gmat quant concepts & formulae

GMAT ® Quant Concepts & Formulae Copyright 2012, by Aristotle Prep ® 2 www.aristotleprep.com Also Check Out: - Aristotle Sentence Correction Grail - Aristotle RC Practice Sets 1 & 2 -Ultimate One minute Explanations to OG12 SC -Aristotle New SC Question bank Available for FREE Download on our website 1) Aristotle CR Question Bank 2) US B-Schools Ranking 2012 3) Quant Concepts & Formulae 4) Global B-School Deadlines 2012 5) OG 12 & 13 Unique Questions’ list 6) GMAT Scoring Scale Conversion Matrix 7) International (non-US) B-Schools Ranking Aristotle Prep ® 3 www.aristotleprep.com 1) Number Properties i) Integers Numbers, such as -1, 0, 1, 2, and 3, that have no fractional part. Integers include the counting numbers (1, 2, 3, …), their negative counterparts (-1, -2, -3, …), and 0. ii) Whole & Natural Numbers The terms from 0,1,2,3,… are known as Whole numbers. Natural numbers do not include 0. iii) Factors Positive integers that divide evenly into an integer. Factors are equal to or smaller than the integer in question. 12 is a factor of 12, as are 1, 2, 3, 4, and 6. iv) Factor Foundation Rule If a is a factor of b, and b is a factor of c, then a is also a factor of c. For example, 3 is a factor of 9 and 9 is a factor of 81. Therefore, 3 is also a factor of 81. v) Multiples Multiples are integers formed by multiplying some integer by any other integer. For example, 6 is a multiple of 3 (2 * 3), as are 12 (4 * 3), 18 (6 * 3), etc. In addition 3 is also a multiple of itself i.e. 3 (1*3). Think of multiples as equal to or larger than the integer in question vi) Prime Numbers A positive integer with exactly two factors: 1 and itself. The number 1 does not qualify as prime because it has only one factor, not two. The number 2 is the smallest prime number; it is also the only even prime number. The numbers 2, 3, 5, 7, 11, 13 etc. are prime. vii) Prime Factorization Prime factorization is a way to express any number as a product of prime numbers. For example, the prime factorization of 30 is 2 * 3 * 5. Prime factorization is useful in answering questions about divisibility. 4 www.aristotleprep.com viii) Greatest Common Factor Greatest Common FACTOR refers to the largest factor of two (or more) integers. Factors will be equal to or smaller than the starting integers. The GCF of 12 and 30 is 6 because 6 is the largest number that goes into both 12 and 30. viii) Least Common Multiple (LCM) Least Common Multiple refers to the smallest multiple of two (or more) integers. Multiples will be equal to or larger than the starting integers. The LCM of 6 and 15 is 30 because 30 is the smallest number that both 6 and 15 go into. ix) Odd & Even Numbers Any number divisible by 2 is even and not divisible by 2 is odd. Odd & Even number Rules Function Result even + even even even + odd odd odd + odd even even - even even even - odd odd odd - odd even even * even even even * odd even odd * odd odd even ÷ even anything (even, odd, or not an integer) even ÷ odd even or not an integer odd ÷ even not an integer odd ÷ odd odd or not an integer Note: Division rules are more complicated because an integer answer is not always guaranteed. If the result of the division is not an integer, then that result cannot be classified as either even or odd. 5 www.aristotleprep.com x) Absolute Value The distance from zero on the number line. A positive number is already in the same form as that number’s absolute value. Remove the negative sign from a negative number in order to get that number’s absolute value. For example the absolute value of - 2 is 2. xi) Positive-Negative Number Rules Function Result positive * positive positive positive * negative negative negative * negative positive positive ÷ positive positive positive ÷ negative negative negative ÷ negative positive xii) Product of n consecutive integers and divisibility The product of n consecutive integers is always divisible by n! Given 5*6*7*8, we have n = 4 consecutive integers. The product of 5*6*7*8 (=1680), therefore, is divisible by 4! = 4*3*2*1 = 24. xiii) Sum of n consecutive integers and divisibility There are two cases, depending upon whether n is odd or even:  If n is odd, the sum of the integers is always divisible by n. Given 5+6+7, we have n = 3 consecutive integers. The sum of 5+6+7 (=18), therefore, is divisible by 3.  If n is even, the sum of the integers is never divisible by n. Given 5+6+7+8, we have n = 4 consecutive integers. The sum of 5+6+7+8 (=26), therefore, is not divisible by 4. xiv) PEMDAS First, perform all operations that are inside parentheses. Absolute value signs also fall into this category. In addition, for any expression with fractions, add parentheses around each distinct fraction. 