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Magoosh Math GRE Formulas
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Gravity
Terms in this set (49)
Prime Numbers Below 60
2, 3 , 5 , 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
Divisibility by 3
Sum of digits divisible by 3
Divisibility by 4
The last two digits of number are divisible by 4
Divisibility by 5
The last digit is either a 5 or a zero
Divisibility by 6
Even number and sum of digits is divisible by 3
Divisibility by 8
If the last three digits are divisible by 8
Divisibility by 9
Sum of digits is divisible by 9
Fast Fractions
1/x + 1/y = (x+y)/xy
Percent Change
Change/Original value (i.e. (52-40)/40)
Ratios
2:5 boys to girls and 2:7 boys to total students
Exponent Laws
x^A * x^B =x^ (A+B)
(x^A)/(x^B) = x^(A-B)
(X^A)^B = x(A*B)
Exponent Laws - 1 and 0 as bases
1 raised to any power is 1.
0 raised to any nonzero power is 0.
Any nonxero number to the power of 0 is 1 (i.e. 7^0 =1)
Exponent Laws-Negative exponents
x^(-1) = 1/x
x^(-2) = 1/(x^2)
Exponent Laws-Negative bases
A negative number raised to an even power is positive; a negative number raised to an odd power is negative
Exponent Laws-Odd/even exponents
x^3=8 therefore x =2 but x^4-16 therefore x=2 and x=-2
Roots-Perfect squares
Numbers with integers as their square roots: 4, 9, 16, etc
To estimate square roots of numbers that aren't perfect squares, examine the nearby perfect squares. For example, to find square root of 50, you know that the square root of 49 is 7 and the square root of 64 is 8 so the square root of 50 must be between 7 and 8
Simplifying roots
Separate the number into its prime factors and take out matching pairs: square root of 20 = square root of 2 x 2 x 5= 2* square root 5
Adding roots
If the radical is the same in each term, combine the term.
sqrt(3) + 4 sqrt(3) =
1 sqrt(3) + 4 sqrt(3) =
(1 + 4) sqrt(3) =
5 sqrt(3)
Algebra-Simplifying Expressions
(6xy-5x)-(4xy-3y)= 2xy+5x+3y
Algebra-Multiplying Monomials
(5y^3)(6y^2)=30y^5
Add exponents when multiplying
Factoring-Quadratic Polynomials
x^2+ax+b= (x+m)(x+n) where a is the sum of m and n and b is their product (i.e. x^2+5x-14=(x+7)(x-2)
Solving Equations-Eliminating Fractions
(a/b)(b/a)=1
2/5x=8
5/2
2/5 x = 5/2
8
x=20
Quadratic Formula
Inequality
Like regular equations with the following exception:
Multiplying or dividing an inequality by a negative number reverses the sign of the inequality.
If w<x and x<y, then w<y
If a<b and c<d, then a+c<b+d (not true for subtracting, multiplying, or dividing
|x| <3 then ,3<x<3; if |x| > 3, then x>3 or x > -3
Geometry-Angles
A right angle is made up of 90 degrees
A straight line is made up of 180 degrees
If two lines intersect, the sum of the resulting for angles is 360 degrees
Geometry-Polygons
Total degrees=180(n-2) where n=# sides
Average degrees per side or degree measure of congruent polygon = 180* (n-2)/n
Geometry-Triangles
Area= 1/2 b*h
An isosceles right triangle (45-45-90) has sides in a ratio of x :x :x square root (2)
A 30-60-90 triangle has sides in a ratio of x:x square root(3): 2x, with the 1x side opposite the 30 degree angle
An equilateral triangle has three equal sides. Each angle is equal to 60 degrees.
Any given angle of a triangle corresponds to the length of the opposite side. The larger the degree measure of the angle, the larger the length of the opposite side.
A right triangle has a right angle (a 90 degree angle); the side opposite the right angle is called the hypotenuse and is always the longest side.
For a right triangle with legs A and B and hypotenuse C: A^2+B^2=C^2. This is called the Pythagorean Theorem.
Memorize Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples are also Pythagorean Triples
The length of the longest side can never be greater than the sum of the two other sides
The length of the shortest side can never be less than the positive difference of the two other sides
Pythagorean Triples
3-4-5, 5-12-13, 8-15-17, 7-24-25
Each side of certain right triangles are integers. These sets of numbers are called Pythagorean triples, and
you should memorize some of them: 3-4-5, 5-12-13, 8-15-17, 7-24-25. A multiple of a Pythagorean triple is
a Pythagorean triple (e.g., 6-8-10).
Geometry: Circles
Area=pi*r^2
Circumference=2
pi
r
A circle has 360 degrees. An arc is the portion of the circumference of a circle in x degrees of the circle.
Arc Length= (x/360)2
pi
r
Area of sector= (x/360)pi*r^2
A fraction of the circumference of a circle is called an arc. To find the degree measure of an arc, look at the
central angle.
A chord is a line segment between two points on a circle. A chord that passes through the middle of the
circle is a diameter.
If two inscribed angles hold the same chord, the two inscribed angles are equal.
An inscribed angle holding the diameter is a right angle (90 degrees).
Geometry-Squares
Perimeter= 4 s, where s=side
Area= s^2
Geometry-Rectangles
Area= lx w, where l=length and w=width
Perimeter=2l+2w
Geometry-Trapezoids
Area= (Base1+Base2)/2 *height
Geometry-Quadrilaterals
The area of a square is s^2 (s = side).
