GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. PRIME NUMBERS POSITIVE INTEGER greater than 1 that has EXACTLY TWO different positive factors – ONE AND ITSELF Note:.


GMAT number properties note: If a positive integer greater than or equal to 2 has only two factors, then the integer must be a prime number TWO is the smallest and only EVEN prime number If the product of two primes is even or if the sum of two primes is odd, one of the primes has to be 2 Conversely, if you know that 2 CANNOT be one of the primes in the sum, then the sum of the two primes must be even Method to determine a prime number Find approximate square root of the number.


GMAT number properties note: Then check if all the prime numbers below the square root are factors of the given number. 91 is not divisible by 2,3 or 5, but it is divisible by 7.


GMAT number properties note: (1) x has exactly 2 factors. (2) When x is divided by 2, the remainder is 0.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. (1) x has exactly 2 factors.


GMAT number properties note: That is also insufficient by itself Taken together, however, the two statements reveal that x must be an even prime-and the only even prime number is 2. BOTH statements TOGETHER are sufficient.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. DIVISIBILITY RULES For any integer n, n3 – n is divisible by 3, n5 – n is divisible by 5, n11 – n is divisible by 11, n13 – n is divisible by 13.


GMAT number properties note: In general, if p is a prime number then for any whole number a, ap – a is divisible by p. The factorial of N, symbolized by N!, is the product of all integers from 1 up to and including N.


GMAT number properties note: = 6 x 5 x 4 x 3 x 2 x 1 = 720. Because it is the product of all the integers from 1 to N, any factorial N!


GMAT number properties note: Must be divisible by all integers from 1 to N. Another way of saying this is that N!


GMAT number properties note: Is a multiple of all the integers from 1 to N.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. PRIME FACTORIZATION COMMON FACTOR Break down both numbers to their prime factors to see what factors they have in common.


GMAT number properties note: Multiply shared prime factors to find all common factors What factors greater than 1 do 135 and 225 have in common? 135 = 3 x 3 x 3 x 5 225 = 3 x 3 x 5 x 5 Both share 3 x 3 x 5 in common—find all combinations of these numbers: 3 x 3 = 9.


GMAT number properties note: 3 x 3 x 5 = 45 GREATEST COMMON FACTOR (GCF) The number that contains all the factors common to both numbers (i.e., biggest number that will divide into both the numbers) Note:. If the numbers do not share any common factors, then the GCF is 1 The GCF for two consecutive numbers is always 1.


GMAT number properties note: The prime factorizations for 2940 & 3150 are: 2940 = 2 x 2 x 3 x 5 x 7 x 7 3150 = 2 x 3 x 3 x 5 x 5 x 7 Therefore, GCF is 2 x 3 x 5 x 7 = 210 LOWEST COMMON MULTIPLE (LCM) The LCM is the smallest number that contains both numbers as factors i.e.


GMAT number properties note: The smallest number that is a multiple of both these values L.C.M of two numbers x and y is maximum power of the prime factors in x and y.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT number properties note: The prime factorizations for 2940 & 3150 are: 2940 = 2 x 2 x 3 x 5 x 7 x 7 3150 = 2 x 3 x 3 x 5 x 5 x 7 Therefore, LCM is 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 = 44,100 LCM * GCF = PRODUCT OF TWO NUMBERS (DOES NOT WORK FOR MORE THAN TWO NUMBERS) GCF of fractions = GCF of numerators ÷ LCM of denominators LCM of fractions = LCM of numerators ÷ GCF of denominators.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. FACTORS AND MULTIPLES FACTORS/DIVISORS X IS A FACTOR OF Y Essentially, a factor is a POSITIVE integer that divides evenly into an integer.


GMAT number properties note: Ex: 1,2,4 and 8 are all the factors of 8 One is a factor of all numbers The largest factor of a number is the number itself MULTIPLES Y IS A MULTIPLE OF X Essentially, a multiple of an integer is formed by multiplying that integer by any integer, so 8, 16,24, and 32 are some of the multiples of 8 For the purpose of the GMAT, multiples are always POSITIVE ** Divisibility can be stated in multiple ways.


GMAT number properties note: This just means that 12 (y) is divisible by 3 (x), and that each of the individuals has 4 items each (n) The smallest multiple of a number is the number itself Y = X*INTEGER Where x and y are both integers, and x is not equal to zero.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. FACTORS AND MULTIPLES – IMPORTANT CONCEPTS PERFECT SQUARES Always have an odd number of factors.


GMAT number properties note: Therefore, if an integer has an odd number of factors, it has to be a perfect square The prime factorization of a perfect square contains only even powers of primes. This is because perfect squares are formed from the product of two copies of the same prime factors.


GMAT number properties note: By contrast, if a number's prime factorization contains any odd powers of primes, then the number is not a perfect square. For instance, 132,300 = 22 x 33 X 52 X 72 is not a perfect square, because the 3 is raised to an odd power.


GMAT number properties note: If this number is multiplied by 3, then the result, 396,900, is a perfect square: 396,900 = 22 x 34 X 52x 72 The sum of all distinct factors of a perfect square is always odd PERFECT CUBES - If a number is a perfect cube, then it is formed from three identical sets of primes, so all the powers of primes are multiples of 3 in the factorization of a perfect cube.


GMAT number properties note: For instance, 903 = (2 X 32 x 5) * (2 X 32 x 5) * (2 X 32 x 5) = 23 x 36 x 53.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Ers of primes are multiples of 3 in the factorization of a perfect cube.


GMAT number properties note: Thus, 4 is a factor of 64 + 44 = 108 and 64 – 44 = 20.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Given that the integer n is divisible by 3, 7, and 11, what other numbers must be divisors (factors) of n?


GMAT number properties note: Since we know that 3, 7, and 11 are prime factors of n, we know that n must also be divisible by all the possible products of the primes (3, 7 and 11): 21, 33, 77, and 231 (given that n > 231) IN ANY LIST OF N CONSECUTIVE INTEGERS, EXACTLY ONE OF THE INTEGERS IS A MULTIPLE OF N.


