GRE Math Formulas

English to Math: plus, and, sum, combined

+

Units Digit Questions

- units digit of any product will be influenced ONLY by the units digit of the factor ex: 57^123 1) look for repeating patterns: 7^1 = 7, 7^2 = 49, 7^3=__3, 7^4=__1, 7^5=__7 --> pattern is 7,9,3,1 and repeats every 4 2) figure out where the pattern will be at your desired power (determine period of power): 120= multiple of 4 so 7^120 = ___1 3) extend the period of the pattern --> 7^121=___7 .... 7^123=__3

Number Properties: 0 and 1

0 - is an integer - neither positive or negative - is even 1 - is odd

1) sum of squares 2) square of a difference 3) difference of 2 squares

1) (a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^ 2 [no way to factor this] 2) (a-b)^2 = (a-b)(a-b) = a^2 - 2ab + b^ 2 3) a^2 - b^2 = (a+b)(a-b)

Probability: "of x and y and z"

1) find the probability of one case/situation that satisfies the question 2) multiply that probability by the total number of possible cases that satisfy the question [i.e. different ways the items that satisfy the question can be arranged - make sure to account for repeats]

Exponential Growth: 2 Cases of Negative Bases Raised to Positive Powers

1) negative base <-1 = absolute value gets larger but +/- signs alternate [ex: (-3)^2 = 9, (-3)^3 = -27] 2) negative base between -1 & 0 = absolute value gets smaller, approaching 0, but +/- signs alternate [ex: (-1/2)^2 = 1/4, (-1/2)^3 = -1/8]

English to Math: equals, is, was, will be, has, costs, adds up to, the same as, as much as

=

Trapezoid: Area

A=1/2h(b1+b2)

Operations with Radicals: Addition, Subtraction, Multiplication, Division

Add & Subtract: - can only add/subtract roots when the parts inside the √ are identical - simplify and combine LIKE terms (think of like variables) Multiply & Divide: - treat whole numbers and Radicals separately.

Scaling Similar Shapes: Ratios of Area + Ratios of Volume

Area - if side length increases by k, area increased by k^2 Volume - if edge length increases by k, volume increases by k^3

Arithmetic vs Geometric Sequences

Arithmetic = each term is the SUM of the preceding term and a constant Geometric = each term is the PRODUCT of the preceding term and a constant

Arithmetic Sequences: definition + finding: average of all terms, # of terms in sequence, and sum of all terms

Arithmetic sequences = have a common difference (d) between items --> methods used to solve this type are applicable for questions asking for the sum of multiples of x from y to z inclusive or sum of all odd/even integers from x to y inclusive - average of all the terms = average the first and last terms (first + last / 2) - number of term = 1) subtract first from last, 2) divide by d, 3) add 1 - sum of all terms = multiple the average by # of terms (avg x # of terms)

Circle: circumference and area

C=2πr or C=πd A=π(r^2)

Motion Equation

D = R x T R = D/T T = D/R Avg velocity = total D / total T

Number Properties: E +/- E , O +/- O, E +/- O, E x E, O x O, E x O

E +/- E = E O +/- O = E E +/- O = O E x E = E O x O = O E x O = E

Pascal's Triangle

Each number in the triangle is the sum of the two numbers above it. For example, there's a 6 in the center of row 4: it's the sum of 3 and 3 in the row above. The very first and very last number in any row are always going to be 1. Pascal's Triangle can be used to find combinations. The top row in Pascal's Triangle is row zero, and the first item in any row (the 1s) are item zero in that row. For example, let's sat we wanted to find 6_C_4. Look in Row 6, at item number 4. the answer is 15.