6 www.aristotleprep.com Second, simplify any exponents that appear in the expression. Third, perform any multiplication and division in the expression; if there are more than one of these, perform the operations from left to right in the expression. Fourth, perform any addition and subtraction in the expression; if there are more than one of these, perform the operations from left to right in the expression. 7 www.aristotleprep.com 2) Base & Exponent In the expression b n , the variable b represents the base and n represents the exponent. The base is the number that we multiply by itself n times. The exponent indicates how many times to multiple the base, b, by itself. For example, 2 3 = 2 * 2 * 2, or 2 multiplied by itself three times. Equations that include an exponent are called as exponential equations. When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions. For example, for x 2 = 25, the two possible solutions are 5 and -5. i) Base of Zero An exponential expression with base 0 yields 0, regardless of the exponent. 0 12 = 0. ii) Base of One An exponential expression with base 1 yields 1, regardless of the exponent. 1 12 = 1. iii) Base of Negative One An exponential expression with base -1 yields 1 when the exponent is even and -1 when the exponent is odd. (-1) 15 = -1 and (-1) 16 = 1. iv) Fractional Base When the base is a fraction between zero and one, the value decreases as the exponent increases. (1/3) 3 = 1/3 * 1/3 * 1/3 = 1/27, which is smaller than the starting fraction, 1/3. v) Compound Base When the base represents a product (multiplication) or quotient (division), we can choose to multiply or divide the base first and then raise the base to the exponent, or we can distribute the exponent to each number in the base. For example (3 * 4) 2 = 12 2 = 144 OR (3*4) 2 = 3 2 * 4 2 = 9 * 16 = 144. vi) Exponent of Zero Any non-zero base raised to the 0 yields 1. Eg. 15 0 = 1. 8 www.aristotleprep.com vii) Exponent of One Any based raised to the exponent of 1 yields the original base. Eg. 15 1 = 15. viii) Negative Exponents Put the term containing the exponent in the denominator of a fraction and make the exponent positive. For example 4 -2 = (1/4) 2 ix) Fractional Exponents If the exponent is a fraction, the numerator reflects what power to raise the base to, and the denominator reflects which root to take. For example 4 2/3 = CUBE ROOT (4 2 ). x) Simplification Rules for Exponents Rule Result 3 4 * 3 3 = Add the exponents 3 7 3 4 / 3 2 = Subtract the exponents 3 2 (3 4 ) 3 = Multiply the exponents 3 12 xi) Root/Radical The opposite of an exponent (in a sense). For example, √25 means what number (or numbers), when multiplied by itself twice, will yield 25? Perfect square roots will yield an integer. Eg. √25 = 5. Imperfect square roots do not yield an integer. √30 is not an integer, but it is between √25 and √36, or between 5 and 6. xii) Simplifying Roots Roots can be combined or split apart if the operation between the terms is multiplication or division. √(4 * 9) = √4 * √9. Note: If the operation between the terms is addition or subtraction, you cannot separate or combine the roots! √(4 + 9) DOES NOT EQUAL √4 + √9. 9 www.aristotleprep.com 3) Equations & Inequalities i) Equation A combination of mathematical expressions and symbols that contains an ‘equals’ sign. Eg. 2 + 5 = 7 is an equation, as is x + y = 5 ii) Linear Equation An equation that does not contain exponents or multiple variables multiplied together. x + y = 5 is a linear equation whereas x*y = 5 and y = x 2 are not. When plotted on a coordinate plane, linear equations will give you straight lines. iii) Simultaneous Equation These are two or more distinct equations containing two or more variables. iv) Quadratic Equation An expression including a variable raised to the second power (and no higher powers). Commonly of the form ax 2 + bx + c, where a, b, and c are constants. v) Special Simplification cases  a 2 – b 2 = (a +b) * (a - b)  (a + b) 2 = a 2 + 2ab + b 2  (a - b) 2 = a 2 – 2ab + b 2 vi) Sequence A sequence is a collection of numbers in a set order. {3, 5, 7, 9, 11, …} is an example of a sequence for which the first five terms are specified (but the sequence continues beyond these five terms, as indicated by the “…”) vii) Linear/Arithmetic Sequence A sequence in which the difference between successive terms is always the same. A constant number (which could be negative!) is added each time. Also called Arithmetic Sequence. Eg. {1,3,5,7, } 10 www.aristotleprep.com viii) Exponential Sequence A sequence in which the ratio between successive terms is always the same; a constant number (which could be positive or negative) is multiplied each time. Also called Geometric sequence. Eg. {2,4,8,16, } ix) Functions A rule or formula which takes an input (or given starting value) and produces an output (or resulting value). For example, f(x) = x + 5 represents