GMAT coordinate geometry note: In order for abc to be even, what must be true? If just one of the numbers is even, then the product will be even.


GMAT coordinate geometry note: The only case that will produce an odd is when all three numbers are odd. Keep this in mind when testing the statements.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. If a + b is even, then the two variables are either both even or both odd: a b c abc even?


GMAT algebra note: Even Even Even Yes Odd Odd Odd No (2) SUFFICIENT: b + c is odd, so one of b and c is odd and the other is even. In other words, it is impossible for all three variables to be odd.


GMAT algebra note: Therefore, the product abc must be even. Set up a table and try to find contradictory scenarios.


GMAT algebra note: 5 6 6 5 6 No 4 6 6 6 6 Yes (2) INSUFFICIENT.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. A B G M S < 5 in one class?A B G M S < 5 in one class?


GMAT sets, probability, and counting note: 6 6 4 6 6 Yes 6 5 5 6 6 No (1) AND (2) SUFFICIENT:. 4 6 6 5 7 Yes Since Geology and Sociology each have to have a minimum of 6 students, and they can’t have the same number of students, one of the two classes has to have at least 7 students.


GMAT sets, probability, and counting note: Between Biology, Geology, and Sociology, then, the minimum number of students is 6 + 6 + 7 = 19, leaving 9 students to be split among the other two classes. In this case, it’s impossible to have 5 or more students in both of those two remaining classes.


GMAT sets, probability, and counting note: The question asks for the probability that a randomly selected integer, from 1 to 100, will be in neither one of the two sets. To calculate this probability, you need to know the total number of integers that are in one or both of the two sets, as well as within the 1 to 100 range.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Jot down the definitions of the two sets.


GMAT number properties note: Integer x is contained in both sets, so, at most, the two sets will contain seven distinct integers between them. However, some of these integers might be less than 1 or greater than 100, and there might be more integers that overlap between the two sets, decreasing the number of possible values that overlap with the selected integer to less than 7.


GMAT number properties note: A statement is sufficient if every possible value of x yields the same number of distinct values between 1 and 100, inclusive. Test a straightforward case first, then test a case that might yield a different value.


GMAT number properties note: The smallest possible value for x that fits the statement is 6. If x = 6, then set A consists of four integers: 6, 62, 63, and 64.


GMAT number properties note: Two of those integers, 6 and 62 = 36, are smaller than 100. As long as you know that 63 and 64 are greater than 100, there is no need to calculate their exact values.


GMAT number properties note: Set B also consists of four integers: 6, 12, 18, and 24. Therefore, the sets contain a total of five distinct integers between 1 and 100, inclusive.


GMAT number properties note: Try an extreme case, such as x = 1,000. In this case, all of the integers in both set A and set B are much greater than 100.


GMAT number properties note: Therefore, the sets contain no integers between 1 and 100.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try another extreme case, such as x = –2.


GMAT number properties note: In this case, the only positive integers in either set will be (–2)2 = 4 and (–2)4 = 16, which are both between 1 and 100. Therefore, there are exactly two distinct integers within the range.


GMAT number properties note: Since the set could contain five integers or two integers within the given range, this statement is also insufficient. (1) AND (2) SUFFICIENT: x = 6 yields five distinct integers that are within the range 1 to 100.


GMAT number properties note: Check to see whether this is also true for x = 7, 8, and For x = 7, 8, or 9, set A contains two integers that are within the range 1 to 100: x and x2. In all three cases, x3 and x4 are outside of the range.


GMAT number properties note: Also, set B contains four integers that are within the range, but one of them, x itself, is also contained in set A. So in every case in which x is between 5 and 10, there are exactly five distinct integers in the range that are contained in one or both sets.


GMAT number properties note: The average price of six coats was originally $85. So the sum of the six prices was originally 6($85) = $510.


GMAT number properties note: Two of the prices were discounted by 20%, resulting in a $9 reduction in the average price, or a 6($9) = $54 reduction in the overall price. Since 20% (or 1/5 of the cost of these two coats came to $54, the original price of these two coat.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. From the question stem, the two discounted coats had a total price of $270, and according to this statement, one of those two coats was the most expensive.


GMAT quant note: Test cases to see whether the other coat could have been the least expensive. Note that the four non- discounted coats had a total price of $510 – $270 = $240.


GMAT quant note: If the discounted coats originally cost $1 and $269 and the other four coats each cost , then the least expensive coat was one of the two that was discounted, and the answer is Yes. If the discounted coats originally cost $130 and $140 and the other four coats each cost , then the least expensive coat was not one of the two that was discounted, and the answer is No.


GMAT quant note: This statement limits the possible original costs of the two discounted coats. Since neither of the coats could have cost more than $180, the less expensive of the two discounted coats must have cost at least $270 – $180 = $90.


GMAT quant note: However, this coat could not have been the one with the lowest original price, as $90 is greater than the original average price. The smallest number in any set cannot be greater than the average of th.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Assign variables to the unknowns in the problem.


GMAT algebra note: C = packages of candy P = bags of popcorn R = number of pretzels Pc = price of a package of candy Pp = price of a bag of popcorn Pr = price of a pretzel Algebraically, the question asks:. What is Pc × C − (Pp × P + Pr × R)?


GMAT algebra note: The question stem indicates that (1) INSUFFICIENT:. Substitute and rephrase the question:.


GMAT algebra note: What is What is What is Pc × C − (Pp × P + Pr × R)?


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Given the relative prices of the candy, popcorn, and pretzels, and the quantity of each purchased, the cost of the candy will always equal the combined cost of the popcorn and pretzels, even though it’s impossible to calculate the exact prices.


GMAT algebra note: Statements (1) and (2) combined are SUFFICIENT to answer the question. What is What is 12Pp − 6Pp − 6Pr?


GMAT algebra note: It’s tempting to rephrase the question as “What is x?” but that is dangerous when the question contains absolute value symbols. It’s possible that two different values of x would resolve to the same value of |x + 4|.


GMAT algebra note: Leave the question as it is. (1) SUFFICIENT: x2 + 8x + 12 = 0.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Plug these answers into the question stem:.


