ACT Math Facts & Formulas


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1 Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as 2, 3 and π Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses / Exponents / Multiply / Divide / Add / Subtract) eac term is equal to te previous term plus d Sequence: t 1, t 1 + d, t 1 + 2d,... Example: d = 4 and t 1 = 3 gives te sequence 3, 7, 11, 15,... Geometric Sequences: eac term is equal to te previous term times r Sequence: t 1, t 1 r, t 1 r 2,... Example: 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 Example: te factors of 52 are 1, 2, 4, 13, 26, and 52 Multiples: te multiples of a number are divisible by tat number witout a remainder Example: te positive multiples of 20 are 20, 40, 60, 80,... Percents: use te following formula to find part, wole, or percent part = percent 100 wole Example: 75% of 300 is wat? Solve x = (75/100) 300 to get 225 Example: 45 is wat percent of 60? Solve 45 = (x/100) 60 to get 75% Example: 30 is 20% of wat? Solve 30 = (20/100) x to get pg. 1
2 Averages, Counting, Statistics, Probability average = sum of terms number of terms average speed = total distance total time Fundamental Counting Principle: sum = average (number of terms) mode = value in te list tat appears most often median = middle value in te list median of {3, 9, 10, 27, 50} = 10 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. (Extend tis for tree or more: N 1 N 2 N 3...) Probability (Optional): probability = number of desired outcomes number of total outcomes Example: eac ACT mat multiple coice question as five possible answers, one of wic is te correct answer. If you guess te answer to a question completely at random, your probability of getting it rigt is 1/5 = 20%. Te probability of two different events A and B bot appening is P(A and B) = P(A) P(B), as long as te events are independent (not mutually exclusive). 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, Solving (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) x 2 + (b + a)x + ab = (x + a)(x + b) Reverse FOIL You can use Reverse FOIL to factor a polynomial by tinking about two numbers a and b wic add to te number in front of te x, and wic multiply to give te constant. For example, to factor x 2 + 5x + 6, te numbers add to 5 and multiply to 6, i.e., a = 2 and b = 3, so tat x 2 + 5x + 6 = (x + 2)(x + 3). To solve a quadratic suc as x 2 +bx+c = 0, first factor te left side to get (x+a)(x+b) = 0, ten set eac part in parenteses equal to zero. E.g., x 2 + 4x + 3 = (x + 3)(x + 1) = 0 so tat x = 3 or x = 1. To solve two linear equations in x and y: use te first equation to substitute for a variable 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. Solving two linear equations in x and y is geometrically te same as finding were two lines intersect. In te example above, te lines intersect at te point (1, 2). Two parallel lines will ave no solution, and two overlapping lines will ave an infinite number of solutions. Functions A function is a rule to go from one number (x) to anoter number (y), usually written y = f(x). Te set of possible values of x is called te domain of f(), and te corresponding set of possible values of y is called te range of f(). For any given value of x, tere can only be one corresponding value y. Absolute value: x = { +x, if x 0; x, if x < 0. pg. 3
