xy = x y 1/x b = x b { ( 1) n +1, if n is even; = 1, if n is odd. pg. 2

3 Factoring, Soving SAT Mat Must-Know Facts & Formuas (x + a)(x + b) = x 2 + (b + a)x + ab FOIL a 2 b 2 = (a + b)(a b) Difference Of Squares a 2 + 2ab + b 2 = (a + b)(a + b) a 2 2ab + b 2 = (a b)(a b) To sove a quadratic suc as x 2 +bx+c = 0, first factor te eft side to get (x+a 1 )(x+a 2 ) = 0, ten set eac part in parenteses equa to zero. E.g., x 2 + 4x + 3 = (x + 3)(x + 1) = 0 so tat x = 3 or x = 1. To sove two inear equations in x and y: use te first equation to substitute for a variabe in te second. E.g., suppose x + y = 3 and 4x y = 2. Te first equation gives y = 3 x, so te second equation becomes 4x (3 x) = 2 5x 3 = 2 x = 1, y = 2. Functions A function is a rue to go from one number (x) to anoter number (y), usuay written y = f(x). For any given vaue of x, tere can ony be one corresponding vaue y. If y = kx for some number k (exampe: f(x) = 0.5 x), ten y is said to be directy proportiona to x. If y = k/x (exampe: f(x) = 5/x), ten y is said to be inversey proportiona to x. Absoute vaue: x = { +x, if x 0; x, if x < 0. Lines (Linear Functions) Consider te ine tat goes troug points A(x 1, y 1 ) and B(x 2, y 2 ). Distance from A to B: Mid-point of te segment AB: Sope of te ine: (x2 x 1 ) 2 + (y 2 y 1 ) 2 ( x1 + x 2 2, y ) 1 + y 2 2 y 2 y 1 = rise x 2 x 1 run pg. 3

4 SAT Mat Must-Know Facts & Formuas Sope-intercept form: given te sope m and te y-intercept b, ten te equation of te ine is y = mx + b. Parae ines ave equa sopes: m 1 = m 2. Perpendicuar ines ave negative reciproca sopes: m 1 m 2 = 1. a a b b a b b a a b m b a Intersecting Lines Parae Lines ( m) Intersecting ines: opposite anges are equa. Aso, eac pair of anges aong te same ine add to 180. In te figure above, a + b = 180. Parae ines: eigt anges are formed wen a ine crosses two parae ines. Te four big anges (a) are equa, and te four sma anges (b) are equa. Trianges Rigt trianges: c a b 30 2x x 3 60 x x 2 45 x 45 x a 2 + b 2 = c 2 Specia Rigt Trianges Note tat te above specia triange figures are given in te test booket, so you don t ave to memorize tem, but you soud be famiiar wit wat tey mean, especiay te first one, wic is caed te Pytagorean Teorem (a 2 + b 2 = c 2 ). A good exampe of a rigt triange is one wit a = 3, b = 4, and c = 5, aso caed a rigt triange. Note tat mutipes of tese numbers are aso rigt trianges. For exampe, if you mutipy tese numbers by 2, you get a = 6, b = 8, and c = 10 (6 8 10), wic is aso a rigt triange. Te Specia Rigt Trianges are needed ess often tan te Pytagorean Teorem. Here, x is used to mean any positive number, suc as 1, 1/2, etc. A typica exampe on te test: you are given a triange wit sides 2, 1, and 3 and are asked for te ange opposite te 3. Te figure sows tat tis ange is 60. pg. 4

5 SAT Mat Must-Know Facts & Formuas A trianges: b Area = 1 2 b Te area formua above works for a trianges, not just rigt trianges. Anges on te inside of any triange add up to 180. Te engt of one side of any triange is aways ess tan te sum of te engts of te oter two sides. Oter important trianges: Equiatera: Tese trianges ave tree equa sides, and a tree anges are 60. Isoscees: Simiar: An isoscees triange as two equa sides. Te base anges (te ones opposite te two sides) are equa. A good exampe of an isoscees triange is te one on page 4 wit base anges of 45. Two or more trianges are simiar if tey ave te same sape. Te corresponding anges are equa, and te corresponding sides are in proportion. For exampe, te triange and te triange from before are simiar since teir sides are in a ratio of 2 to 1. Circes (, k) r r n Arc Sector Area = πr 2 Circumference = 2πr Fu circe = 360 (Optiona) Lengt Of Arc = (n /360 ) 2πr Area Of Sector = (n /360 ) πr 2 pg. 5

6 Rectanges And Friends SAT Mat Must-Know Facts & Formuas w Rectange Paraeogram (Optiona) (Square if = w) (Rombus if = w) Area = w Area = Te formua for te area of a rectange is given in te test booket, but it is very important to know, so you soud memorize it anyway. Soids w r w Rectanguar Soid Voume = w Rigt Cyinder Voume = πr 2 Note tat te above soids figures are given in te test booket, so you don t ave to memorize tem, but you soud be famiiar wit wat tey mean. pg. 6

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