Cube: Space Diagonal

Each side of the cube is x units long. Use 45-45-90 degree angle ratio ( 1 - 1 - √ 2 ) OR use the Pythagorean theorem twice to get the face diagonal. Diagonal between opposite vertices = s√3

Negative Fractions

For all integers c and d: -c/d , c/-d , - (c/d) ARE EQUIVALENT

Coordinate Geometry: Graphing functions - f(x) +/- d, -f(x), f(x +/- d), f(-x)

If original y = f(x) f(x) +/- d --> shift up/down by d -f(x) --> flipped over x axis f(x + d) --> shift left by d f(x - d) --> shift right by d f(-x) --> flipped over y axis

What must be an integer vs what can be a non integer

Must be an integer = multiple, factor, prime, even, odd Can be a non integer = positive, negative

Rectangle: perimeter and area

P=2l+2w A=lw (A=bh)

Coordinate Geometry: Slope: positive vs negative, steep vs gentle

Positive slope = m>0 Negative slope = m<0 Steep slope = m>1 Gentle slope = 0 < m < 1

Cylinder Inscribed in Cube / Cube Inscribed in Cylinder

The height of the cylinder = side of cube Radius of the cylinder = side of the cube/2

Overlapping Sets: 2 Sets + NOT Given 4/5 elements of formula

Use table

Cylinders: Volume, Surface Area, Outside surface area , Area of base

V = π(r^2)h Sa= 2π(r^2 )+ 2πrh = 2πr (r+h) Outside surface area = 2πrh Area of one base = π(r^2)

Rectangular Solids: Volume, Surface Area

V= l x w x h Sa= 2wh + 2lw + 2lh

Cube: Volume, Surface Area

V=s^3 Sa=6s^2

Mixed Number to Improper Fractions

numerator = denominator x integer + original numerator denominator remains the same Ex: 2 4/7 7 x 2 + 4 = 18 18/7

Special Right Triangles

refers to the right triangles: 45-45-90 (1:1:√2) [isosceles right triangle] and 30-60-90 (1:√3:2) [leg:leg:hyp] [1/2 of equilateral triangle]

English to Math: a number, how much, how many, what

x, n, etc

Coordinate Geometry: Quadratics

y = ax^2 + bx + c Graph of ^ where a, b, and c are constants = parabola x-intercepts = the solutions to the equation ax^2 + bx + c = 0 y-intercepts = solved by plugging 0 in for x term In the form of x^2 + ax + b can sometimes be solved via factoring, needs two numbers that add to a and multiply to b

English to Math: per, out of, each, ratio

÷

Overlapping Sets: 2 Sets + Given 4/5 elements of formula

use formula: total = group A + group B - both + neither

English to Math: times, of, product of, by

x

Coordinate Geometry: Slope-intercept form, Slope formula, How to find x and y intercepts, Midpoint formula

y=mx+b, where m is the slope and b is the y-intercept of the line. Slope of line passing through 2 points Q(x1,y1) and R (x2,yz) is y2-y1/x2-x1 - to find x-intercept, set y=0 - to find y-intercept (if you have slope), plug in coordinates for x and y in equation midpoint=(x1+x2/2,y1+y2/2)

Absolute Value of |x| and |-x|

|x| = x if x > 0 |x| = -x if x < 0 |x| = 0 if x=0 |-x| = -x if x>0 --> -x > 0 |-x| = -(-x) if x<0 --> -x < 0 |-x| = 0 if x=0

Quadrilateral Properties: 1) Parallelograms, 2) Rhombuses, 3) Rectangles, 4) Squares

1) a. opposite sides are parallel, b. opposite sides are equal, c. opposite angles are equal, d. diagonals bisect each other 2) are parallelograms so a,b,c,d apply + e. all 4 sides are equal, f. diagonals are perpendicular 3) are parallelograms so a,b,c,d apply + g. all angles = 90 degrees, h. diagonals are congurent 4) is a parallelogram, rhombus, and rectangle so all properties (a,b,c,d,e,f,g,h) apply

QC Strategies - Algebra: 1) if a variable has a unique value, 2) if a variable has a defined range, 3) if a variable has a relationship with another variable, 4) if a variable has no constraints, 5) if a variable has specific properties

1. if a variable has a unique value (e.g. x+3 =-5) THEN solve for value of the variable 2. if a variable has a defined range (e.g. -4 ≤ w ≤ 3) THEN test the boundaries 3. if a variable has a relationship with another variable (e.g. 2p=r) THEN simplify the equation and make a direct comparison of the variables 4. if a variable has no constraints THEN try to prove (D) 5. if a variable has specific properties (e.g. x is negative) THEN try to prove (D)