GMAT algebra note: (2) INSUFFICIENT: x2 + 6x = 0. Plug both values into the question stem:.


GMAT algebra note: If abcd ≠ 0, then none of the variables equals 0. In order for the product abcd to be negative, you would need to have an odd number of negatives in the mix.


GMAT quant note: Scenario Product abcdScenario Product abcd All 4 positive or all 4 negative + 3 positive, 1 negative − 2 positive, 2 negative + 1 positive, 3 negative − (1) INSUFFICIENT:. If b and d are both positive or both negative, then ad > bc.


GMAT quant note: If one is positive and one is negative, then ad < bc. This statement indicates nothing about a and c, though.


GMAT quant note: If a and c are both positive or both negative, then bc > ad. If one is positive and one is negative, then bc < ad.


GMAT quant note: This statement indicates nothing about b and d, though.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Here’s where the trick comes in.


GMAT number properties note: The two statements allow two possible scenarios:. If ad > bc, then b and d have the same sign but a and c have opposite signs.


GMAT number properties note: In this case, three signs are the same and one is not, so there are an odd number of negatives in the mix, and the product of all four variables must be negative. If ad < bc, b and d have opposite signs, but a and c have the same signs.


GMAT number properties note: In this case, three signs are the same and one is not. As before, the product of all four variables must be negative.


GMAT number properties note: Either way, the two statements together are sufficient to answer the question. In other words, n is the product of four consecutive integers.


GMAT number properties note: In any four consecutive integers, one number must be divisible by 3 and two numbers must be even. Furthermore, of the two even numbers, one must be divisible by 4.


GMAT number properties note: As a result, the product of any four consecutive integers is divisible by (3)(2)(4) = 24.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Alternatively, you could figure this out by testing a few cases: (2) INSUFFICIENT:.


GMAT algebra note: If m = 1, then n could equal 5, which is not a multiple of 24. If m = 20, then n could equal 24, which is a multiple of 24.


GMAT algebra note: M = 1: (8)(7)(6)(5), which is divisible by (8)(3) = 24 m = 2: (9)(8)(7)(6), which is divisible by (8)(3) = 24 m = 3: (10)(9)(8)(7), which is divisible by (8)(3) = 24 15. This algebra problem includes a special quadratic: (1 – y)(1 + y) = 1 – y2.


GMAT algebra note: Rephrase the question as “What is the value of x(1 – y2)?” (1) INSUFFICIENT:. This statement appears to provide little information.


GMAT algebra note: However, it significantly limits the possible values of x. If x = 1, the answer to the question is the value of 1 – y2, which is unknown.


GMAT algebra note: If x = 0, the answer to the question is 0. If y2 = x, then the question simplifies to “What is the value of x(1 – x)?” Different values of x produce different answers, so this statement is insufficient.


GMAT quant note: In this case, the answer is x(1 – y2) = 0(1 – 0) = 0. In this case, the answer is x(1 – y2) = 1(1 – 1) = 0.


GMAT quant note: Since the answer to the question is always 0, the two statements together are sufficient.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. PART TWO Strategies for All Problem Types.


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


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Pattern Recognition Problems Sequence Problems Units (Ones) Digit Problems Remainder Problems Other Pattern Problems.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If you can grasp the rule, you can predict numbers that appear later in the series.


GMAT sequences and patterns note: The series may be part of a defined sequence or it may arise from a general list of possibilities. Either way, if you can spot the pattern, you can eliminate a lot of unnecessary calculation.


GMAT sequences and patterns note: In fact, o en the only feasible way to get the answer in two minutes (e.g., finding the 100th number in some series) is to recognize the underlying pattern. Try-It #5-1 Each number in a sequence is 3 more than the previous number.


GMAT sequences and patterns note: If the first number is 4, what is the value of the 1,000th term in the sequence?


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Obviously, finding the 1,000th number the long way (by computing every intervening number) is impossible in the time allotted on the GMAT.


GMAT sequences and patterns note: You can solve this problem in several ways, but one powerful way is to compute the first several terms, spot the underlying pattern, figure out the rule, and then apply it. The first five terms of the sequence are 4, 7, 10, 13, 16.


GMAT sequences and patterns note: Notice that you repeatedly add 3 to get the next value. Repeated addition is just multiplication, so match these numbers to the multiples of 3.


GMAT sequences and patterns note: The first five multiples of 3 are 3, 6, 9, 12, 15—all 1 less than the numbers in the sequence. Thus, the rule for generating the sequence is “take the corresponding multiple of 3, and add 1.” Therefore, the 1,000th term of the sequence is 1 more than the 1,000th multiple of 3, which is 3,000 + 1 = 3,001.


GMAT sequences and patterns note: Once you spot the pattern, you can skip over vast amounts of unnecessary work. The two most basic patterns are these: integers.


GMAT sequences and patterns note: As simple as this pattern may seem, it is the basis for many other patterns. For instance, the sequence of the multiples of 7 (7, You can write this sequence as Sn = 7n, where n is the basic sequence of positive integers.


GMAT sequences and patterns note: A repeating cycle of numbers. For instance, the sequence 4, 2, 6, 4, cycles can be derived in various ways from the counting numbers (e.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. When you are examining a string of numbers for a pattern, follow these steps: to think of repeating cycles on their own terms, separately from the counting numbers and related patterns.


GMAT sequences and patterns note: Compute the first five to eight terms and try to match them to a pattern that you already know. The most basic pattern is the counting numbers, but you should also have these related patterns up your sleeve:.


GMAT sequences and patterns note: For each of these sequences, notice exactly where the counting numbers come into play. For instance, in the squares series, the counting numbers are the bases, but in the powers series, the counting numbers are the exponents.


GMAT sequences and patterns note: Multiples (e.g., 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, 7 × 5 = 35 a. As soon as you generate a repeated term, see whether the sequence will repeat itself from that point onward.


GMAT sequences and patterns note: Repeating cycles on the GMAT typically begin repeating every four terms (or fewer), so five to eight terms should be sufficient to identify the pattern. Of course, some cycles repeat every single term —that is, it’s the same number over and over again!