GMAT number properties note: AS A RESULT, THE PRODUCT OF N CONSECUTIVE INTEGERS IS ALWAYS A MULTIPLE OF N (OR, IT IS DIVISIBLE BY N) In any list of three consecutive integers, one number (33) is a multiple of 3  33, 34, 35 Products of 113 × 114 × 115 and (y + 5)(y + 6)(y + 7) are both divisible by 3.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Ee consecutive integers, one number (33) is a multiple of 3  33, 34, 35 Products of 113 × 114 × 115 and (y + 5)(y + 6)(y + 7) are both divisible by 3 THE NUMBER OF FACTORS A NUMBER HAS (INCLUDING 1 AND THE NUMBER ITSELF) CAN BE COMPUTED BY FOLLOWING THE STEPS DETAILED BELOW:.


GMAT number properties note: First, determine all the prime factors of the number through LCM Second, determine the number of copies of each of the prime factors and express them in the following format - p^a * q^b *....


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. QUOTIENTS AND REMAINDERS Y = (QUOTIENT * DIVISOR) + REMAINDER.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT number properties note: EVEN AND ODD INTEGERS ONLY integers can be even/odd An odd integer can be expressed as 2n + 1 and an even integer can be expressed as 2n Sum of two different prime numbers will ALWAYS be EVEN, unless one of the numbers is 2 If two even integers are multiplied, the result will be divisible by 4, and if three even integers are multiplied, the result will be divisible by 8 etc.


GMAT number properties note: If x is even, and N = (x)(x+1)(x+2), is N divisible by 24? The terms x and x+2 must each have a factor of 2, and the product (x)(x+1)(x+2) must have factors 1, 2, 3 and 4 Therefore, the prime factorization of N must be a multiple of 1*2*3*2*2 = 24.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. POSITIVE AND NEGATIVE INTEGERS ZERO is neither positive nor negative – be very careful!


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Average the first and last term to find the precise "middle" of the set  (100+20)/2 = 60 Count the number of terms: 100 - 20 = 80, plus 1 yields 81 Multiply the "middle" number by the number of terms to find the sum: 60 x 81 = 4,860.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. BASIC EQUATIONS (THOSE WITHOUT EXPONENTS) SIMULTANEOUS EQUATIONS – THREE VARIABLES Look for ways to simplify the work.


GMAT algebra note: What is the sum of x, y and z? Y + z = 7 In this case, DO NOT try to solve for x, y, and z individually.


GMAT algebra note: Instead, notice the symmetry of the equations-each one adds exactly two of the variables-and add them all together. Therefore, 2x + 2y + 2z = 26.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ce the symmetry of the equations-each one adds exactly two of the variables-and add them all together.


GMAT algebra note: Therefore, 2x + 2y + 2z = 26. However, the question does NOT ask you to solve for all three variables.


GMAT algebra note: It only asks you to solve for x, which IS possible.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. N gives us a value for 3y + 5z, which we can substitute into the first equation in order to solve for x.


GMAT algebra note: Thus, BOTH statements TOGETHER are sufficient COMBO PROBLEMS – MANIPULATION The GMAT often asks you to solve for a combination of variables, called COMBO problems. In these cases, since you are not asked to solve for one specific variable, you should generally NOT try to solve for the individual variables right away.


GMAT algebra note: Instead, you should try to manipulate the given equation(s) so that the COMBO is isolated on one side of the equation. There are four easy manipulations that are the key to solving most COMBO problems:.


GMAT algebra note: Multiply or divide the whole equation by a certain number A:. Add or subtract a number on both sides of the equation D:.


GMAT algebra note: Distribute or factor an expression on ONE side of the equation. Then, if the other side of an equation from a statement contains a VALUE, that equation is.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. ABSOLUTE VALUE EQUATIONS Absolute value refers to the POSITIVE value of the expression within the absolute value brackets.


GMAT algebra note: Equations that involve absolute value generally have TWO SOLUTIONS. In other words, there are TWO numbers that the variable could equal in order to make the equation true.


GMAT algebra note: The reason is that the value of the expression inside the absolute value brackets could be POSITIVE OR NEGATIVE. For instance, if we know |x| = 5, then x could be either 5 or -5, and the equation would still be true If |x| > x then x must be negative The following three-step method should be used when solving for a variable expression inside absolute value brackets.


GMAT algebra note: Solve for w, given that 12 + |w – 4| = 30. Isolate the expression within the absolute value brackets: |w – 4| = 18 Remove the absolute value brackets and solve the equation for 2 different cases (positive and negative) w – 4 = 18.


GMAT algebra note: Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal. The possibility of a failed solution is a peculiarity of absolute value equations.


GMAT algebra note: For most other types of equations, it is good to check your solutions, b.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. INEQUALITIES COMBINING INEQUALITIES Many GMAT inequality problems involve more than one inequality.


GMAT statistics note: First, solve any inequalities that need to be solved.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ferably to the left (less than): 8 < x, x <17, x < 14 Third, combine the inequalities by taking the more limiting upper and lower extremes: 8 < x < 14 Another helpful approach is to combine inequalities by adding the inequalities together.


GMAT algebra note: That would be a mistake, however, because both m and n could be negative numbers that yield a number larger than 10 when multiplied together. The two statements together are INSUFFICIENT Consider the following variation: if both m and n are positive, is mn < 10 given that (1) m < 2 (2) n < 5.


GMAT algebra note: Since the variables are positive, we can multiply these inequalities together and conclude that mn < 10.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Since the variables are positive, we can multiply these inequalities together and conclude that mn < 10.


GMAT algebra note: You can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality, not just the outside terms.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In order to add inequalities, we must make sure the inequality signs are facing the same direction.


GMAT algebra note: One way to see this clearly is by realizing that only the right side of the equation contains the variable d. Perhaps ab2 – b is greater than b2c – d simply because of the magnitude of d.