Compound Interest Formula

A = P(1 + r/n)^(n x t) r is the rate, n is the number of times compounded, t is time

Work Equation

A = R x T R = A/T T = A/R Combined work = sum of individual work rates

Operations with Inequalities: Add, Sub, Mult, Div, Squaring, Square Rooting

Adding two inequalities = signs must be in the same direction ex: a>b + c>d = a+c > b+d Subtracting two inequalities = signs must be in the opposite direction ex: a>b - d<c = a-d > b-c = big-small > small-big d<c = c>d = not changing the value of the inequality itself just rearranging so the sign is opposite of a>d Multiplying/Dividing = doesn't change sign but --> whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign. Squaring - CANNOT DO IT UNLESS WE KNOW THE SIGNS OF BOTH SIDES -if both sides neg = flip the sign - if both sides pos = dont flip sign - if one neg and one pos = CANNOT square Square Rooting - wont change the inequality only when a and b are greater than or equal to 0 - a ≤ b --> √a ≤ √b

Probability: Exactly X Events (Binomial)

Binomial probability refers to the probability of exactly x successes on n repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment). Method 1: If the probability of success on an individual trial is p, then the binomial probability is nCx⋅p^x⋅(1−p)^(n−x) Here nCx indicates the number of different combinations of x objects selected from a set of n objects. Note that if p is the probability of success of a single trial, then (1−p) is the probability of failure of a single trial. ** If the outcomes of the experiment are more than two, but can be broken into two probabilities p and q such that p+q=1 , the probability of an event can be expressed as binomial probability.** Method 2: 1-(p(both) + p(neither))

Geometry: Complementary vs Supplementary angles

Complementary = the sum of their measures is 90 degrees Supplementary = the sum of their measures is 180 degrees

Absolute Value Equations

Equations |x+3| = 12 --> 2 equations: |x+3| = 12 OR |x+3| = -12, Absolute value equations always have 2 solutions and cannot be negative (|x| = 5 is 5 or -5) --> same in absolute value inequalities: |x-4| < 3, x-4 = 3 -> x=7 -> x<7, x-4=-3 -> x=1 -> x>1 [must reverse sign]

Probability: A or B or both (A and B) happen

Events A or B: A happens, B happens, or both A and B happen. If NOT mutually exclusive: P (A or B) = P(A) + P(B) - P(A and B) If ARE mutually exclusive: P (A or B) = P(A) + P(B) **THIS IS THE SAME AS ASKING P(AT LEAST ONE OF A OR B OCCUR)** P(at least A or B occurs) = 1 - P(A and B DON'T OCCUR or (1-P(A)) x (1-P(B))) == P (A or B) = P(A) + P(B) - P(A and B) ** BUT if question asks for at least one but not both: 1- p(both or neither) = 1 - p(p(both) + p(neither)) THEN ITS AN 'EXACTLY ONE' PROBLEM**

Probability: At Least One

Find the probability that among several trials, we get at least one of some specified event. P(at least one) = 1 - P(no successes)

Triangle: Area [General], Area [Equilateral]

General - A=1/2bh Equilateral - where a = side length A = a^2(√3) / 4

Coordinate Geometry: Circles in the xy plane

General equation for a circle located at point (0,0) is x^2 + y^2 = r^2 where x and y are coordinates of points on the circle For circle not located at origin, shift center to any point (h,k) (x-h)^2 + (y-k)^2 = r^2

Finding the percent of a number

Given a number A. It is known that the number B is p% of the number A. Then the number B is equal to B=A⋅p/100

Median: if # of observations is odd formula vs if even

If the number of observations is odd, the number in the middle of the list is the median. This can be found by taking the value of the (n+1)/2 -th term, where n is the number of observations. Else, if the number of observations is even, then the median is the simple average of the middle two numbers