GMAT sequences and patterns note: If you are stuck, look for patterns within differences between terms or sums across terms.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Look at the difference between consecutive terms.


GMAT sequences and patterns note: For instance, this process can help you spot linear sequences (sequences of multiples plus a constant): a. Also look at the cumulative sum of all the terms up to that point.


GMAT sequences and patterns note: This is helpful if the terms get closer to zero or alternate in sign:. Notice that the cumulative sum for this sequence approaches 1.


GMAT sequences and patterns note: Some sums involve matching pairs that sum to the same number (or even cancel each other out). Be on the lookout for such matching pairs.


GMAT sequences and patterns note: You can, of course, sum these numbers in order, but look to make natural intermediate sums (subtotals) with matching pairs. Working outward, 10 and 12 sum to 22 as.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. O en, the pattern will be completely hidden until you actually compute the first several values of the sequence or other initial results.


GMAT sequences and patterns note: Several types of problems frequently involve underlying patterns. When you see these types of problems on the GMAT, be ready to analyze the pattern so you can find the rule: well.


GMAT sequences and patterns note: So do 8 and 14, 5 and 17, and 1 and 21. In all, there are five subtotals of 22, for a grand total of 110.


GMAT sequences and patterns note: Look at characteristics of the numbers: positive/negative, odd/even, integer/non-integer, etc. Once you have extended the pattern for several terms, these characteristics will generally repeat or alternate in some predictable way.


GMAT sequences and patterns note: Nearly all sequence problems involve a pattern in the elements (or terms) of the sequence. Sequences can be defined either directly (i.e., each value in the series is a function of its location in the order of the sequence) or recursively (i.e., each value is a function of the previous items in the sequence).


GMAT sequences and patterns note: Units (Ones) Digit Problems:. Questions involving the last digit (sometimes called the units or ones digit) of an integer almost always involve some sort of repeating cycle pattern that can be.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Integer is a special case of a remainder: it’s the remainder a er division by 10.


GMAT number properties note: Some pattern problems do not involve deciphering a string of numbers and discovering the rule. For instance, you may have to count a set of numbers that all fit some constraint.


GMAT number properties note: The point is to discover a simple rule or group of rules that let you account for all the numbers—and therefore count them —without having to generate each one. Break the problem into sub-problems.


GMAT number properties note: For instance, a sum may be split into several smaller sums. Or you might count a larger total, then subtract items that do not fit the constraint.


GMAT number properties note: You even might multiply a larger total by the proportion of suitable items, if that fraction is easy to calculate. Recall counting and summing methods from Manhattan Prep’s All the Quant guide:.


GMAT number properties note: When you have a series of successive decisions, you multiply the number of choices you have at each stage to find the number of total choices you have. For instance, if you can choose 1 appetizer out of 6 possible appetizers and 1 main course out of 7 possible main courses, then you could have 6 × 7 = 42 possible meals.


GMAT number properties note: Number of Items in a Consecutive Set of Integers:. The number of integers in a consecutive set of integers equals.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Average of the largest integer and the smallest integer.


GMAT statistics note: This is also equal to the median, or “middle,” number in the set. As you go, always check that the extreme cases are still valid.


GMAT statistics note: Two or three constraints can interact in surprising ways, eliminating some of the values that would seem to work otherwise.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Sequence Problems Any question that involves the definition of a sequence (usually involving subscripted variables, such as An and Sn) is very likely to involve patterns.


GMAT sequences and patterns note: These patterns can range from relatively straightforward linear patterns to much more complicated ones. When you are given a sequence definition, list a few terms of the sequence, starting with any particular terms you are given, and look for a pattern.


GMAT sequences and patterns note: Do not be intimidated by a recursive definition for a sequence, in which each term is defined using earlier terms. (By contrast, a direct definition defines each term using the position or index of the term.) To illustrate the difference, here are two ways to define the series of positive odd integers {1, 3, 5, 7, 9, etc.}:.


GMAT sequences and patterns note: Recursive Definition Direct Definition An = An − 1 + 2 where n > 1 and A1 = 1 Translation: “This term = the previous term + 2, and the first term is 1.” An = 2n − 1, where n ≥ 1 Translation: “This term = the index number × 2, minus 1. Thus, the first term is (2)(1) − 1 = 1.”.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Try-It #5-2 The sequence Xn is defined as follows:.


GMAT sequences and patterns note: Xn = 2Xn − 1 − 1 whenever n is an integer greater than 1. If X1 = 3, what is the value of X20 − X19?


GMAT sequences and patterns note: The pattern underlying this sequence is not obvious, so begin computing a few of the terms in the set: n Xn 2 2(3) − 1 = 5 3 2(5) − 1 = 9 4 2(9) − 1 = 17 5 2(17) − 1 = 33 6 2(33) − 1 = 65 7 2(65) − 1 = 129.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. You might notice that there appears to be a repeating pattern among the not help to answer the question, which asks about the difference between two consecutive elements later in the set.


GMAT sequences and patterns note: Instead, look at the differences between consecutive elements: n Xn Xn − Xn − 1 2 2(3) − 1 = 5 5 − 3 = 2 3 2(5) − 1 = 9 9 − 5 = 4 4 2(9) − 1 = 17 17 − 9 = 8 5 2(17) − 1 = 33 33 − 17 = 16 6 2(33) − 1 = 65 65 − 33 = 32 7 2(65) − 1 = 129 129 − 65 = 64 The pattern quickly emerges: the difference between consecutive terms in the sequence appears to always be a power of 2.


GMAT sequences and patterns note: Specifically, X2 − X1 = 2 = 21, X3 − X2 = 4 = 22, X4 − X3 = 8 = 23, etc.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. You can determine the pattern for the sequence:.


GMAT sequences and patterns note: This is a difference pattern—a pattern or rule that exists among the differences between consecutive terms in the sequence. Be careful at this last step!


GMAT sequences and patterns note: When you extrapolate the pattern, you might accidentally think that the number you want is 220. Always explicitly match to the index, and realize that you might be slightly shi ed.