GMAT algebra note: Therefore there is no way to draw any conclusions about the relationship between a and c. By adding the two inequalities from statements (1) and (2) together, we can come to the conclusion that a > c.


GMAT algebra note: Two inequalities can always be added together as long as the direction of the inequality signs is the same: ab2 – b > b2c – d (+) b > d.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. INEQUALITIES – USING EXTREME VALUES One effective technique for solving GMAT inequality problems is to focus on the EXTREME VALUES of a given inequality.


GMAT algebra note: This is particularly helpful when solving the following types of inequality problems:. Problems with multiple inequalities where the question involves the potential range of values for variables in the problem Problems involving both equations and inequalities.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. INEQUALITIES AND ABSOLUTE VALUES Solving inequalities √x2 = |x| It is often helpful to try to visualize the problem with a number line.


GMAT algebra note: One way to understand this inequality is to say "x must be less than 5 units from zero on the number line." Indeed, one interpretation of absolute value is simply distance on the number line. For a simple absolute value expression such as lx|, we are evaluating distance from zero.


GMAT algebra note: The range therefore, would be -5 < x < 5 Equations involving absolute value require you to consider two scenarios: one where the expression inside the absolute value brackets is positive, and one where the expression is negative. The same is true for inequalities.


GMAT algebra note: Note that you should never change I x – 5 |to x + 5. Remember, when you drop the absolute value signs, you either leave the expression alone or enclose the ENTIRE expression in parentheses and put a negative sign in front.


GMAT algebra note: Given that | x – 2 | < 5, what is the range of possible values of x? To work out the FIRST scenario, we simply remove the absolute value brackets and solve: x – 2 < 5.


GMAT algebra note: X < 7 To work out the SECOND scenario, we reverse the signs of the terms inside the absolute value brackets, remove the brackets, and solve again: - (x - 2) < 5.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. The constant factor is called the common ratio and is formed by dividing any term by the term which precedes it To answer a question about a sequence, we not only need to know the rule the sequence follows, but also at least one term in the sequence.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Should be taken as a – 3d, a – d, a + d, a + 3d A series of quantities is said to be in harmonic progression when their reciprocals are in A.P.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. PROGRESSIONS AND SEQUENCES – GEOMETRIC PROGRESSION Geometric progression is a series of numbers in which each term is formed from the preceding by multiplying it by a constant factor (r).


GMAT sequences and patterns note: The constant factor is called the common ratio and is formed by dividing any term by the term which precedes it. Should be taken as a/r, a, ar Four numbers in a G.P.


GMAT sequences and patterns note: Should be taken as a/r3, a/r, ar, ar3.


GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. FUNCTIONS The "domain" of a function indicates the possible inputs and the "range" of a function indicates the possible outputs.


GMAT coordinate geometry note: For instance, the function f(x) = x2 can take any input but never produces a negative number. So the domain is all numbers, but the range is f(x) > = O COMPOUND FUNCTIONS – work from the INSIDE out If f(x) = x3 + √x and g(x) = 4x – 3, what is f(g(3))?


GMAT coordinate geometry note: First, solve g(3): 4.3 – 3 = 9.


GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. First, solve g(3): 4.3 – 3 = 9.


GMAT coordinate geometry note: Now, plug-in 9 on f(x): 93 + √9 = 729 + 3 DETERMINING WHETHER A FUNCTION IS EVEN/ODD - To do this, you take the function and plug –x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(–x) = f(x), so all of the signs are the same), then the function is even.


GMAT coordinate geometry note: If you end up with the exact opposite of what you started with (that is, if f(–x) = –f(x), so all of the "plus" signs become "minus" signs, and vice versa), then the function is odd. So I'll plug –x in for x, and simplify:.


GMAT coordinate geometry note: Determine algebraically whether f(x) = –3x2 + 4 is even, odd, or neither f(–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 Since the new expression is the same as the initial expression, the function is even.


GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. N, odd, or neither f(–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 Since the new expression is the same as the initial expression, the function is even COMMON FUNCTION TYPES i) Direct proportionality means that the two quantities always change by the same factor and in the same direction.


GMAT fractions, decimals, and percents note: For instance, tripling the input will cause the output to triple as well. Direct proportionality relationships are of the form Y = kx, where x is the input value, y is the output value and k is the proportionality constant.


GMAT fractions, decimals, and percents note: The maximum height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?


GMAT fractions, decimals, and percents note: Typically with direct proportion problems, you will be given "before" and "after" values. We then write y1/x1 = y2/x2, since both ratios are equal to the same constant k.


GMAT fractions, decimals, and percents note: In the problem given above, note that the direct proportion is between.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. COMBINATORICS PERMUTATION ORDER MATTERS - a permutation is nothing but an ordered combination.


GMAT sets, probability, and counting note: Essentially, permutation questions are about taking a group of objects and totaling the number of ways in which they can be arranged in specific/pre-defined ways The number of permutations of r items, chosen from a pool of n items, is n! The number of permutations when things are not all different:.


GMAT sets, probability, and counting note: If there be n things, p of them of one kind, q of another kind, r of still another kind and so on, then the total number of permutations is given by n! R!...) As a special case, the number of permutations of all n items in a pool of n items is just n!, and the number of ways to arrange n distinct objects along a fixed circle is: (n – 1)!


GMAT sets, probability, and counting note: COMBINATION ORDER DOES NOT MATTER (i.e., whether the events are taking place simultaneously or sequentially). Since the order does not matter, we need to remove all the “duplicate” cases The number of combinations of r items, chosen from a pool of n items, is n!


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Does not matter, we need to remove all the “duplicate” cases The number of combinations of r items, chosen from a pool of n items, is n!


GMAT sets, probability, and counting note: Figure out the combinations from each group and then multiply them together.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Combinations are being drawn from several groups to form a complete set.


GMAT sets, probability, and counting note: If each player shakes hands with every other player NOT on his own team, how many handshakes take place? How many people take part in a handshake?


GMAT sets, probability, and counting note: There are 35 people who could be in the first spot, but that person cannot shake hands with anyone on his own team. So that person has only 30 people who’s hands he can shake.