4 Logaritms (Optional): Logaritms are basically te inverse functions of exponentials. Te function log b x answers te question: b to wat power gives x? Here, b is called te logaritmic base. So, if y = log b x, ten te logaritm function gives te number y suc tat b y = x. For example, log 3 27 = log3 33 = log 3 3 3/2 = 3/2 = 1.5. Similarly, log b b n = n. A useful rule to know is: log b xy = log b x + log b y. Complex Numbers A complex number is of te form a + bi were i 2 = 1. Wen multiplying complex numbers, treat i just like any oter variable (letter), except remember to replace powers of i wit 1 or 1 as follows (te pattern repeats after te first four): i 0 = 1 i 1 = i i 2 = 1 i 3 = i i 4 = 1 i 5 = i i 6 = 1 i 7 = i For example, using FOIL and i 2 = 1: (1 + 3i)(5 2i) = 5 2i + 15i 6i 2 = i. Lines (Linear Functions) Consider te line tat goes troug points A(x 1, y 1 ) and B(x 2, y 2 ). Distance from A to B: Midpoint of te segment AB: Slope of te line: (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 Pointslope form: given te slope m and a point (x 1, y 1 ) on te line, te equation of te line is (y y 1 ) = m(x x 1 ). Slopeintercept form: given te slope m and te yintercept b, ten te equation of te line is y = mx + b. To find te equation of te line given two points A(x 1, y 1 ) and B(x 2, y 2 ), calculate te slope m = (y 2 y 1 )/(x 2 x 1 ) and use te pointslope form. Parallel lines ave equal slopes. Perpendicular lines (i.e., tose tat make a 90 angle were tey intersect) ave negative reciprocal slopes: m 1 m 2 = 1. pg. 4
5 a a b b a b b a a b m l b a Intersecting Lines Parallel Lines (l m) Intersecting lines: opposite angles are equal. Also, eac pair of angles along te same line add to 180. In te figure above, a + b = 180. Parallel lines: eigt angles are formed wen a line crosses two parallel lines. Te four big angles (a) are equal, and te four small angles (b) are equal. Triangles Rigt triangles: c a b 30 2x x 3 60 x x 2 45 x 45 x a 2 + b 2 = c 2 Special Rigt Triangles A good example of a rigt triangle is one wit a = 3, b = 4, and c = 5, also called a rigt triangle. Note tat multiples of tese numbers are also rigt triangles. For example, if you multiply tese numbers by 2, you get a = 6, b = 8, and c = 10 (6 8 10), wic is also a rigt triangle. All triangles: b Area = 1 2 b pg. 5
6 Angles on te inside of any triangle add up to 180. Te lengt of one side of any triangle is always less tan te sum and more tan te difference of te lengts of te oter two sides. An exterior angle of any triangle is equal to te sum of te two remote interior angles. Oter important triangles: Equilateral: Tese triangles ave tree equal sides, and all tree angles are 60. Isosceles: Similar: An isosceles triangle as two equal sides. Te base angles (te ones opposite te two sides) are equal (see te 45 triangle above). Two or more triangles are similar if tey ave te same sape. Te corresponding angles are equal, and te corresponding sides are in proportion. For example, te triangle and te triangle from before are similar since teir sides are in a ratio of 2 to 1. Trigonometry Referring to te figure below, tere are tree important functions wic are defined for angles in a rigt triangle: ypotenuse θ adjacent opposite sin θ = opposite ypotenuse SOH cos θ = adjacent ypotenuse CAH tanθ = opposite adjacent TOA (te last line above sows a mnemonic to remember tese functions: SOHCAHTOA ) Optional: A useful relationsip to remember wic works for any angle θ is: sin 2 θ + cos 2 θ = 1. For example, if θ = 30, ten (refer to te Special Rigt Triangles figure) we ave sin 30 = 1/2, cos 30 = 3/2, so tat sin cos 2 30 = 1/4 + 3/4 = 1. pg. 6
7 Circles (, k) r r n Arc Sector Area = πr 2 Circumference = 2πr Full circle = 360 Lengt Of Arc = (n /360 ) 2πr Area Of Sector = (n /360 ) πr 2 Equation of te circle (above left figure): (x ) 2 + (y k) 2 = r 2. Anoter way to measure angles is wit radians. Tese are defined suc tat π radians is equal to 180, so tat te number of radians in a circle is 2π (or 360 ). To convert from degrees to radians, just multiply by π/180. For example, te number of radians in 45 is 0.785, since 45 π/180 = π/4 rad rad. Rectangles And Friends Rectangles and Parallelograms: Trapezoids (Optional): l w Rectangle Parallelogram (Square if l = w) (Rombus if l = w) Area = lw Area = l l w base 2 base 1 ( ) base1 + base 2 Area of trapezoid = 2 pg. 7
8 Solids (Optional) r l w Rectangular Solid Volume = lw Rigt Cylinder Volume = πr 2 pg. 8
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