Statistics: Normal Distribution

In a normal distribution, the mean = median = 50th percentile (wherein 50% of values are above and 50% are below) In a bellcurve values tend to be bunched towards the middle so you would 'pass' more scores/values increasing score from mean of 72 to score of 78 than one would 'pass' if score was increased from 78 to 84

Absolute Value Inequalities

Inequalities = distance from x to positive p = |x-p|, distance from x to negative p = |x+p| To express |x-7| ≤ 3 as a regular inequality: 7-3=4, 7+3=10, 4 ≤ x ≤ 10 To express 5 < x < 17 as absolute value, 5+17/2 = 11 = midpoint -> |x-11| < 6 [ both 5 and 17 are 6 away from 11]

Multiple vs Factor / LCM vs GCF

Multiple: -product obtained after multiplying # by an integer - # reached by multiplying given # by another # - outcome = greater than or equal to given # - infinite --> LCM = least positive integer thats a multiple of all the integers [LCM of 36 and 90 is 180] Factor: - # that leaves behind no remainder after it divides into specific # - a # that can be multiplied to get another # - outcome - less than or equal to given # - finite --> GCF = greatest positive integer thats a factor of all integers [GCF of 36 and 90 is 18] **IF GCD OF POSITIVE INTEGERS A AND B IS H, AND IF LCM OF POSITIVE INTEGERS A AND B IS L, THEN H x L = A x B --> PRODUCT OF GCD AND LCM OF TWO NUMBERS IS = TO THE PRODUCT OF THE TWO NUMBERS**

Exponent Rules: Multiplying & Dividing with different bases AND different powers

Multiplying When you multiply expressions with different bases and different exponents, there is no rule to simplify the process. For example, suppose you want to multiply 2^3*5^2. You can see that 2^3 = 8 and 5^2 = 25. Thus 8*25 = 200. But, if you tried (2*5)^3+2, you would get 105, which is incorrect. Dividing The expression b^4 / a^2 is equivalent to (b * b * b * b) / (a * a). Nothing cancels here, but you can transform the expression by grouping by exponents. For example, b^4 / a^2 = (b / a)^2 * b^2, or (b^2 / a)^2.

Polygons: diagonals, triangles, sum of angles

N sided polygon has: n(n-3)/2 diagonals (n-2) triangles (n-2) x 180 sum of angles

Exponent Rules: b^-n, -b^-n, (p/q)^-n,

Negatives: b^-n = 1 / b^n -b^-n = -(1 / b)^n = 1/(-b)^n (p/q)^-n = (q/p)^n (-a)^12 = positive -(a^11) = negative -(-a^11) = positive = only time a negative makes positive product with exponents

Coordinate Geometry: Reflections over the x-axis, over the y-axis, over y = x, & over y = -x

Reflect a point across the x-axis = the x-coordinate remains the same, but the y-coordinate is transformed into its opposite (its sign is changed). The reflection of the point (x,y) across the x-axis is the point (x,-y) Reflect a point across the y-axis = the y-coordinate remains the same, but the x-coordinate is transformed into its opposite (its sign is changed). The reflection of the point (x,y) across the y-axis is the point (-x,y). Reflect a point across the line y = x - the x-coordinate and y-coordinate change places. If you reflect over the liney = -x, the x-coordinate and y-coordinate change places and are negated (the signs are changed). The reflection of the point (x,y) across the line y = x is the point (y, x) & reflection of the point (x,y) across the line y = -x is the point (-y, -x).

Coordinate Geometry: Distance Formula

To find the distance between two points (𝑥1,𝑦1) and (𝑥2,𝑦2) Distance =√‾((𝑥2−𝑥1)^2+(𝑦2−𝑦1)^2)

Rectangular Solid: Face diagonal, space diagonal

Use Pythagorean theorem OR 30-60-90 degree angle ratio(1 -√ 3 - 2) to figure out the face diagonal AND Pythagorean theorem twice to figure out the space diagonal.

