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GRE Quantitative Reasoning Prep
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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
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 - 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
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 Use this to determine the value of variables in quadratic equations. Quadratic equations have at most two real solutions FOIL - 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.
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 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.
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 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 =
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!) Common right triangle length ratios - 1: 1 :√ 1: 2 :√ 3: 4 : 5: 12 : 8: 15 : 7: 24 : 9: 40 : 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
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) Volume of a cube - V = e³ (where e is any edge of the cube) Surface area of a cube - S = 6e² (where e is any edge of the cube) Surface area of a cylinder (bases incl.) - S = 2πrh(h + r) Surface area of a cylinder (without bases) - S = 2πrh Volume of a sphere - V = 4/3πr²
Surface area of a sphere - S = 4πr² Volume of a hemisphere - V = 2/3πr² Surface area of a hemisphere - S = 2πr² (without base); S = 3πr² (with base) Area of an equilateral triangle - √3s² / 4 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
Permutation formula - P = (n)(n - 1)(n - 2)...(n - r+1) | | - Absolute value sign -- that is, the numerical value regardless of plus or minus sign. All absolute values are positive The union of sets A and B - A ∪ B (do not repeat items/digits shared between the two sets when unifying them) The intersection of sets A and B - A ∩ B (this is the list of members shared between the two sets) Subset - a set, all of whose members comprise part of a larger set. so (1, 2, 4) is a subset of (1, 2, 3, 4, 7, 9, 19) number of subsets in a set with n items - Set with n items has 2ⁿ subsets Set of ordered pairs - A relation, denoted by (x, y). The order of the elements in the pair matters Domain of a relation - the set of the first components of the ordered pairs Range of a relation - Set of the second components of the ordered pairs Function (set theory) - a relation in which each element of the domain occurs only once as a first component Solution sets - the set of solutions to an equation or inequality Closed set - a set in which, under an operation, any two members of the set constitute an element of the set (i.e., if multiplying two members of the set gives you another element of the same set). Sets are closed and open not absolutely, but in relation to these specific equations
- Diagonal of a square - d = s√
- ounces in a pound -
- pints in a quart -
- feet in a yard -
- feet in a mile -
- Area of an equilateral triangle with side s - A = s²√3 /