GMAT sets, probability, and counting note: Therefore, the number of combinations is (35*30)/2.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. COMBINATIONS One concept that you need to know for the exam is that when dealing with combinations and permutations, each result corresponds to a unique set of circumstances.


GMAT sets, probability, and counting note: No other value of z would yield exactly 15 different groups of two. So if you know how many subgroups of a certain size you can choose from an unknown original larger group, you can deduce the size of the larger group Seating 5 people in 3 chairs, is the same as seating 3 people in 5 chairs.


GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. If you have 4 types of bread, 3 types of cheese and 2 types of ham and wish to make a sandwich, you can make it in 4*3*2 = 24 different ways.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. How many codes are possible?


GMAT sequences and patterns note: In this problem, you do not want to choose the five digits in order. Instead, you should start by picking the first and last digits (which must be odd), because these digits are the most restricted.


GMAT sequences and patterns note: Because there are 5 different odd digits (1, 3, 5, 7, and 9), there are 5 ways of picking the first digit. Since no repetition is allowed, there are only 4 odd digits left for the last digit.


GMAT sequences and patterns note: You can then pick the other three digits in any order, but make sure you account for the lack of repetition. For those three choices, you have only 8, 7, and 6 digits available.


GMAT sequences and patterns note: Therefore, the total number of lock codes is 4 x 5 x 8 x 7 x 6 = 6,720.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. For those three choices, you have only 8, 7, and 6 digits available.


GMAT statistics note: Therefore, the total number of lock codes is 4 x 5 x 8 x 7 x 6 = 6,720 SIMPLE FACTORIALS The number of ways in which n different things can be arranged in a straight line, such that no one thing is allowed to appear more than once in any of the arrangements (i.e. When repetitions are not allowed) is n!.


GMAT statistics note: In how many different orders can she arrange the books? Using the Fundamental Counting Principle, we see that we have 6 choices for the book that goes first, 5 choices for the book that goes next, and so forth.


GMAT statistics note: Ultimately, we have this total: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 different orders.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Hat goes first, 5 choices for the book that goes next, and so forth.


GMAT statistics note: Ultimately, we have this total: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 different orders ANAGRAMS An anagram is a rearrangement of the letters in a word or phrase.


GMAT statistics note: For instance, the word DEDUCTIONS is an anagram of DISCOUNTED, and so is the gibberish "word“ CDDEINOSTU Anagrams with repeated words The number of anagrams of a word is the factorial of the total number of letters, divided by the factorial(s) corresponding to each set of repeated letters.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Possible combinations from the 7 letters in the word PIZZAZZ.


GMAT statistics note: However, there are only 210. Why are there, relatively speaking, so few anagrams?


GMAT statistics note: The answer lies in repetition: the four Z's are indistinguishable from each other. If the four Z's were all different letters, then we would have 7!


GMAT statistics note: To picture that scenario, imagine labeling the Z's with subscripts:. We could then list the 5,040 anagram.


GMAT statistics note: Now erase the subscripts from those 5,040 anagrams. You will notice that many "different“ arrangements-like AIPZ1Z2Z3Z4, AIPZ4Z2Z3Z1' and so on-are now the same.


GMAT statistics note: In fact, for any genuinely unique anagram-like AIPZZZZ-there are now 4! = 24 identical copies in the list of 5,040 anagrams, because there are 4!


GMAT statistics note: = 24 ways to rearrange the four Z's in the word PIZZAZZ without changing anything. Because this 24-fold repetition occurs for every unique anagram of PIZZAZZ, we take 7!


GMAT statistics note: (which counts the arrangements as if the letters were all distinct) and divide by 4!


GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. SOLVING COMBINATORICS PROBLEMS – ADVANCED STRATEGIES.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. PROBABILITY PROBABILITY = NUMBER OF FAVORABLE OUTCOMES TOTAL NUMBER OF OUTCOMES PROBABILITY OF MULTIPLE EVENTS AND If two events have to occur together, generally an "and" is used.


GMAT sets, probability, and counting note: If a coin is tossed twice, what is the probability that on the first toss the coin lands heads and on the second toss the coin lands tails? The probability that the coin will land on heads is 1/2.


GMAT sets, probability, and counting note: The probability that the coin will land on Tails is also 1/2. Since we want them to happen together, we multiply individual probabilities 1/2 × 1/2 = 1/4 OR If both events do not necessarily have to occur together, an “or” may be used.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. John will win $100 if, from a deck of 52 standard playing cards, he chooses either a 7 or a 9 when pulling a single card from the deck.


GMAT sets, probability, and counting note: What is the probability that John will win $100? He can win by pulling out either a 7 or a 9.


GMAT sets, probability, and counting note: His chances of doing that are higher than if he could win only by pulling out a 7. In that case, he’d only have 4 cards that would make him win $100 (because there are 4 7's in a standard deck), now he has 8 cards.


GMAT sets, probability, and counting note: To find the total probability, we need to figure out the probability of each event and then add them together. Drawing from a pack of cards without replacing the card drawn MUTUALLY EXCLUSIVE EVENTS.


GMAT sets, probability, and counting note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. PROBABILITY – 2 The greatest probability- the certainty that an event will occur-is 1.


GMAT sets, probability, and counting note: Thus, a probability of 1 means that the event must occur The lowest probability-the impossibility that an event will occur is 0. Thus, a probability of 0 means that an event will NOT occur.


GMAT sets, probability, and counting note: What is the probability that, on three rolls of a single fair die, at least of the rolls will be a six? What is the probability that NONE of the rolls will yield a 6?


GMAT sets, probability, and counting note: On each roll, there is a 5/6 probability that the die will not yield a 6. Thus, the probability that on all 3 rolls the die will not yield a 6 is (5/6)*(5/6)*(5*6) = 125/216 Now, we originally defined success as rolling at least one six.


GMAT sets, probability, and counting note: Since we have found the probability of failure, we answer the original question by subtracting this probability from 1: 1 – (125/216) = 91/256.


