GRE Quantitative Reasoning Prep

a^2 - b^2 =

(a + b) (a - b)

sum of the measures of the interior angles of an n-sided polygon

(n - 2)(180 degrees)

graph of a circle

(x - a)^2 + (y - b)^2 = r^2 (centre is at point a, b and radius of r)

x^0 =

1

x^-1 =

1/x

x^-n =

1/x^n

Sum of the measures of the interior angles of a triangle

180 degrees

first ten prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, and 29

The first ten composite numbers

4, 6, 8, 9, 10, 12, 14, 15, 16, and 18

Inequality

< > ≤ ≥

surface area of rectangular solid

A = 2(lw + lh + wh) -- the sum of the areas of the six faces

surface area of a right circular cylinder

A = 2(Πr^2) + 2Πrh

area of a quadrilateral

A = bh (or lw): the base times height or length times width

area of a sector

A = ∏r² (c/360), where c = the central angle)

central angle

A central angle of a circle is an angle with its vertex at the center of the circle.

circular cylinder

A circular cylinder consists of two bases that are congruent circles and a lateral surface made of all line segments that join points on the two circles and that are parallel to the line segment joining the centers of the two circles. The latter line segment is called the axis of the cylinder. A right circular cylinder is a circular cylinder whose axis is perpendicular to its bases.

linear equation

A linear equation is an equation involving one or more variables in which each term in the equation is either a constant term or a variable multiplied by a coefficient. None of the variables are multiplied together or raised to a power greater than 1

proportion

A proportion is an equation relating two ratios; for example, 9 / `2 = 3 / 4. To solve a problem involving ratios, you can often write a proportion and solve it by cross multiplication

parallelogram

A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In a parallelogram, opposite sides are congruent and opposite angles are congruent

rectangle / square

A quadrilateral with four right angles is called a rectangle. Opposite sides of a rectangle are parallel and congruent, and the two diagonals are also congruent. A rectangle with four congruent sides is called a square.

rectangular solid

A rectangular solid has six rectangular surfaces called faces, as shown in the figure below. Adjacent faces are perpendicular to each other. Each line segment that is the intersection of two faces is called an edge, and each point at which the edges intersect is called a vertex. There are 12 edges and 8 vertices. The dimensions of a rectangular solid are the length l, the width w, and the height h.

sector

A sector of a circle is a region bounded by an arc of the circle and two radii

identity

A statement of equality between two algebraic expressions that is true for all possible values of the variables involved

right triangle

A triangle with an interior right angle is called a right triangle. The side opposite the right angle is called the hypotenuse; the other two sides are called legs.

isosceles triangle

A triangle with at least two congruent sides is called an isosceles triangle. If a triangle has two congruent sides, then the angles opposite the two sides are congruent. The converse is also true.

equilateral triangle

A triangle with three congruent sides is called an equilateral triangle. The measures of the three interior angles of such a triangle are also equal, and each measure is 60 degrees.

area of a triangle

A=½bh or bh/2

area of a circle

A=∏r²

length of diagonal in rectangular prism

A^2+B^2+C^2 = D^2 or L^2+W^2+H^2 = D^2 (A is not area, just a side length)

function

An algebraic expression in one variable can be used to define a function of that variable. Usually denoted by letters such as f, g, and h. For example, the algebraic expression 3x+5 can be used to define a function f by: f(x) = 3x+5

length of an arc

An arc is a piece of the circumference. If n is the degree measure of the arc's central angle, then the formula is: Length of an Arc = 1 (n/360) (2πr)

quadratic equation

An equation that can be written in the form ax^2 + bx + c = 0, where a,b,and c are real numbers and a ≠ 0

composite number

An integer greater than 1 that is not a prime number

Order of operations

BEDMAS (brackets, exponents, division / multiplication, addition / subtraction)

Opposite/vertical angles

Created when two lines intersect at a point. Opposite angles have equal measures, and angles that have equal measures are called congruent angles. Hence, opposite angles are congruent. The sum of the measures of the four angles is 360.

graph of an equation

Equations in two variables can be represented as graphs in the coordinate plane. In the xy-plane, the graph of an equation in the variables x and y is the set of all points whose ordered pairs (, xy satisfy the equation.

prime factorization

Every integer greater than 1 either is a prime number or can be uniquely expressed as a product of factors that are prime numbers, or prime divisors

arc

Given any two points on the outside edge of a circle, an arc is the part of the circumference containing the two points and all the points between them. Two points on a circle are always the endpoints of two arcs. It is customary to identify an arc by three points to avoid ambiguity.

Graphing linear inequalities

Graphs of linear equations can be used to illustrate solutions of systems of linear equations and inequalities. Solve each equation for y in terms of x, then graph each. The solution of the system of equations is the point at which the two graphs intersect.

percent change

If a quantity increases from 600 to 750, then the percent increase is found by dividing the amount of increase, 150, by the base, 600, which is the initial number given

compound interest (compounded more than once annually)

If the amount P is invested at an annual interest rate of r percent, compounded n times per year, then the value V of the investment at the end of t years is given by the formula v = p (1 + r/100n)^nt

compound interest

In the case of compound interest, interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly. Each time interest is added to the principal, the interest is said to be compounded. After each compounding, interest is earned on the new principal, which is the sum of the preceding principal and the interest just added. If the amount P is invested at an annual interest rate of r percent, compounded annually, then the value V of the investment at the end of t years is given by the formula v = p (1 + r/100)^t

FOIL

Multiply the First, Outer, Inner, and Last terms of a pair of binomials

cumulative percent change

Must calculate each successive percent change by using the result of the previous change as the new original

Simple interest

Simple interest is based only on the initial deposit, which serves as the amount on which interest is computed, called the principal, for the entire time period. If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formula v = p (1 + rt / 100) (v and p in dollars)

circumference

The distance around a circle. C = 2(pi)r

frequency/count

The frequency, or count, of a particular category or numerical value is the number of times that the category or value appears in the data. A frequency distribution is a table or graph that presents the categories or numerical values along with their associated frequencies.