GMAT sequences and patterns note: In this case, the difference you want is not 2n. Try-It #5-3 If for all positive integers n, what is the sum of the first 100 elements of An?


GMAT sequences and patterns note: Once again, compute the first few elements of An. Because you need to know the sum of the first 100 elements, also track the cumulative sum: n An Sum through An.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. N An Sum through An The sum of the first n terms of An equals .


GMAT sequences and patterns note: Therefore, the sum of the first 100 terms is . This is a summing pattern—a pattern or rule that exists among the cumulative sum of the terms in the sequence.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Units (Ones) Digit Problems When you raise an integer to a power, the units digit always displays some kind of pattern as you increase the power.


GMAT sequences and patterns note: Try-It #5-4 What is the units digit of 4674?


GMAT sequences and patterns note: Observe what happens to the units digit of the consecutive powers of 4, starting with 41: last digit of 4 = 4 last digit of 4(4) = last digit of 16 = 6 last digit of 4(6) = last digit of 24 = 4 last digit of 4(4) = last digit of 16 = 6 Because the computations (4 × 4 = 16) and (4 × 6 = 24) keep repeating, the units digit will continue to alternate [4, 6].


GMAT sequences and patterns note: Thus, 4x will have a units digit of 4 whenever x is odd and a units digit of 6 whenever x is even (assuming, of course, that x is positive). The units digit of 4674 is therefore 6.


GMAT sequences and patterns note: Also notice that in determining the value of the units digit of a product, all of the other digits besides the units digit are irrelevant.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. And 3,184674 will both also have units digits of 6.


GMAT sequences and patterns note: This is also true for multiplication of any two integers, as well as the addition of any integers:. As mentioned earlier, every integer raised to different positive exponents has a units digit pattern.


GMAT sequences and patterns note: As an exercise, derive several of the patterns yourself for the units digits 2, 3, 5, 7, and 8. You can check your work using the table below:.


GMAT sequences and patterns note: Series Consecutive Powers Units Digit Pattern 1x 1.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Series Consecutive Powers Units Digit Pattern 7x 7.


GMAT sequences and patterns note: [0] You can either memorize this chart or know how to regenerate these patterns quickly. The units digits 1, 5, 6, and 0 just repeat the same digit forever.


GMAT sequences and patterns note: There’s no real pattern to memorize. That leaves you with six possibilities to memorize (or to re-create when you need them).


GMAT sequences and patterns note: Note that no pattern goes beyond four numbers before repeating, so you don’t have to check beyond the first four terms of any pattern. Try-It #5-5 What is the units digit of 1940?


GMAT sequences and patterns note: Therefore, the pattern is a two-term just like odd and even integers. Since 40 is an even number, the units digit of 1940 will equal 1.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. What is the remainder when 1940 is divided by 10?


GMAT sequences and patterns note: This alternative question is asking the exact same thing as the original question. The remainder whenever an integer is divided by 10 will always be the same as the units digit of the original number.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Remainder Problems In general, remainders provide a means by which the GMAT can disguise an underlying pattern.


GMAT number properties note: The number of terms in the repeat equals the divisor in this case. Conversely, you can calculate all of the numbers that have a certain remainder when divided by a certain value, because they appear at regular intervals as well.


GMAT number properties note: For instance, the numbers that have a remainder of 3 when divided by 4 are 3, 7, 11, 15, 19, 23, etc. Notice that those numbers are evenly spaced exactly 4 apart.


GMAT number properties note: When a problem discusses remainders, look for patterns and take advantage of them. Try-It #5-6 If x and y are positive integers, what is the remainder when 5x is divided by y?


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The problem provides no information about the value of y, however.


GMAT sequences and patterns note: If y = 6, then the remainder is 5. This statement may not initially appear to be sufficient, but test some different values for x: x 5x Remainder of The pattern is clear: no matter what exponent 5 is raised to, the remainder when divided by 4 will always equal 1—a fact that you probably did not expect before testing the rule for this problem.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If you used both statements, you could calculate the exact remainder.


GMAT sequences and patterns note: However, that would be too easy for a tough GMAT Quant problem. You don’t actually need both statements to determine that the answer to the question is 1.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Other Pattern Problems Many questions will not at first glance demonstrate an obvious pattern.


GMAT number properties note: The point is to discover a simple rule or group of rules that let you account for all the possibilities—and therefore count or maximize them—without having to generate each possibility separately. Try-It #5-7 How many of the integers between 1 and 400, inclusive, are not divisible by 4 and do not contain any 4’s as a digit?


GMAT number properties note: This problem involves a counting pattern. It’s clear that there are 400 integers between 1 and 400, inclusive.


GMAT number properties note: You’ll need to subtract the integers that are divisible by 4 or contain a 4 as a digit. The tricky part is the overlap: some numbers, such as 64 and 124, violate both constraints.


GMAT number properties note: It is easier to determine the number of multiples of 4. Since there are 400 integers in the set, and those 400 integers are consecutive, there must be integers that are divisible by 4.


GMAT number properties note: There are 400 − 100 = 300 integers remaining.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Next, consider the remaining numbers that have a 4 among their digits.


GMAT number properties note: These numbers have a 4 among their digits but are not themselves multiples of 4. Only one of the integers between 1 and 400, inclusive, has a 4 in the hundreds place: 400.


GMAT number properties note: You have already eliminated that one from the count because it is a multiple of 4. To eliminate integers with a 4 in the tens place, of the form x4y, count only those that are not multiples of 4.


GMAT number properties note: These are the numbers whose last two digits are 41, 42, 43, 45, 46, 47, or 49. There are seven such numbers in each set of “hundreds,” that is, the 300’s, the 200’s, the 100’s, and the no hundreds.


GMAT number properties note: That is a total of 7 × 4 = 28 terms. There are 300 − 28 = 272 integers remaining.


GMAT number properties note: Last, count and eliminate the integers with 4 in the units digit, of the form xy4, that have not already been subtracted. These are the numbers whose last two digits are 14, 34, 54, 74, or 94 (numbers with an even integer in the tens place and 4 in the units are all divisible by 4, so they have been eliminated already).