GMAT geometry note: LINES AND ANGLES Two straight lines which meet at a point form an angle between them Acute angle is between 0 and 90 degrees Right angle is 90 degrees Obtuse angle is between 90 degrees and 180 degrees Reflex angle is between 180 degrees and 360 degrees A straight line is the shortest distance between two points and measures 180 degrees Parallel lines are lines that never intersect, no matter how far they are stretched, and are equidistant Perpendicular lines intersect at 90 degrees (may or may not bisect the line segment) The sum of angles at a point is 360 degrees INTERSECTING LINES EXTERIOR ANGLES OF A TRIANGLE.


GMAT geometry note: LINES AND ANGLES PARALLEL LINES CUT BY A TRANSVERSAL GMAT can disguise the lines, by not drawing them completely. Extending them can help you to find out whether they are transversal or parallel.


GMAT geometry note: GMAT uses the symbol II to indicate 2 lines are parallel. MN II OP means those segments are parallel Properties Alternate interior angles are equal Angles 3 and 6 are equal.


GMAT geometry note: Angles 4 and 5 are equal Alternate exterior angles are equal Angles 1 and 8 are equal.


GMAT geometry note: Angles 2 and 7 are equal Corresponding angles are equal (corresponding angles are one interior and one exterior angle that are on the same side of the transversal) Angles 1 and 5 Angles 2 and 6 Angles 3 and 7 Angles 4 and 8 It is important to note that if two lines cut by a transversal have any of the above properties, then the two lines must be parallel.


GMAT geometry note: TWO – DIMENSIAL POLYGONS TWO DIMENSIONAL POLYGONS TRIANGLES – 3 SIDED QUADRILATERALS – 4 SIDED OTHERS – N SIDED Equilateral Isosceles Scalene Pentagon – 5 sides Hexagon – 6 sides Comprised of 6 equilateral triangles.


GMAT geometry note: Therefore, the area of a hexagon will be 6 * area of an equilateral triangle Octagon – 8 sides Irregular quadrilaterals Parallelogram Rectangle Rhombus Square All quadrilaterals are not necessarily parallelograms.


GMAT geometry note: For a quadrilateral to be a parallelogram opposite sides and opposite angles have to be equal and parallel Trapezium SUM OF THE INTERIOR ANGLES OF A POLYGON – This depends on the number of sides (n) a polygon has: (n-2)*180 SUM OF THE EXTERIOR ANGLES OF A POLYGON = 360.


GMAT geometry note: TRIANGLES EQUILATERAL ISOSCELES SCALENE TYPES OF TRIANGLES – SIDES TWO SIDES ARE EQUAL AND THE THIRD SIDE IS LESS THAN THE SUM OF THE OTHER TWO SIDES All THREE SIDES ARE EQUAL ALL THREE SIDES ARE DIFFERENT ACUTE OBTUSE RIGHT – ANGLED TYPES OF TRIANGLES – ANGLES ONE ANGLE IS GREATER THAN 90 DEGREES In a triangle ABC with the vertex at B, AB2 + BC2 < AC2 ALL ANGLES ARE LESS THAN 90 DEGREES In a triangle ABC with the vertex at B, AB2 + BC2 > AC2 ONE ANGLE IS 90 DEGREES The 45-45-90 Triangle is known as a right-angled isosceles triangle In a triangle ABC with the vertex at B, AB2 + BC2 = AC2.


GMAT geometry note: BASIC PROPERTIES OF TRIANGLES – B SUM OF THE ANGLES OF A TRIANGLE = 180 DEGREES When one side is extended in any direction, an angle is formed with another side. This is called the exterior angle.


GMAT geometry note: There are six exterior angles in a triangle ANGLES CORRESPOND TO THEIR OPPOSITE SIDE This means that the largest angle is opposite the longest side, while the smallest angle is opposite the shortest side.


GMAT geometry note: Additionally, if two angles are equal, their sides are also equal SUM OF ANY TWO SIDES OF A TRIANGLE MUST BE GREATER THAN THE THIRD SIDE DIFFERENCE BETWEEN ANY TWO SIDES IS LESS THAN THE THIRD SIDE THE LENGTH OF THE THIRD SIDE MUST LIE BETWEEN THE SUM AND THE DIFFERENCE OF THE TWO GIVEN SIDES.


GMAT geometry note: SIMILAR AND CONGRUENT TRIANGLES Similarity – if any of the conditions are met – Angle, angle, angle – if the corresponding angles of each triangle have the same measurement. In other words, the above triangles are similar if:.


GMAT geometry note: Angle M = Angle P Side, angle, side - An angle in one triangle is the same measurement as an angle in the other triangle and the two sides containing these angles have the same ratio. Side LM/Side OP = Side LN/Side OQ Side, side, side - Each pair of corresponding sides have the same ratio.


GMAT geometry note: Side LM/Side OP = Side LN/Side OQ = Side MN/Side PQ.


GMAT geometry note: RIGHT- ANGLED TRIANGLE AND THE PYTHAGOREAN THEOREM Pythagorean Theorem: c2 = a2 + b2 a and b are the legs, while c is the hypotenuse THE 45-45-90 ISOSCELES TRIANGLE The 45-45-90 triangle is a special triangle with 2 equal sides and a relation between each side. If you are given one dimension on a 45-45-90 triangle, you can find the others.


GMAT geometry note: Relationship between sides 45° - 45° - 90° Leg - leg - hypotenuse x: x: x√2 Therefore, in a 45-45-90 degree triangle, you only need to know the value of one side to determine the others A 45-45-90 is exactly half of a square. Two 45-45-90 triangles put together make up a square.


GMAT geometry note: So, if you are given the diagonal of a square, you can find the side by using the relation above.


GMAT geometry note: HYPOTENUSE HYPOTENUSE EQUILATERAL TRIANGLES AND THE 30-60-90 TRIANGLE An equilateral triangle is one in which all three sides and all three angles are equal Equilateral Triangles and the 30-60-90 Triangle:.