Graph of a quadratic equation

The graph of a quadratic equation of the form y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0 is a parabola

parabola

The graph of a quadratic equation of the form y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0 is a parabola The x-intercepts of the parabola are the solutions of the equation ax^2 + bx + c = 0. If a is positive, the parabola opens upward and the vertex is its lowest point. If a is negative, the parabola opens downward and the vertex is the highest point. Every parabola is symmetric with itself about the vertical line that passes through its vertex. In particular, the two x-intercepts are equidistant from this line of symmetry.

measure of an arc

The measure of an arc is the measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc. An entire circle is considered to be an arc with measure 360 degrees

ratio

The ratio of one quantity to another is a way to express their relative sizes, often in the form of a fraction, where the first quantity is the numerator and the second quantity is the denominator. Thus, if s and t are positive quantities, then the ratio of s to t can be written as the fraction .st The notation "s to t" or "s : t" is also used to express this ratio. For example, if there are 2 apples and 3 oranges in a basket, we can say that the ratio of the number of apples to the number of oranges is 2/3 or that it is 2 to 3 or that it is 2:3.

relative frequency

The relative frequency of a category or a numerical value is the associated frequency divided by the total number of data. Relative frequencies may be expressed in terms of percents, fractions, or decimals. A relative frequency distribution is a table or graph that presents the relative frequencies of the categories or numerical values

interval

The set of all real numbers that are between, say, 5 and 8 is called an interval, and the double inequality is often used to represent that interval: 5 < x < 8

add two fractions with different denominators

To add two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator. So: 1/3 + -2/5 = 5/15 + -6/15 = -1/15

graphing a function in the xy-plane

To graph a function in the xy-plane, you represent each input x and its corresponding output (f)x as a point (x, y) where y = f(x). In other words, you use the x-axis for the input and the y-axis for the output.

volume of a right circular cylinder

V = (pi)r^2h

volume of rectangular solid

V = lwh

Adding a positive or negative constant to both sides of inequality

When the same constant is added to or subtracted from both sides of an inequality, the direction of the inequality is preserved and the new inequality is equivalent to the original. When both sides of the inequality are multiplied or divided by the same nonzero constant, the direction of the inequality is preserved if the constant is positive but the direction is reversed if the constant is negative. In either case, the new inequality is equivalent to the original.

Ratio Box

X item Y item Total Ratio Multiply by Real

(√a)^2

a

√a^2

a

(a + b)^2 =

a^2 + 2ab + b^2

Pythagorean theorem

a^2 + b^2 = c^2

(a - b)^3

a^3 - 3a^2b + 3ab^2 - b^3

add two fractions with the same denominator

add the numerators and keep the same denominator. For example, - 8 / 11 + 5 / 11 = -8 + 5 / 11 = -3 / 11

prime number

an integer greater than 1 that has only two positive divisors: 1 and itself

percent change formula

difference / original (100) = % increase

even + even =

even

even - even =

even

even × even =

even

even × odd =

even

odd + odd =

even

odd - odd =

even

weighted average

example: 2 (x) + 1 (y) / 2 + 1 = a (where 2 and 1 represent the ratio of each entity)

To divide one fraction by another

first invert the second fraction—that is, find its reciprocal—then multiply the first fraction by the inverted fraction. So (3/10)/(7/13) = (3/10)(13/7) = 39/70

Area of a trapezoid

half the product of the sum of the lengths of the two parallel sides b1 and b2 and the corresponding height h: a = 1/2 (b1 + b2)(h)

To multiply two fractions

multiply the two numerators and multiply the two denominators. So: (10/7) (-1/3) = (10)(-1) / (7)(3) = -10/21

negative number raised to odd power =

negative

even + odd =

odd

even - odd =

odd

odd × odd =

odd

percentage

part / whole (100) = %

negative number raised to even power =

positive

slope (m)

rise/run, y2-y1/x2-x1

greatest common divisor (or greatest common factor)

the greatest positive integer that is a divisor of both a and b. For example, the greatest common divisor of 30 and 75 is 15. This is because the positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, and the positive divisors of 75 are 1, 3, 5, 15, 25, and 75. Thus, the common positive divisors of 30 and 75 are 1, 3, 5, and 15, and the greatest of these is 15.

least common multiple

the least positive integer that is a multiple of both a and b. For example, the least common multiple of 30 and 75 is 150. This is because the positive multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, etc., and the positive multiples of 75 are 75, 150, 225, 300, 375, 450, etc. Thus, the common positive multiples of 30 and 75 are 150, 300, 450, etc., and the least of these is 150.

radius

the length of a line segment between the center and circumference of a circle or sphere (r)

diameter

the length of a straight line passing through the center of a circle and connecting two points on the circumference (d)

x^1 =

x

quadratic formula

x = -b ± √(b² - 4ac)/2a Use this to determine the value of variables in quadratic equations. Quadratic equations have at most two real solutions

x^30 - x^29 =

x(x^29) - x^29

x^m/x^n =

x^m-n (also = 1 / x^m-n)

(x^m)^n =

x^mn

(xy)^n =

x^n y^n

(x/y)^n =

x^n/y^n

x^m x^n =

xm+n

(x^a)(y^a) =

xy^a

equation of a line

y = mx + b b is the y-intercept, y is the point on the y axis, x is the point on the x axis.

√a/√b

√ab

√a√b

√ab