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GRE Quantitative Reasoning Prep
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
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
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 *** 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/ negative number raised to even power = *** positive
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)^3 *** 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 *** 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 quadratic formula *** x = -b ± √(b² - 4ac)/2a
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^
- 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 *** 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. Pythagorean theorem *** a^2 + b^2 = c^ area of a triangle *** A=½bh or bh/ 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
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 average (arithmetic mean) *** To calculate the average of n numbers, take the sum of the n numbers and divide it by n. weighted average/mean *** When several values are repeated in a list, it is helpful to think of the mean of the numbers as a weighted mean of only those values in the list that are different. The number of times a value appears in the list, or the frequency, is called the weight of that value. median *** To calculate the median of n numbers, first order the numbers from least to greatest. If n is odd, then the median is the middle number in the ordered list of numbers. If n is even, then there are two middle numbers, and the median is the average of these two numbers mode *** The mode of a list of numbers is the number that occurs most frequently in the list range *** The range of the numbers in a group of data is the difference between the greatest number G in the data and the least number L in the data; that is, G-L interquartile range *** The difference between the scores (or estimated scores) at the 75th percentile and the 25th percentile. Used more than the range because it eliminates extreme scores. Formula: IQR = Q3-Q standard deviation *** The standard deviation of a group of n numerical data is computed by (1) calculating the mean of the n values, (2) finding the difference between the mean and each of the n
number of combinations of n objects taken k at a time *** n! / k!(n-k)!, sometimes notated as nCk probability *** probability of event occurring is defined by the ratio P(E) = number of outcomes that satisfy event E / the number of possible outcomes probability of two or more events BOTH occurring *** P(A and B) = P(A) x P(B) probability of EITHER one or another event occurring *** P(A) + P(B) - P(AB) probability of neither of multiple events occurring *** the product of 1 - P(A), 1 - P(B), etc. group equation *** T = G1 + G2 - B + N (T is total, groups G, B is members of both group, N is members of neither) probability of event E AND F *** E x F (if E and F are independent) probability of event E OR F *** E + F (if E and F are mutually exclusive) probability of event E OR F but not both *** E + F - P(E and F) continuous probability distribution *** relative frequency distributions are often approximated using a smooth curve—a distribution curve or density curve—for the tops of the bars in the histogram. The region below such a curve represents a distribution, called a continuous probability distribution. There are many different continuous probability distributions, but the most important one is the normal distribution, which has a bell-shaped curve
length of a diagonal in a parallelogram *** p^2 + q^2 = 2((a^2) + (b^2)), where p and q are the diagonals and a and b are sides. You may need to construct a right triangle by connecting a top corner with the baseline and then finding its hypotenuse (which will serve as the length of the angled side). average of two averages *** find total amount for each average (a = total / number of items), then determine the new average, deriving your new total from the sum of these totals. harmonic mean formula *** n / ((1/a1)+(1/a2)+(1/an)) Formula for "n percent greater/less than x" *** x ± (n/100)x x² - y² *** (x + y) (x - y) x² + 2xy + y² *** (x + y) (x + y) or (x + y)² x² - 2xy + y² *** (x - y) (x - y) or (x - y)² (x + y) / xy *** 1/x + 1/y if x, y ≠ 0 (x - y) / xy *** 1/x - 1/y if x, y ≠ 0 xy + xz *** x (y + z) xy - xz *** x (y - z) If x > y, then *** x + z > y + z
measurement of angle x originating on the edge of a circle *** 1/2 the arc it cuts (between the points of the two lines extending from it across the circle) units digit of 3^x *** Will always end in 3, 9, 7, 1, in that sequence (a + b) (a -b) *** a² - b²
- (a - b) *** (b - a) y x 10^x *** move decimal point x digits to the left/right a² x b² *** (ab)² central angle of sector *** arc/ solve the percentage of circumference covered by an arc in terms of the central angle *** x/360 = % of circumference Measure of any inscribed angle (within a circle) whose triangle base is a diameter *** 90 degrees Inscribed angle y in terms of arc *** y = arc/ Adding fractions with different denominators *** cross multiply (bottom to top, top to bottom), taking those values as your new numerator, and then also multiply the denominators and use that as your new denominator
- (- y < - x) *** y > x (note the reversal of the inequality
(a + b) (c + d) *** ac + ad + bc + bd x¹ *** x x⁰ *** 1 (ab)ⁿ *** aⁿbⁿ a³/a² *** a³−² Multiplying Decimals *** Work as if they were whole integers. Then, count the number of digits to the right of the decimal place in each factor, combine them, and place the point that many digits to the left of your new product Distance formula *** speed x time = distance Work formula *** rate x time = work/output mixture formula *** concentration x amount of solution = amount of ingredient cost *** rate x number of items = value Area of square calculated in relation to its diagonal *** a = 1/2d² Area of a parallelogram *** a = bh (do not mistake with the formula for the height of a triangle. note also that "height" must be a straight line drawn from the base, not one of the diagonal sides)
Area of a hexagon *** a = (3√3 / 2)t (where t is the side length) diagonal of a square *** d = s√2 (where s equals the length of a side) area of a triangle *** ab sin C / 2, where a and b are any two sides and C is the angle between them Relationship between diagonal of a hexagon and side *** The longest diagonal is 2s (where s is the length of a side) perimeter of a hexagon *** p = 6r (where r is a given radius) formula for distance between two points on a coordinate graph *** d = √(x₂ - x₁)² + (y₂ - y₁)² (NB that the sqrt sign extends across the entire formula Coordinates for the midpoint of the line segment joining 2 points *** (x₁ + x₂ / 2, y₁ + y₂ / 2) (an average of the coordinates of the endpoints) Subtracting from both sides of an inequality *** reverse the central sign Adding to both sides of an inequality *** central sign remains the same multiplying or dividing by a negative number in an inequality *** reverse the central sign multiplying or dividing by a positive number in an inequality *** central sign remains the same Types and characteristics of triangles *** Scalene: no two sides or angles equal Isosceles: two equal sides and angles
Equilateral: All three sides and all angles equal Each angle must be 60 degrees Right: one angle is a right angle (90) Congruent triangles *** 1. each side of the first triangle equals the corresponding sides of the second triangle
- Two sides of the first triangle equal the corresponding angles of the second triangle, and their included angles are equal. The included angle is formed by the two sides of the triangle
- Two angles of the first triangle equal the corresponding angles of the second triangle, and any pair of corresponding sides are equal Median of a triangle *** A line drawn from a vertex (point) to the midpoint of its opposite side. The medians of a triangle cross at a point that divides each median into two parts: one part of one third the length of the median and the other part of two thirds the length Angle bisectors of a triangle *** Lines that divide each angle of a triangle into two equal parts; they cross in the middle of a circle inscribed in the center of the triangle Sum of any two sides of a triangle *** Greater than the length of the third side Angle inscribed in a semicircle *** Must be a right angle converting diameter to radius *** d = 2r; so d² = 4r² Combination formula *** C = (n)(n - 1)(n - 2)...(n - r+1) / (r)(r - 1)(r - 2)...(1) Permutation formula *** P = (n)(n - 1)(n - 2)...(n - r+1)