GMAT number properties note: There are 5 such numbers in each set of “hundreds,” so that is a total of 5 × 4 = 20 terms. There are 272 − 20 = 252 integers remaining.


GMAT number properties note: Try-It #5-8 x = 1010 − z, where z is a two-digit integer.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The first thing to do in solving this problem is to subtract any two-digit number from 1010 and look for a pattern.


GMAT sequences and patterns note: Notice the pattern: the first eight digits of x are all 9’s, so those digits sum to 72. This will be true no matter which two-digit integer you try for z.


GMAT sequences and patterns note: Therefore, the final two digits of x must sum to 84 − 72 = 12. What possibilities would work for these final two digits of x?


GMAT sequences and patterns note: 39, 48, 57, 66, 75, 84, and 93 all add to 12. When subtracted from 100, these numbers will have to produce a two-digit integer.


GMAT sequences and patterns note: Try subtracting from 100 to find the pattern:. Each successive number is larger, so it will result in a smaller two- digit integer.


GMAT sequences and patterns note: Do the numbers actually drop to one digit at some point? Jump to the other end of the scale and try 93:.


GMAT sequences and patterns note: In order for x to end in 93, z would have to be 7, which is not a two-digit integer. According to the GMAT, a two-digit integer must have a nonzero.


GMAT sequences and patterns note: Track mean, median, range, distribution, and how operations change spread. Tens digit and zeros for all higher places.


GMAT sequences and patterns note: What about the next number up? If x ends in 84, then it would result in the two-digit integer 100 − 84 = 16.


GMAT sequences and patterns note: All of the other numbers are two-digit integers. Therefore, the final two digits of x can only be 39, 48, 57, 66, 75, and 84, resulting in six possible values for z.


GMAT sequences and patterns note: As the two previous problems demonstrate, unusual patterns can appear in problems on the GMAT. Sometimes you must “think outside the box” to identify the wanted pattern.


GMAT sequences and patterns note: O en, these repeating patterns can only be identified by listing out a few values in the pattern. An = n + k, where n and k are integers.


GMAT sequences and patterns note: Note that the average term = the median term = (First + Last).


GMAT sequences and patterns note: Track mean, median, range, distribution, and how operations change spread. An = 7n, where n is a set of consecutive integers.


GMAT sequences and patterns note: The evens are just a special case (multiples of 2, or 2n). Note that the average term = the median term = × (First + Last).


GMAT sequences and patterns note: Evenly Spaced Sets 9, 16, 23, 30, etc. (Constant difference of 7 between consecutive terms) When dividing this series by 7, each of the terms leaves a remainder of 2.


GMAT sequences and patterns note: Can be defined as a multiple plus/minus a constant:. The odds are a special case (multiples of 2, plus 1, or 2n + 1).


GMAT sequences and patterns note: Note that the average term = the median term = × (First + Last), as for consecutive multiples. Non- Uniform Spacing that Itself Follows a Pattern 0, 1, 3, 6, 10, 15, etc.


GMAT sequences and patterns note: (Spacing between terms follows 1, 2, 3, 4, 5, etc. (Spacing between squares = 1, 3, 5, 7, 9, etc.


GMAT sequences and patterns note: = the odd integers!) Alternating Sign −3, 3, etc. Can result from a (−1)n term in a direct sequence definition, or a (−(An − 1)) term in a recursive sequence definition.


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


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. What is the remainder when the 75th term is divided by 9?


GMAT sequences and patterns note: If x and y are integers between 0 and 9, inclusive, and the units digit of xy is 5, what are the possible values of x and y? What is the remainder when 1317 + 1713 is divided by 10?


GMAT sequences and patterns note: If y is a positive integer, what is the units digit of y? The units digit of y2 equals 6.(1) The units digit of (y + 1)2 equals 5.(2) 5.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The units digit of y2 = 1.(1) The units digit of y does not equal 1.(2) 6.


GMAT sequences and patterns note: In sequence A, A1 = 1, A2 = 100, and the value of An is strictly between the values of An – 1 and An – 2 for all n ≥ 3. Which of the following must be true?


GMAT sequences and patterns note: If x is an integer, what is the remainder when x is divided by 5? X2 has a remainder of 4 when divided by 5.(1) x3 has a remainder of 2 when divided by 5.(2) 8.


GMAT sequences and patterns note: If x and y are positive integers, what is the remainder when 5x is divided by y? X is an even integer.(1) y = 3(2).


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. A, b, c, and d are positive integers.


GMAT sequences and patterns note: If has a remainder of 9 and has a remainder of 10, what is the minimum possible value for bd? What is the sum of the numbers in the grid below?


GMAT sequences and patterns note: 7 −8 −4 5 12 9 −10 −5 6 15 11 −12 −6 7 18 all n > 1. If a1 = 10, what is the value of a11?


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. In sequence Q, the first number is 3, and each subsequent number in the sequence is determined by doubling the previous number and then adding 2.


GMAT sequences and patterns note: How many times does the digit 8 appear in the units digit of the first 10 terms of the sequence? If ♦ K♦ is divisible by 411, what is the smallest possible value for K?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Mitchell plans to work at a day camp over the summer.


GMAT fractions, decimals, and percents note: Each week, he will be paid according to the following schedule:. At the end of the first week, he will receive $1.


GMAT fractions, decimals, and percents note: At the end of each subsequent week, he will receive $1, plus an additional amount equal to the sum of all payments he’s received in previous weeks. How much money will Mitchell be paid in total during the summer, if he works for the entire duration of the 8-week-long camp?


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The sequence starts with the number 4, then adds 5 to each subsequent term.


GMAT sequences and patterns note: The math plays out in this way:. The sequence pattern is 4 + 5(n − 1), where n is the number of the term.


GMAT sequences and patterns note: The 75th term in the sequence is therefore 4 + 5(75 − 1) = 4 + 5(74) = 374. In order to find the remainder, first find the multiple of 9 closest to 374 but smaller than that number.


GMAT sequences and patterns note: 360 is a multiple of 9, and so is 369. Therefore, has a remainder of 0, so has a remainder of 5.