GMAT geometry note: INSCRIBED AND CIRCUMSCRIBED TRIANGLES AND CIRCLES A right triangle inscribed in a semi- circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT geometry note: QUADRILATERALS How to prove a quadrilateral is a parallelogram If both pairs of opposite sides of a quadrilateral are equal If both pairs of opposite angles of a quadrilateral are equal If all pairs of consecutive angles of a quadrilateral are supplementary If one pair of opposite sides of a quadrilateral is both equal and parallel If the diagonals of a quadrilateral bisect each other The area of a parallelogram is twice the area of a triangle created by one of its diagonals Of all quadrilaterals with a given perimeter, the square has the largest area.


GMAT geometry note: Conversely, of all the quadrilaterals with a given area, the square is the one with the smaller perimeter The diagonal formed by joining vertices of two smaller angles will be greater than the diagonal formed by joining vertices of two greater angles in a rhombus The diagonals of a kite are perpendicular bisectors of each other.


GMAT geometry note: The kite also has two pairs of adjacent, congruent sides (i.e., all sides are not equal)– THE KITE IS NOT A PARALLELOGRAM.


GMAT geometry note: QUADRILATERALS TRAPEZIOD One pair of opposite sides is parallel, but not equal Area = 1/2 * (a+b)*h Isosceles Trapezoid What is the area of the trapezoid as show? Angle A = 120 degrees The perimeter of the trapezoid ABCD = 36 The area of a trapezoid is equal to the average of the bases multiplied by the height.


GMAT geometry note: In this problem, you are given the top base (AB = 6), but not the bottom base (CD) or the height. (Note: 8 is NOT the height!) In order to find the area, you will need a way to figure out this missing data.


GMAT geometry note: Drop 2 perpendicular lines from points A and B to the horizontal base CD, and label the points at which the lines meet the base E and F.


GMAT geometry note: Drop 2 perpendicular lines from points A and B to the horizontal base CD, and label the points at which the lines meet the base E and F. The congruent symbols in the drawing tell you that Angle A and Angle B are congruent, and that Angle C and Angle D are congruent.


GMAT geometry note: This tells you that AC = BD and CE = FD. Therefore, since the sum of all 4 angles must yield 360 (which is the total number of degrees in any four-sided polygon), we know that Angle B = 120, Angle C = 60, and Angle D = 60.


GMAT geometry note: This means that triangle ACE and triangle BDF are both 30-60-90 triangles. The relationship among the sides of a 30-60-90 triangle is in the ratio of x: x √3: 2x, where x is the shortest side.


GMAT geometry note: The same measurements hold for triangle BFD. Thus we have the length of the bottom base (4 + 6 + 4) and the height and we can calculate the area of the trapezoid.


GMAT geometry note: Asurements hold for triangle BFD. We already know that the lengths of sides AB (6), AC (8), and BD (8) sum to 22.


GMAT geometry note: Further, since EF = 6, we can determine that CE = FD = 4. From this information, we can work with either Triangle ACE or Triangle BDF, and use the Pythagorean theorem to figure out the height of the trapezoid.


GMAT geometry note: Now, knowing the lengths of both bases, and the height, we can calculate the area of the trapezoid EACH STATEMENT ALONE IS SUFFICIENT.


GMAT geometry note: CIRCLES Circumference of a circle = 2πr Circumference of a semi-circle = πr Perimeter of a semi-circle = πr + D (Diameter of the circle) Area of a circle = πr2 Area of a semi-circle = πr2 / 2 Length of the arc = (θ/360) × (2 π r) Area of the sector = (θ/360) × πr2 Distance travelled by a wheel in n revolutions = n x circumference Perimeter of a sector = (θ/360) × (2 π r) + 2r Circles, when graphed on the coordinate plane, have an equation of x2 + y2 = r2 where r is the radius (standard form) when the center of the circle is the origin.


GMAT geometry note: When the center of the circle is (h, k) and the radius is of length r, the equation of a circle (standard form) is (x - h)2 + (y - k)2 = r2.


GMAT geometry note: CIRCLES, ARCS AND SECTORS A circle is a set of points that are equidistant from a fixed point called the centre. RADIUS: distance from the centre of the circle to any point on the circle.


GMAT geometry note: Can be shown as a line segment connecting the centre to a point on the circle – OC DIAMETER: is a line segment that connects two points on the circle and goes through the centre of the circle - AE Diameter = 2r CHORD: is any line segment whose endpoints are any two points on the circle – BD The tangent line and the radius of the circle that has an endpoint at the point of tangency are perpendicular to each other – AO IS PERPENDICULAR TO XY ARC – a portion/distance on the circle A C B E O D Y X A B O C.


GMAT geometry note: INSCRIBED CIRCLES When a circle is inscribed inside a square, the side equals the diameter.


GMAT geometry note: CO-ORDINATE GEOMETRY – GENERAL CONCEPTS CO-ORDINATE POINTS Points in the plane are identified by using an ordered pair of numbers, (x, y), where x is the x co-ordinate and y is the y co-ordinate.


GMAT geometry note: The point where a line intersects a co-ordinate axis is called an intercept X intercept: the point at which the line intersects the x axis The x-intercept is expressed using the ordered pair (x, 0), where x is the point where the line intersects the x-axis.


GMAT geometry note: The x-intercept is the point on the line at which y = O Y intercept: the point at which the line intersects the y axis The y-intercept is expressed using the ordered pair (0, y), where y is the point where the line intersects the y-axis. The y-intercept is the point on the line at which x = O.


GMAT algebra note: Ssed using the ordered pair (0, y), where y is the point where the line intersects the y-axis.


GMAT algebra note: The y-intercept is the point on the line at which x = O LINE A line can be formed in any of the three ways By knowing the co-ordinates of any two points on the line By knowing the co-ordinates of one point on the line and the slope of the line By knowing the equation of the line SLOPE OF A LINE:.