Overlapping Sets: 3 Sets

Use venn diagram a = only group A, b = only group B, c = only group C, w = group A and group B, y = group A and group C, x = group B and group C, z = groups A, B, and C Group A = a + w + z + y Group B = b + w + z + x Group C = c + y + z + x

Coordinate Geometry: Intersection of 2 lines

When 2 lines intersect it means that at the point of intersection, both equations representing the lines are true. The pair of #s (x,y) that represents the point of intersection solve BOTH equations Finding this point = solving a system of two linear equations Lines that don't intersect = parallel

Percent change

(Original - new / original) x 100 Original = the percent after the "than" in a problem " the number of x sold by Store A is what percent less THAN the number of x sold by Store B?" (Store B - Store A / Store B) x 100

Circles: Arc Length & Area of Sector

*where x = central angle* arc length = x /360 * 2πr area of sector = x/360 * πr^2

English to Math: less than, difference between, fewer, decreased by

-

Finding LCM

- create factor trees for both #s - spot prime #s that are present in both factor trees - bring that/those #(s) down - cross out that pair - bring down any prime #s the trees don't share - multiply to get LCM

Exponent Rules: Addition and Subtraction with same base and different power, or diff base and same power

- there is no law of exponents for adding and subtracting powers --> there is no convenient way to combine a sum or difference of powers into a single power expression - in other words, 𝑎^𝑛 plus or minus 𝑎^𝑚 is not going to equal a-to-the-power-of-anything-in-particular. BUT we in some cases we can use 'factoring out' EX: simplify 2^17-2^13 --> since both terms are powers of two, they share several common factors --> GCF of both of these terms is 2^13 2^17=(2^4)×(2^13) 2^13=(1)×(2^13) Now, we can write the difference of powers as 2^17-2^13=(2^4)×(2^13)-(1)×(2^13)= (2^13)×[(2^4)-1] = 2^13 x [16-1] = 2^13 x 15 EX: 3^4 + 12^4 = 3^4 + (2x2x3)^4 = 3^4 (1+4^4)

Arithmetic Sequences: formula to find nth term, average of evenly spaced terms

- to find nth term = An = A1 + (n-1)d - The average of a set of evenly spaced numbers is equal to the median of that set

Factorization of Large Numbers [Finding total number of factors]

1) Find the prime factorization of a number (each one of the number's prime factors raised to the appropriate power). 2) List all of the exponents. 3) Add one to each of the exponents. (Remember, it's possible to raise the prime factor to the zero power.) 4)Multiply the resulting numbers.

Exponential Growth + Relation to √: 2 Cases of Positive Bases Raised to Positive Powers vs √

1) positive base b>1 - get larger quickly [ex: 7^2 = 49, 7^3 = 343, 7^4 = 2401] - √ makes it smaller [√b < b] - if b>1 and if n>m then 1 < n√b < m√b < b 2) positive base 0<b<1 - get smaller quickly [ex: (1/2)^2 = 1/4, (1/2)^3 = 1/8, (1/2)^4 = 1/16] - √ makes it bigger [√b>b] - if 0<b<1 and if n>m then 0 < b < m√b < n√b < 1

Statistics: Quartiles

3 #s that divide the list into 4 smaller equal lists Q1 = median of lower list = divides bottom 25% from rest Q2 = median = divides lower 50% from upper 50% Q3 = median of upper list = divides lower 75% from upper 25% IQR = Q3 - Q1 = size of middle 50%

Probability: A and B

Events A and B (if they are independent events): P(A and B) = P(A) x P(B) Events A and B (if A and B are dependent events): P(A and B) = P(A) x P(B|A)

Statistics: Measures of spread vs Measures of central tendency

Measures of spread: range, quartiles, SD = summarise the data in a way that shows how scattered the values are and how much they differ from the mean value. Central Tendency: mode, median, and mean = summarise the data into a single value that is typical or representative of all the values in the dataset, but this is only part of the 'picture' that summarises a dataset

Exponent Rules: a^n ⋅ a^m, a^n ⋅ b^n, a^n / a^m, a^n / b^n, (a^n)^m, m√(b^n), b^1/n

Product: a^n ⋅ a^m = a^n+m a^n ⋅ b^n = (a * b)^n Quotient: a^n / a^m = a^n-m a^n / b^n = (a / b)^n Powers: (a^n)^m = a^n*m m√(b^n) = b^n/m b^1/n = n√b