GMAT sequences and patterns note: Only the integer 5 can be raised to a power to result in a units digit of 5. Any power of 5 will have a units digit of 5, other than zero, because 50 = 1.


GMAT sequences and patterns note: The integer y, on the other hand, can have any value except for 0. The remainder when dividing an integer by 10 always equals the units digit.


GMAT sequences and patterns note: Ignore all but the units digits and rephrase the question: “What is the units digit of 317 + 713?”.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. The pattern for the units digits of 3 is [3, 9, 7, 1].


GMAT sequences and patterns note: Every fourth term is the same. The 17th power is 1 past the end of the repeat.


GMAT sequences and patterns note: Since 316 ends in 1, 317 must end in 3. The pattern for the units digits of 7 is [7, 9, 3, 1].


GMAT sequences and patterns note: The 13th power is 1 past the end of the repeat. Since 712 ends in 1, 713 must end in 7.


GMAT sequences and patterns note: The sum of these units digits is 3 + 7 = 10. For problems 4 and 5, reference the following chart:.


GMAT sequences and patterns note: For integers ending in a certain digit, the table shows the units digit pattern. Integers ending in 9 have a units digit pattern of 9, 1.


GMAT sequences and patterns note: 1 2 3 4 5 6 7 8 9 0 y1 1 2 3 4 5 6 7 8 9 0 y2 1 4 9 6 5 6 9 4 1 0 y3 1 8 7 4 5 6 3 2 9 0 y4 1 6 1 6 5 6 1 6 1 0 4. According to the Units Digit Patterns chart above, both 4 and 6 yield a units digit of 6 when raised to the second power.


GMAT sequences and patterns note: (Notice that you would only need to check even numbers for this.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Statement, as odd numbers to any power cannot end in a 6, an even number.) (2) SUFFICIENT:.


GMAT sequences and patterns note: Only 5 yields a units digit of 5 when raised to any power. Since the units digit of y + 1 is 5, the units digit of y must be 4.


GMAT sequences and patterns note: The statement indicates only that the units digit does not equal 1, but it could still be any other digit. Because both 1 and 9 yield a units digit of 1 when raised to the second power, there are two possible values for the units digit.


GMAT sequences and patterns note: The two statements together indicate that the units digit of y must be 9. Since there isn’t time to calculate that many terms, there must be a pattern.


GMAT sequences and patterns note: Write down what you know about the first few terms in order to identify that pattern. Each term in the sequence is strictly between the two terms that come immediately before it.


GMAT sequences and patterns note: To keep track of this, draw a number line on your.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Put the third term of the sequence somewhere between the first two terms:.


GMAT sequences and patterns note: Continue to add terms to the number line, making sure to put each term between the two that came before it. The first eight terms have been plotted on the number line shown below:.


GMAT sequences and patterns note: The odd-numbered terms of the sequence appear in increasing order, so that A1 < A3 < A5 < A7, and so on. The even-numbered terms are in decreasing order so that A2 > A4 > A6 > A8, etc.


GMAT sequences and patterns note: Because the question is asking about the remainder when x is divided by 5, you only need to check the units digits 0 through 4. The pattern will recycle for the next set of 5 (5 through 9) and for every set of 5 a er that.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. X x2 = remainder of 4 = remainder of?


GMAT sequences and patterns note: 2 4 Yes = 0 remainder 2 3 9 Yes = 0 remainder 3 There are at least two possible values for the remainder. (2) SUFFICIENT: x x3 = remainder of 2 = remainder of?


GMAT sequences and patterns note: 3 27 Yes = 0 remainder 3 There is only one possible remainder: 3. If you aren’t sure that you only need to test these five cases, try the next five values for x.


GMAT sequences and patterns note: Note the repeat of the remainder pattern. 0 0 No (invalid) 1 1 No (invalid) 0 0 No (invalid) 1 1 No (invalid) 2 8 No (invalid) 4 64 No (invalid).


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. If x is even, then 5x = 25, 625, and so on.


GMAT sequences and patterns note: This statement provides no information about y, though. Test some different values for x.


GMAT sequences and patterns note: Because you will have to test the two statements together, continue testing another couple of numbers to see whether there is a pattern. When 5 is raised to an odd power, the remainder is 2, but when 5 is raised to an even power, the remainder is 1.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. When 5even integer is divided by 3, the remainder is always 1.


GMAT number properties note: The remainder must always be smaller than the divisor. Thus, b must be at least 10, and d must be at least 11.


GMAT number properties note: Therefore, bd must be at least 110. The purpose of this problem is to remind you of these constraints on remainders.


GMAT number properties note: Each column contains a series of evenly spaced integers. To avoid calculation, cancel out as many values as possible before summing the remaining values.


GMAT number properties note: In each row, the sum of the integers in the first two columns is −1, and the sum of the integers in the third and fourth columns is 1. Therefore, the sum of the integers in the first four columns in each row is −1 + 1 = 0.


GMAT number properties note: Only the integers in the fi h column need to be considered. The integers in the fi h column can be rewritten as follows:.


GMAT number properties note: The sum of the integers is therefore 3(21) = 63. Write out the first few terms of the sequence to find the pattern.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. To simplify the calculation, these terms can be rewritten as follows:.


GMAT number properties note: Therefore, the nth term in the sequence can be found by adding 10 a total of n times, then adding the sum of the integers from 1 to n – 1, which equals 10(11) + 55 = 165. Compute the first few elements of Sn.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. N Sn Use a number line to track the sum:.


GMAT word problem note: Place the first term, , on the number line. The second term is + so the sum will move to the right (closer to 0) on the number line.


GMAT word problem note: The third term is so the sum will move to the le, but it can’t go as far as again, because you’re only subtracting this time—the distance is smaller than the first hop that you made. Each subsequent hop flips back and forth between positive and negative but also keeps getting smaller and smaller, so you’ll never “break out” of the range to 0.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Calculate the first several terms of the sequence to find the pattern:.


GMAT sequences and patterns note: The pattern should continue, so 8 will be the units digit 9 out of the first 10 times. This is a counting pattern problem.


