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A comprehensive overview of various mathematical concepts and formulas relevant to the gre quantitative reasoning section. It covers a wide range of topics, including arithmetic operations with even and odd numbers, prime numbers and prime factorization, fractions and their addition, compound interest, linear equations and graphs, quadratic equations and parabolas, circle equations, common right triangle ratios, and various geometric formulas for areas, volumes, and perimeters. The document also includes explanations of important mathematical concepts such as slope, reciprocals, and the properties of zero. This resource would be highly valuable for students preparing for the gre quantitative reasoning exam, as it provides a concise and organized summary of the key mathematical knowledge and skills required for success on this section of the test.
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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 -
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 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. Ratio Box - X item Y item Total Ratio 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
Multiply the First, Outer, Inner, and Last terms of a pair of binomials Inequality - < > ≤ ≥ 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. 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+ 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) 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 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 -
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 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. area of a quadrilateral - A = bh (or lw): the base times height or length times width 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) 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) circumference - The distance around a circle. C = 2(pi)r 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.
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 -
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 values, (3) squaring each of the differences, (4) finding the average of the n squared differences, and (5) taking the nonnegative square root of the average squared difference sample standard deviation - computed by dividing the sum of the squared differences by instead of n. The sample standard deviation is only slightly different from the standard deviation but is preferred for technical reasons for a sample of data that is taken from a larger population of data. Sometimes the standard deviation is called the population standard deviation to help distinguish it from the sample standard deviation Set -
The objects of a set are called members or elements. Some sets are finite, which means that their members can be completely counted. Finite sets can, in principle, have all of their members listed, using curly brackets, such as the set of even digits {}0,2,4,6,8. list - A list is like a finite set, having members that can all be listed, but with two differences. In a list, the members are ordered; that is, rearranging the members of a list makes it a different list. Thus, the terms "first element," "second element," etc., make sense in a list. Also, elements can be repeated in a list and the repetitions matter. For example, the lists 1, 2, 3, 2 and 1, 2, 2, 3 are different lists, each with four elements, and they are both different from the list 1, 2, 3, which has three elements multiplication principle - Suppose there are two choices to be made sequentially and that the second choice is independent of the first choice. Suppose also that there are k different possibilities for the first choice and m different possibilities for the second choice. The multiplication principle states that under those conditions, there are km different possibilities for the pair of choices. permutation - The number of ways in which a set of values can be ordered. Formula: n(n-1)(n-2)(n-3) etc. Symbolized by n! number of permutations of n objects taken k at a time - n! / (n-k)! combination - In contrast with permutation, this is the number of ways in which a set of values can be ordered but without counting different orders for the same values. Formula: number of ways to select with order / number of ways to order = 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
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 If x > y and w > z, then - x + w > y + z If w > 0 and x > y, then - wx > wy If w < 0 and x > y, then - wx < wy If x > y > 0 and w > z > 0, then - xw > yz If x < 0 and z = x + y, then - z > y If xy > 0, then - x > 0 and y > 0 or x < 0 and y < 0 If xy < 0, then - x > 0 and y < 0 or x < 0 and y > 0 If a vehicle travels a certain distance at a mph and travels the same distance at b mph, the average rate is - 2ab / a + b (only works when the distance is the same at both speeds!)
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
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 an equilateral triangle - a = 1/4s²√ Area of a trapezoid - a = 1/2h(B + b), where B and b represent the "bases" (i.e. typically the straight lines at the bottom and the top of the figure, between which the height is drawn and measured) Perimeter of a semicircle - P = d(1/2π + 1)