Coordinate Geometry: Quadrants

Quadrant I = x>0, y>0 Quadrant II = x<0, y>0 Quadrant III = x<0, y<0 Quadrant IV = x>0, y<0

Geometry: triangles inscribed in circles (right & equilateral), circle in square, square in a circle, quadrilateral in circle

Right Triangle inscribed in circle: if one side of triangle is a diameter of the circle, then the triangle is a right triangle and if triangle is a right triangle, then one of its sides is a diameter of the circle. Equilateral Triangle inscribed in circle: R = radius of circle, a = side of equilateral triangle (also side of square), R = a(√3/3) Circle inscribed in a square: d of circle = s of square Square inscribed in a circle: diagonal of square [s√2] = d of circle Quadrilateral inscribed in circle: opposite angles of quadrilateral must add up to 180 degrees

Probability: Sets vs Lists

Sets - order does NOT matter, can't have repeats Lists - order DOES matter, allows repeats

Square: perimeter and area

Square with side length "s" Perimeter: P = 4s Area: A = s^2 Diagonal = s√2

Tangent of a circle

Tangent = a line that is in the same plane as a circle and intersects the circle at exactly one point If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency. Line m is tangent and creates 90 degree angel at point of tangency

Triangles: Pythagorean Theorem & Pythagorean Triplets

a2+b2=c2 This theorem can only be used for right triangles (triangles with a 90-degree angle). a and b are the two shorter sides, or "legs," and c is the hypotenuse (the longest side of a right triangle). Certain triangle-side combinations (a:b:c), are called Pythagorean triples: (a-leg:b-leg:c-hyp) 3:4:5 5:12:13 8:15:17 7:24:25

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. If we know sides P & Q, p-q < 3rd side < p+q

Exponent Rules: Base^0, 0^n, Base^1, 1^n, (1/b)^n, b^m/n

Zero: b^0 = 1 0^n = 0 , for n>0 Ones: b^1 = b 1^n = 1 Fraction: (1/b) ^n = 1^n/b^n b^m/n = (b^m)^1/n = (b^1/n)^m

Rebuilding the dividend formula

dividend = (integer quotient)*(divisor) + remainder In remainder problems, remainder is always the same no matter what multiple is applied to the variable EX: what is the remainder of x when divided by 24 [x=6y+10, y=8z+4]? --> plug in 1 for z, y=12, x=82 --> 82/24 = 3 +10r --> 10 will always be the remainder when x/24

Geometry: inscribed shapes definition

drawing one shape inside another so that it just touches; just touching sides but never crossing; shape fits snugly inside other shape

Coordinate Geometry: Slope: horizontal vs vertical lines, parallel vs perpendicular

horizontal has slope = 0, so y=b (y-intercept) vertical slope = undefined since 'run' = 0, so x=a lines are parallel if they have same slope AND different y-intercepts [line y= 2x +4 is parallel to y= 2x -4 but NOT y = 2x +4] lines are perpendicular if their slopes are negative reciprocals [line y = -7x + 5 is perpendicular to y = 1/7 + 10 ]

Combinations Formula

nCr = n!/r!(n-r)! When order does NOT matter, we only care about the elements selected and not the order -> different orders of the same items = repetitions Can also think of it as starting with FCP and dividing off repetitions [ 10C4 = 10x9x8x7/4x3x2x1] - draw as many slots as there are items being selected - above the slots = count down the number of possible items to select - below the slots = count down the number of items being selected - multiply and reduce EX: My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad

Permutation Formula + 3 examples of Permutations

nPr = n!/(n-r)! when order DOES matter --> P(ermutation)=P(osition) Can also think of it as breaking selection into stages + using FCP = permutation of N different items = N! 3 Examples: 1) # of ways to arrange 26 distinct letters of the alphabet = 26! 2) # of ways to arrange 26 letters of the alphabet if any number of repetitions allowed (could be 26 As) = n^r = 26^26 3) # of ways to arrange 26 distinct letters of the alphabet if Z was replaced with another A (so two As) = 26!/2!