GMAT sequences and patterns note: In order for ♦ K♦ to be divisible by 411, it must be divisible by (22)11 = 222. Thus, ♦ K♦ must contain 22 twos in its prime factorization.


GMAT sequences and patterns note: The number 2 has one 2 in its prime factorization. This amounts to a total of only eight 2’s:.


GMAT sequences and patterns note: Number 2 4 6 8 10 Prime Factor(s) 2 2, 2 2, 3 2, 2, 2 2, 5 Total 2’s in PF 1 2 1 3 1 Cumulative 2’s 1 3 4 7 8.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Keep adding even numbers to the result until you get to 22 twos in total:.


GMAT number properties note: Number 2 4 6 8 10 12 14 16 18 20 22 24 Prime Factor(s) Total 2’s in PF 1 2 1 3 1 2 1 4 1 2 1 3 Cumulative 2’s 1 3 4 7 8 10 11 15 16 18 19 22 Thus, the smallest possible value for K is 24. At the end of the first week, Mitchell receives $1.


GMAT number properties note: At the end of the second week, he gets $1, plus $1 for the total he had been paid up to that point, for a total of $2. At the end of the third week, he gets $1, plus ($1 + $2), or $3, for the total he had been paid up to that point, so this third week’s total is $4.


GMAT number properties note: Week # Paid This Week($) Cumulative Pay Including This Week ($).


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


GMAT sequences and patterns note: Week # Paid This Week($) Cumulative Pay Including This Week ($) 2 1 + 1 = 2 1 + 2 = 3 3 1 + 3 = 4 3 + 4 = 7 4 1 + 7 = 8 7 + 8 = 15 5 1 + 15 = 16 15 + 16 = 31 6 1 + 31 = 32 31 + 32 = 63 7 1 + 63 = 64 63 + 64 = 127 8 1 + 127 = 128 127 + 128 = 255 This calculation is not so bad, but you may notice that this payment schedule is a geometric sequence, 2n − 1, where n is the number of the week in which Mitchell is being paid.


GMAT sequences and patterns note: Summing that sequence is equivalent to 2t − 1, where t is the total number of weeks. In other words, the cumulative pay is one less than the next power of 2.


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. Common Terms Quadratic Templates Quadratic Templates in Disguise.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. But of all the steps you could take, how do you decide which steps you should take?


GMAT algebra note: Two indicators can o en help you on the GMAT:. You can substitute one expression for another if they are equal.


GMAT algebra note: You can add some number to one side of an equation as long as you do the same on the other side of the equation. You can cross-multiply to simplify equations with fractions on each side of the equals sign.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Common Terms If you spot common terms, you can o en spot the path all the way to the solution.


GMAT algebra note: Common terms appear on the GMAT in three typical ways. ALGEBRA Look for terms that appear in the same form more than once.


GMAT algebra note: Those recurring expressions might also appear in slightly modified form such as reciprocal, negative, or raised to a power: is the reciprocal of is negative Is ? Where and 3y appear both squared and multiplied together.


GMAT algebra note: This expression is of the form: (More on Quadratic Templates later in this chapter).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #6-1 If y ≠ 3, simplify as much as possible:.


GMAT number properties note: Spot the common term (3 − y). Note that (−3 + y) is −(3 − y), or −1 × (3 − y).


GMAT number properties note: Factor out the common term (3 − y) and cancel:. By the way, the condition that y could not equal 3 just prevented you from dividing by 0 and ending up with an undefined number.


GMAT number properties note: EXPONENTS Exponents can be manipulated when either bases or exponents are common. Also look for bases that have common factors, such as 3 and 12 (common factor of 3).


GMAT number properties note: You can o en create a common base. Similarly, 4, 16, and 256 are all powers of 4.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #6-2 If 3x + 243 = 2(3x), what is the value of x?


GMAT algebra note: Note the common term 3x, and note the fact that 243 = 35: 3. FACTORS AND MULTIPLES When many terms share a factor, pull that shared factor out to the side.


GMAT algebra note: These can appear in algebraic or numerical expressions: → get common denominators, then cross them all off → x18 + 2x16 + x14 → x14 is a factor of each term → x14(x4 + 2x2 + 1) = x14(x2 +.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Factorials are particularly noteworthy, as they o en have an abundance of shared factors.


GMAT number properties note: For any integer n, the factorial n! Is calculated as follows: n!


GMAT number properties note: = (6)(5)(4)(3)(2)(1) = (6)(5)(4!). More generally, factorials are “super multiples.” Without ever computing their precise value, you can tell that they’re divisible by all sorts of numbers.


GMAT number properties note: If x is an integer between 7! X is one of the following integers: x has one of the following factors, if x is: 7!


GMAT number properties note: + 3 = 3 × Integer not prime! Sometimes a common factor is just a random number buried inside a couple of larger numbers.


GMAT number properties note: Find it and pull it out: 10’s in the numerator line up with 5’s in the corresponding digit place of the denominator.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Try-It #6-3 If n is a positive integer and is a positive integer, what is the value of n?


GMAT number properties note: What would need to be true in order for the square root to be a positive integer? The number under the square root symbol would have to be a perfect square.


GMAT number properties note: Rearrange the expression to determine whether there are any restrictions that could help narrow down the possibilities before going to the statements. Try to break the numbers down into primes to locate and pull out any existing perfect squares: n is prime.(1) n < 3(2).


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. It turns out that you can also pull out the term 7n − 1.


GMAT quant note: The term 7n + 1 = (7n − 1) The first two terms are both perfect squares and can be pulled out of the square root sign:. What would need to be true in order for 7n − 1 to be pulled out of the square root sign?


GMAT quant note: It would also have to be a perfect square, so n – 1 must be even and n itself must be odd.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. The question is: “If n is odd, what is the value of n?” (1) INSUFFICIENT:.


GMAT quant note: This statement allows multiple possible odd values of n. The question stem indicates that n is a positive integer, and this statement specifies that n < 3.


GMAT quant note: The only odd, positive integer less than 3 is the number 1.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Quadratic Templates On the GMAT, quadratic expressions take three common forms called the Quadratic Templates.


GMAT algebra note: Memorize these templates, and get comfortable transforming back and forth between factored and distributed form:.