The diagonals of a square bisect one another, forming four 90 degree angles
The diagonals of a rhombus bisect one another, forming four 90 degree angles
The perimeter of a rectangle is twice its height plus twice its length (or, the sum of all its sides).
The area of a parallelogram can be found multiplying base x height (the base always forms a right angle with
the height).
Geometry-Cubes
Volume=s^3
Surface Area=6s^2
The volume of a cube and the surface area of a cube are equal when s = 6
Geometry-Rectangular Solids (including cubes)
V=height x depth x width
Surface Area=2
height
width +2
width
depth+2
depth
height
Geometry-Cylinders
Volume=r^2pi*h
Surface Area=2*pi*r^2 + 2*pi*r*h = 2
pi
pi
ace Area=2*pi*r^2 + 2*pi*r
h = 2
pi*r (r+h)
Coordinate Geometry-Lines
Any line can be represented by y=mx+b, where m is the slope and b is the y-intercept. This is called slope-intercept form.
The slope of a line can be found subtracting the y values of a pair of coordinates and dividing it by the difference in the x values: slope=m= (y2-y1)/(x2-x1)
To find the y-intercept plug in zero for x and solve for y
To find the x-intercept, plug in zero for y and solve for x
An equation like x = 3 is a vertical line at x = 3; an equation like y = 4 is a horizontal line at y = 4.
If given two points and asked to find the equation of a line that passes through them, first find the slope
using the above formula, then plug one of the points into y = mx+b and solve for b.
The slopes of two lines which are perpendicular to each other are in the ratio of x : -1/x, where x is the slope of one of the lines (think: negative reciprocal)
The Distance Formula
square root ([x2-x1]^2 + (y2-y1]^2)
For finding the distance between (x1, y1) and (x2, y2)
Quadratics
This is the format of a quadratic equation: y=ax^2 +bx +c
The graph of a quadratic equation is a symmetrical shape called a parabola, which open upwards if a > 0 and
down if a < 0.
Word Problems-Distance, Rate, and Time
D=R*T
R=D/T
T=D/R
Average Speed= Total distance traveled/total time
Word Problems-Work Rate
1/total work = (1/work rate 1) + (1/work rate 2)
output=rate*time
Word Problem-Sequences
1+2+3+...+n = [n(n+1)/(2)]
Word Problems-Interest
Simple Interest: V=P(1+(rt/100)), where P is principal, r is rate, and t is time
Compound Interest: V=P(1+ (r/100n)^nt) , where n is the number of times compounded per year
Statistics-Mean, Median, Mode
Average or mean:
For a set of n numbers: total sum / n
Median:
Middlemost value when numbers are arranged in ascending order; for an even amount of numbers,take the average of the middle two
Mode:
The number that occurs most frequently
Example:
2, 3, 3, 4, 5, 6, 6, 6, 7: Mean = 42/9, Median = 5, Mode = 6
If the numbers in a set are evenly spaced, then the mean and median of the set are equal: {30, 35,
40, 45, 50, 55}
Stats-Weighted Average
(proportion) x (group A average) + (proportion) x (group B average) + ...
Stats-Range
Greatest value - least value
Stats-Standard Deviation
If you're given a set of n numbers a, b, c, ... with a mean m:
SD=
The standard deviation represents the average distance the data values are away from the mean.
Variance is the value inside the square root of the standard deviation =SD^2
If the standard deviation of a set of numbers is k, then k = 1 unit of standard deviation.
1 SD=68
2SD=95
3SD= 99.7
Counting
Fundamental Counting Principle: If a task is comprised of stages, where...
One stage can be accomplished in A ways
Another can be accomplished in B ways
Another can be accomplished in C ways
...and so on, then the total number of ways to accomplish the task is A x B x C x ...
When tackling a counting problem:
Identify/list possible outcomes
Determine whether the task can be broken into stages
Determine the number of ways to accomplish each stage, beginning with the most restrictive stage(s)
Apply the Fundamental Counting Principle
Factorial notation: n!= n x (n-1) x (n-2) x...x3x2x1
n unique objects can be arranged in n! ways. Example: There are 9 unique letters in the word
wonderful, so we can arrange its letters in 9
8
7*... = 362,880 ways.
Restrictions: number of ways to follow a rule=number of ways to ignore the rule-number of ways to break the rule
Arranging objects when some are alike: n!/(A!) (B!) (C!)...
Given n objects where A are alike, another B are alike, another C are alike and so on.
Combinations:
nCr=n!/(r!(n-r)!
When the order does not matter - for example, picking any 3 friends from a group of 5.
Permutations: nPr= n!/(n-r)!
When the order does matter - for example, how many ways you could order 3 letters from the word
PARTY?
Probability
The probability of an event:
0 = the event definitely won't occur
1 = the event definitely will occur
0.5 = there is a 50/50 chance the event will occur
Probability that event A will happen:
The complement of an event:
The chance the event doesn't occur--so the complement of drawing a green ball is drawing a ball
that isn't green.
P(event happens) + P(event does not happen) = 1
Mutually exclusive events:
Two events are mutually exclusive if they can't happen together: P (A and B) = 0
Events A and B (if they are independent events):
P(A and B) = P(A) x P(B)
Events A or B:
A happens, B happens, or both A and B happen.
P (A or B) = P(A) + P(B) - P(A and B)
Events A and B (if A and B are dependent events):
P(A and B) = P(A) x P(B|A)
P(B|A) is the probability that B occurs given that A occurs (example: the probability of drawing a
heart, assuming you already drew a spade).
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