GMAT algebra note: Y2 – Y1 / X2 – X1 A line in the plane is formed by the connection of two or more points. Therefore, in the equation y = 4x + 8.


GMAT algebra note: The slope is 8 If the equation of the line is given in the general form ax + by + c = 0. Then the slope is –a/b and the intercept is –c/b.


GMAT algebra note: The slope is 8 If the equation of the line is given in the general form ax + by + c = 0. Then the slope is –a/b and the intercept is –c/b The slope of all points/ co-ordinates on a line will ALWAYS be the same.


GMAT algebra note: However, just because two points have the same slope, doesn’t necessarily mean that the lie on the same line and vice versa. For instance, parallel lines have the same slope.


GMAT algebra note: Conversely, if two lines have the same slope they could either be parallel or lie on each other If the slope is 1, the angle formed by the slope is 45 degrees Parallel lines have equal slopes Perpendicular lines have negative reciprocal slopes SLOPE DIRECTION - There are four types of slope an equation can have Positive – rises upward from left to right Negative – falls downward from left to right Zero – parallel to the x axis Undefined – parallel to the y axis Every line (but the one that crosses the origin OR the one that is parallel to X or Y axis OR the X and Y axis themselves) crosses three quadrants.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. CO-ORDINATE GEOMETRY – EQUATION OF A LINE/ SLOPE-INTERCEPT LINE What is the slope-intercept form for a line with the equation 6x + 3y = 18?


GMAT algebra note: Instead, they are expressed as simple, one-variable equations. Horizontal lines are expressed in the form: y = some number, such as y = 3 or y = 5 (i.e., equation of the line parallel to the x axis is 3 or 5).


GMAT algebra note: Vertical lines are expressed in the form: x = some number, such as x = 4 or x = 7 (i.e., equation of the line parallel to the y axis is 4 or 7) Equation of the X axis is Y = 0 Equation of the Y axis is X = 0.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


GMAT word problem note: Draw a right triangle connecting the points Find the lengths of the two legs of the triangle by calculating the rise and the run The y-coordinate changes from 3 to -5, a difference of 8 (the vertical leg) The x-coordinate changes from 1 to 7, a difference of 6 (the horizontal leg) Use the Pythagorean Theorem to calculate the length of the diagonal, which is the distance between the points 62 + 82 = c2 36 + 64 = c2 100 = c2 c = 10 POSITIVE AND NEGATIVE QUADRANTS Quadrant I: x and y are both positive Quadrant II: x is negative, y is positive Quadrant III: x is negative, y is negative Quadrant IV: x is positive, y is negative.


GMAT word problem note: Nd y are both positive Quadrant II: x is negative, y is positive Quadrant III: x is negative, y is negative Quadrant IV: x is positive, y is negative The GMAT sometimes asks you to determine which quadrants a given line passes through. Which quadrants does the line 2x + Y = 5 pass through?


GMAT word problem note: 2x + y = 5 y = 5 – 2x Since b = 5, the y-intercept is the point (0, 5). The slope is -2, so the line slopes downward steeply to the right from the y-intercept.


GMAT word problem note: If Point A is (5,20) and Point B is (30,7), define a line through point C (12,10) that is parallel to the line that passes through both A and B Slope of AB = (7-20)/(30-5)  -13/25  -0.52 Substituting in equation y-y1 = m(x-x1)  y = -0.52(x – 12) + 10 y = -0.52x + 16.24 Distance between two parallel lines y = mx + b and y = mx + c can be found by the formula D = | b – c |/ √m2 + 1.


GMAT coordinate geometry note: CO-ORDINATE GEOMETRY – 4 PERPENDICULAR BISECTORS The perpendicular bisector of a line segment forms a 90° angle with the segment and divides the segment exactly in HALF. The key to solving perpendicular bisector problems is remembering this property: the perpendicular bisector has the negative reciprocal slope of the line segment it bisects.


GMAT coordinate geometry note: If the coordinates of point A are (2, 2) and the coordinates of point B are (0, -2), what is the equation of the perpendicular bisector of line segment AB? Slope of AB is 2 Slope of the perpendicular bisector is – (1/2) Now we know that the equation of the perpendicular bisector has the following form: y = (-1/2)x + b.


GMAT coordinate geometry note: However, we still need to find the value of b To do this, we will need to find one point on the perpendicular bisector The perpendicular bisector passes through the midpoint of AB. Thus, if we find the midpoint of AB, we will have found a point on the perpendicular bisector.


GMAT coordinate geometry note: The x co-ordinate of the perpendicular bisector will be the average of the x co-ordinates of A and B. Similarly, the y co-ordinate of the perpendicular bisector of AB.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Dinate of the perpendicular bisector of AB will be the average of the y co-ordinates of A and B.


GMAT algebra note: Plug in the values of x (1) and y (0) to get b = 1/2 INTERSECTION OF TWO LINES Recall that a line in the coordinate plane is defined by a linear equation relating x and y. That is, if a point (x, y) lies on the line, then those values of x and y satisfy the equation.


GMAT algebra note: For instance, the point (3, 2) lies on the line defined by the equation y = 4x - 10, since the equation is true when we plug in x = 3 and y = 2. On the other hand, the point (7, 5) does not lie on that line, because the equation is false when we plug in x = 7 and y = 5 So, what does it mean when two lines intersect in the coordinate plane?


GMAT algebra note: It means that at the point of intersection, BOTH equations representing the lines are true. That is, the pair of numbers (x, y) that represents the point of intersection solves BOTH equations.


GMAT algebra note: At what point does the line represented by y = 4x - 10 intersect the line represented by 2x + 3y = 26? Solving the equations simultaneously, we get x = 4 and y = 6 Therefore, the point of intersection is (4,6) If two lines in a plane do not intersect, then the lines are parallel.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.


GMAT algebra note: In this case, infinitely many points (x, y) along the line satisfy the two equations (which must actually be the same equation in two disguises) The coordinates of the point P(x, y) which divides the line segment joining the points A(x1, y1) and B(x2, y2), internally, in the ratio m:n are (nx1 + mx2)/m+n and (ny1 + my2)/m+n.