a function, where x is the input, f(x) is read as "f as a function of x" or "f of x" and refers to the output (also known as the "y" value), and x + 5 is the rule for what to do to the x input. Eg. f(4) = x + 5 = 4 + 5 = 9. x) Domain All of the possible inputs, or numbers that can be used for the independent variable, for a given function. In the function f(x) = x 3 , the domain is all numbers. xi) Range All of the possible outputs, or numbers that can be used for the dependent variable, for a given function. In the function f(x) = x 3 , the range is f(x) >= 0. xii) Compound/Composite Functions Two nested functions which are to be solved starting from the inner parentheses. For example, f(g(x)) is an example of a compound function and is read as "f of g of x." Given f(x) = x + 5 and g(x) = 3x, g(x) is substituted first followed by f(x). Eg. f(g(2)), g(2) = 3*2 = 6 and now f(6) = 6+5 = 11. xiii) Direct Proportion Two given quantities are said to be "directly proportional" if the two quantities always change by the same factor and in the same direction. For example, doubling the input causes the output to double as well. The standard formula is y = kx, where x is the input, y is the output, and k is the proportionality constant (or the factor by which the numbers change). xiv) Inverse Proportion Two given quantities are said to be "indirectly proportional" if the two quantities change by reciprocal factors. For example, doubling the input causes the output to halve. Tripling the input cuts the output to one-third of its original value. The standard formula is y = (k/x), where x is the input, y is the output and k is the proportionality constant [...]... center of the cube; the diagonal of a face of a cube is not the main diagonal The main diagonal of any cube can be found my multiplying the length of one side by the square root of 3 Circles & Cylinders i) Circle & Semi-Circle A circle is a set of points in a plane that are equidistant from a fixed center point Half of a circle = a semi-circle A semicircle contains 180º, exactly half of the 360º in a... (n - 2) * 180 = sum of interior angles, where n = the number of sides in the shape For example, the interior angles of a four-sided closed shape will always add up to (4 - 2)*180 = 360° iii) Perimeter & Area In a polygon, the sum of the lengths of the sides is called the perimeter The formula depends on the specific shape iv) Triangle A three-sided closed shape composed of straight lines; the interior...11 xv) Inequality A comparison of quantities that have different values There are four ways to express inequalities: less than () Inequalities can be manipulated... www.aristotleprep.com 18 xiii) Volume of a Cylinder V = πr2h, where V is the volume, r is the radius of the cylinder, h is the height of the cylinder, and π is a constant that equals approximately 3.14 Lines & Angles i) Straight Line, or Line A line measuring 180º Extends indefinitely in both directions ii) Line Segment A line segment has a finite length iii) Parallel Lines Lines that lie in a plane and never... A or event B occurs, find the individual probabilities of each event, add them together, and then subtract the probability that the two events will happen together www.aristotleprep.com 25 5) Decimals & Fractions i) Decimal Numbers that fall in between integers; expresses a part-to-whole relationship in terms of place value Example: 1.2 is a decimal The integers 1 and 2 are not decimals An integer written... fraction expresses a part-to-whole relationship in terms of a numerator (the part) and a denominator (the whole) Example: (7/2) is equivalent to the decimal 3.5 www.aristotleprep.com 28 xii) Numerator & Denominator The top part of a fraction is called the numerator and the bottom part the denominator In the fraction (7/2), 7 is the numerator and 2 is the denominator xiii) Proper Fraction Fractions that... 10 Multiply the first fraction by 2 and the second fraction by 5 to get (6/10) + (5/10) Next, add the numerators and keep the same denominator to get (11/10) www.aristotleprep.com 30 5) Miscellaneous Concepts i) Percent Literally, “per one hundred” Percent expresses a special part-to-whole relationship between a number (the part) and one hundred (the whole) A special type of fraction or decimal that . < x < 5 is a compound inequality, as is a < b < c < d. When manipulating, the same operations must be done to every term in the inequality. For example, given x < y + 2 <. function, where x is the input, f(x) is read as "f as a function of x" or "f of x" and refers to the output (also known as the "y" value), and x + 5 is the rule for what. Question Bank 2) US B-Schools Ranking 2012 3) Quant Concepts & Formulae 4) Global B-School Deadlines 2012 5) OG 12 & 13 Unique Questions’ list 6) GMAT Scoring Scale Conversion Matrix 7)