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GRE Quantitative Reasoning Prep
Original even + even = even even - even = even even + odd = odd even - odd = odd odd + odd = even odd - odd = even odd × odd = odd even × odd = even even × even = even 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.
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. prime number an integer greater than 1 that has only two positive divisors: 1 and itself first ten prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 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 composite number An integer greater than 1 that is not a prime number The first ten composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18 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 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/ To multiply two fractions multiply the two numerators and multiply the two denominators. So: (10/7) (- 1/3) = (10)(-1) / (7)(3) = -10/ To divide one fraction by another
Multiply by Real 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 percentage part / whole (100) = % 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 percent change formula difference / original (100) = % increase cumulative percent change Must calculate each successive percent change by using the result of the previous change as the new original Order of operations BEDMAS (brackets, exponents, division / multiplication, addition / subtraction) x^1 = x x^0 = 1 x^-1 = 1/x x^m x^n = xm+n 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^-n = 1/x^n (x^a)(y^a) = xy^a identity A statement of equality between two algebraic expressions that is true for all possible values of the variables involved (a + b)^2 = a^2 + 2ab + b^ (a - b)^ a^3 - 3a^2b + 3ab^2 - b^ a^2 - b^2 = (a + b) (a - b) x^30 - x^29 = x(x^29) - x^ 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 quadratic equation
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 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 slope (m) rise/run, y2-y1/x2-x 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. 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. 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. 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. graph of a circle (x - a)^2 + (y - b)^2 = r^2 (centre is at point a, b and radius of r) 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. weighted average example: 2 (x) + 1 (y) / 2 + 1 = a (where 2 and 1 represent the ratio of each entity) 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. Sum of the measures of the interior angles of a triangle 180 degrees sum of the measures of the interior angles of an n-sided polygon (n - 2)(180 degrees) 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. 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. right triangle
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. 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 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) central angle A central angle of a circle is an angle with its vertex at the center of the circle. area of a circle A=∏r² sector A sector of a circle is a region bounded by an arc of the circle and two radii area of a sector A = ∏r² (c/360), where c = the central angle) 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. volume of rectangular solid V = lwh surface area of rectangular solid
A = 2(lw + lh + wh) -- the sum of the areas of the six faces 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) 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. volume of a right circular cylinder V = (pi)r^2h surface area of a right circular cylinder A = 2(Πr^2) + 2Πrh 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. 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
