Download GRE MATH FORMULAS REVIEW and more Exams Advanced Education in PDF only on Docsity!
GRE MATH FORMULAS
Square: perimeter and area
Square with side length "s"
Perimeter: P = 4s
Area: A = s^
Diagonal = s√
Rectangle: perimeter and area
P=2l+2w
A=lw (A=bh)
Circle: circumference and area
C=2πr or C=πd
A=π(r^2)
Triangles: Pythagorean Theorem & Pythagorean Triplets
a2+b2=c
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)
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]
Triangle: Area [General], Area [Equilateral]
General - A=1/2bh
Equilateral - where a = side length
A = a^2(√3) / 4
Trapezoid: Area
A=1/2h(b1+b2)
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
Circles: Arc Length & Area of Sector
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:
of ways to arrange 26 distinct letters of the alphabet = 26!
of ways to arrange 26 letters of the alphabet if any number of repetitions allowed
(could be 26 As) = n^r = 26^
of ways to arrange 26 distinct letters of the alphabet if Z was replaced with another
A (so two As) = 26!/2!
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: 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)
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
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/
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.
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
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
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^
Coordinate Geometry: Slope: positive vs negative, steep vs gentle
Positive slope = m>
Negative slope = m<
Steep slope = m>
Gentle slope = 0 < m < 1
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 ]
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 line y = -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 ).
sum of squares
square of a difference
difference of 2 squares
(a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^ 2 [no way to factor this]
(a-b)^2 = (a-b)(a-b) = a^2 - 2ab + b^ 2
a^2 - b^2 = (a+b)(a-b)
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
Quadrilateral Properties: 1) Parallelograms, 2) Rhombuses, 3) Rectangles, 4) Squares
- a. opposite sides are parallel, b. opposite sides are equal, c. opposite angles are
equal, d. diagonals bisect each other
Avg velocity = total D / total T
Work Equation
A = R x T
R = A/T
T = A/R
Combined work = sum of individual work rates
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)
English to Math:
equals, is, was, will be, has, costs, adds up to, the same as, as much as
English to Math:
times, of, product of, by
x
English to Math:
per, out of, each, ratio
÷
English to Math:
plus, and, sum, combined
English to Math:
less than, difference between, fewer, decreased by
English to Math:
a number, how much, how many, what
x, n, etc
Absolute Value of |x| and |-x|
|x| = x if x > 0
|x| = -x if x < 0
|x| = 0 if x=
|-x| = -x if x>0 --> -x > 0
|-x| = -(-x) if x<0 --> -x < 0
|-x| = 0 if x=
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]
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)
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
Negative Fractions
For all integers c and d:
-c/d , c/-d , - (c/d) ARE EQUIVALENT
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
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>
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
Number Properties: 0 and 1
- is an integer
- neither positive or negative
- is even
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
Factorization of Large Numbers [Finding total number of factors]
- Find the prime factorization of a number (each one of the number's prime factors
raised to the appropriate power).
List all of the exponents.
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.
Overlapping Sets: 2 Sets + Given 4/5 elements of formula
use formula:
total = group A + group B - both + neither
Overlapping Sets: 2 Sets + NOT Given 4/5 elements of formula
Use table
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
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
Units Digit Questions
- units digit of any product will be influenced ONLY by the units digit of the factor
ex: 57^
- 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
- figure out where the pattern will be at your desired power (determine period of
power): 120= multiple of 4 so 7^120 = ___
- extend the period of the pattern --> 7^121=_7 .... 7^123=
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√
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.
Mixed Number to Improper Fractions
numerator = denominator x integer + original numerator
denominator remains the same
Ex: 2 4/
7 x 2 + 4 = 18
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%
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.
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^
--> since both terms are powers of two, they share several common factors --> GCF of
both of these terms is 2^
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)
Probability: Sets vs Lists
Sets - order does NOT matter, can't have repeats
Lists - order DOES matter, allows repeats
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]
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
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
Geometry: Complementary vs Supplementary angles
Complementary = the sum of their measures is 90 degrees
Supplementary = the sum of their measures is 180 degrees
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
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
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