GMAT geometry note: ADVANCED CONCEPTS EQUATION OF A CIRCLE The equation of a circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2 The equation of a circle with centre (0, 0) and radius r is x2 + y2 = r2 In the coordinate plane, a circle has center (2, -3) and passes through the point (5, 0). What is the area of the circle?


GMAT geometry note: (5 – 2)2 + (0 – (-3))2 = r2 (3)2 + (3)2 = r2 18 = r2 Therefore, area = 18 π Find the equation of a circle whose center is at (4, 2) and is tangent to y-axis Since the circle is tangent to y-axis, the radius of the circle is perpendicular to y-axis.


GMAT geometry note: It also means that the length of the radius is also the length of the perpendicular segment from the center of the circle to y –axis. The point of tangency is at (0, 2).


GMAT geometry note: To find the length of the radius, use the distance formula: (0 – 4)2 + (2 – 2)2 = r2 r = 4.


GMAT geometry note: The point of tangency is at (0, 2). To find the length of the radius, use the distance formula: (0 – 4)2 + (2 – 2)2 = r2 r = 4 Write the equation of the circle with the given condition: (10, 8) and (4, -2) are the endpoints of the diameter In a circle, the radius is one-half of the diameter.


GMAT geometry note: Since the given are the endpoints of the diameter, the center of the circle is the midpoint of the diameter The next step is to get the length of the radius. Since radius is one-half of the circle, it will equal the distance from the center to any one end point of the diameter: (10 – 7)2 + (8 – 3)2 = r2  r = root 34 Equation = (x – 7)2 + (y – 3)2 = 34.


GMAT word problem note: X and y are co-ordinates of the point Therefore, A = 3, B = -4 and C = -24, x = 1, y = 0 Radius = 21/5.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. ADVANCED CONCEPTS PARABOLAS The equation y = Ax2 + Bx + C is the equation of the quadratic graph which is a parabola with axis parallel to y axis C is the y intercept (i.e., the value of y when x is 0) If A > 0, the parabola opens upwards.


GMAT algebra note: If A < 0, the parabola opens downwards If B2 > 4AC, the parabola cuts the x-axis at 2 different points If B2 = 4AC, the parabola touches the x-axis at one point (the two points become co-incident) If B2 < 4AC, the parabola does not cut the x-axis at all The vertex of a parabola represents the maximum or minimum value of the function.


GMAT algebra note: The vertex is located at point (-b/2a, c – (b2/4a)).


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. (1997 is not a leap year When Mark purchased Product X in January 1997, the warranty did not expire until March 1997 When Santos purchased Product X in May 1997, the warranty expired in May 1997 Rephrase the two statements in terms of extreme possibilities:.


GMAT fractions, decimals, and percents note: Statement 1 Shortest possible warranty period:. 1 (29 days later) Longest possible warranty period:.


GMAT fractions, decimals, and percents note: 31 (89 days later) Statement 2 Shortest possible warranty period:. May 1 to May 2, or similar (1 day later) Longest possible warranty period:.


GMAT fractions, decimals, and percents note: May 1 to May 31 (30 days later).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ement 2 Shortest possible warranty period:.


GMAT algebra note: May 1 to May 2, or similar (1 day later) Longest possible warranty period:. May 1 to May 31 (30 days later) Even taking both statements together, there are still two possibilities-29 days and 30 days -so both statements together are still insufficient Note that, had the given year been a leap year, the two statements together would have become sufficient!


GMAT algebra note: In general optimization problems, the objective is to maximize or minimize some quantity, given constraints on other quantities. These quantities are all related through some equation.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The objective is to maximize or minimize some quantity, given constraints on other quantities.


GMAT algebra note: If no individual guest consumed more than 2.5 pounds of food, what is the minimum number of guests that could have attended the banquet? Begin by considering the extreme case in which each guest eats as much food as possible, or 2.5 pounds apiece.


GMAT algebra note: The corresponding number of guests at the banquet works out to 401/2.5 = 160.4 people You obviously cannot have a fractional number of guests at the banquet. Thus the answer must be rounded.


GMAT algebra note: To determine whether to round up or down, consider the explicit constraint: the amount of food per guest is a maximum of 2.5 pounds per guest. Therefore, the minimum number of guests is 160.4 (if guests could be fractional), and we must round up to make the number of guests an integer: 161 Note the careful reasoning required!


GMAT algebra note: Although the phrase "minimum number of guests" may tempt you to round down, you will get an incorrect answer if you do so. In general, as you solve this sort of problem, put the extreme case into the underlying equation, and solve.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In general, as you solve this sort of problem, put the extreme case into the underlying equation, and solve.


GMAT algebra note: Then round appropriately In grouping problems, you make complete groups of items, drawing these items out of a larger pool. The goal is to maximize or minimize some quantity, such as the number of complete groups or the number of leftover items that do not fit into complete groups.


GMAT algebra note: As such, these problems are really a special case of optimization. One approach is to determine the limiting factor on the number of complete groups.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. As such, these problems are really a special case of optimization.


GMAT algebra note: Each group must have exactly one person from Division A, two people from Division B, and three people from Division C. There are 20 people from Division A, 30 people from Division B, and 40 people from Division C at the conference.


GMAT algebra note: What is the smallest number of people who will not be able to be assigned to a group? The first step is to find out how many groups you can make with the people from each division separately, ignoring the other divisions.


GMAT algebra note: There are enough Division A people for 20 groups, but only enough Division B people for 15 groups and only enough Division C people for 13 groups. So the limiting factor is Division C.


GMAT algebra note: Therefore the total number of people left over will be 7 from A + 4 from B + 1 from C = 12 COMPUTATION Very occasionally, the GMAT features problems centered on computation problems that contain no variables at all. If the wire thus made is exactly 1 meter long, how long is each of the identical pieces?


GMAT quant note: Identify the concept first, then apply the matching rule or setup.