GMAT algebra note: Factored Distributed Square of a Sum (a + b)2 = a2 + 2ab + b2 Square of a Difference (a − b)2 = a2 − 2ab + b2 Difference of Two Squares (a + b) (a − b) = a2 − b2 QUICK MANIPULATION Expressions with both squared and non-squared common terms should make you suspect that you are looking at a Quadratic Template.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This problem requires you to manipulate a rather complicated expression.


GMAT algebra note: However, by using the common terms, you can put the problem in the more basic template form to solve:. Once you are comfortable with Quadratic Templates, you can manipulate even complicated expressions quickly, as in the middle box above.


GMAT algebra note: Until then, write down the templates and the substitution of the common terms, as in the box on the right. The very same problem could have been presented in disguise:.


GMAT algebra note: Factor The common terms are slightly harder to spot in this form. In such a case, start with the squared terms, and 25y2.


GMAT algebra note: Then, try to untangle their square roots, and 5y, from the remaining term.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. THE MIDDLE TERM: 2AB The square of a sum and square of a difference templates have something in common: the middle term is ±2ab.


GMAT algebra note: The only difference is the sign of that middle term. When you add these two templates, the middle terms cancel, leaving the end terms:.


GMAT algebra note: Factored Distributed Square of a Sum (a + b)2 = a2 + 2ab + b2 + Square of a Difference (a − b)2 = a2 − 2ab + b2 Addition: (a + b)2 + (a − b)2 = 2a2 + 0 + 2b2 2(a2 + b2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. In contrast, when you subtract these two templates, the end terms cancel, leaving the middle term:.


GMAT algebra note: Factored Distributed Square of a Sum (a + b)2 = a2 + 2ab + b2 + Square of a Difference (a − b)2 = a2 − 2ab + b2 Subtraction: (a + b)2 − (a − b)2 = 0 + 4ab + 0 This is handy for simplification. Also, whenever you see the sum of two squares (a2 + b2), which is not itself a Quadratic Template, remember that it can be derived from this sum of two templates.


GMAT algebra note: Try It #6-5 What is the sum of 9,9992 and 10,0012?


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. These numbers all look seriously cumbersome.


GMAT algebra note: If only they had given the When wishful thinking pops up, try to use it to make the problem easier. If you changed both of these numbers to the form 10,000 plus or minus a number, what would that be?


GMAT algebra note: These are the Quadratic Templates! In this case, a = 10,000 and b = 1.


GMAT algebra note: If you’ve memorized the sum term, plug these in:. Now, notice something: the part in the parentheses is going to have a units digit of 1.


GMAT algebra note: Multiply the number by 2, and the end result will have a units digit of 2. Even if you don’t memorize the sum term of the Quadratic Templates, the math is still far easier to do in this rewritten form.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Quadratic Templates in Disguise Quadratic Templates can be disguised in arithmetic computations.


GMAT algebra note: Try-It #6-6 What is 198 × 202? You can round each number and quickly estimate the result to be about 2002.


GMAT algebra note: Or, you could laboriously multiply two 3-digit numbers by hand to get an exact result. But if you need an exact result quickly, you can use a Quadratic Template, as shown in the previous section.


GMAT algebra note: You just need to turn 198 into (200 − 2) and 202 into (200 + 2) as shown here:. Another place to hide a Quadratic Template is in an advanced right-triangle problem.


GMAT geometry note: You know that and d2 = b2 + h2 (by the Pythagorean theorem). Do the common terms b2, h2, and bh look familiar?


GMAT geometry note: Use the Square of a Sum template:. Likewise, there is a similar relationship based on the Square of a Difference template:.


GMAT geometry note: An advanced GMAT problem can draw on these complicated relationships. For instance, you can compute the area of a right triangle directly from the sum of the shorter sides and the hypotenuse.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. You should absolutely not memorize these particular formulas.


GMAT algebra note: Rather, be able to recognize when the GMAT is indirectly testing these generic Quadratic Templates.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. For problems 2–4, if x < –1, which of the following inequalities must be true?


GMAT fractions, decimals, and percents note: If, and |x| ≠ |y|, what is the ratio of x to y?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If , what is the value of 7.


GMAT number properties note: If (ax + by)2 = (bx + ay)2, what is the value of a – b? X2 > y2(1) a and b are positive integers.(2) 9.


GMAT number properties note: If x and k are both integers, x > k, and x–k = 625, what is x? |k| is a prime number.(1) x + k > 20(2).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Distribute the expression in problems 10–14 without FOILing (doing the math the long way).


GMAT algebra note: Use the Quadratic Templates. Factor problems 15–19 according to the Quadratic Templates.


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


GMAT geometry note: In the right triangle below, side a is 7 inches longer than side b. If the area of the triangle is 30 inches2, what is the length of hypotenuse c?


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. From xy > 0, you know that neither x nor y equals 0, and they must have the same sign (++ or − −).


GMAT algebra note: Combine the exponent in the question to get What is the significance of the inequality in the question stem? If the exponent equals 2, then 52 = 25.


GMAT algebra note: In order to be greater than 25, the exponent has to be greater than 2. If, on the other hand, y is negative, then (1) INSUFFICIENT:.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, y3 must equal 8, so y = 2.


GMAT algebra note: This provides no information about x, however. For questions 2–4: since x ≠ 0, divide by the common terms, making sure to flip the inequality sign if the common term is negative.


GMAT algebra note: Divide both sides by x2, leaving the sign as it is. The square of a number smaller than −1 will be greater than positive 1.


GMAT algebra note: (Or, you might have subtracted x3 immediately.) TRUE:. Divide both sides by x5, flipping the inequality sign.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Try to find like terms in order to simplify the le -hand side of the equation:.


GMAT algebra note: The ratio of x to y is 2: 1. Since n is an integer, 4n is even.


GMAT algebra note: An even exponent “hides the sign” of the base, so you can treat the (−3) base as a (3).


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Begin by simplifying the equation in the question stem, making it look as similar as possible to the question itself:.


GMAT algebra note: Do not divide by x2 – y2, since it may equal 0. Instead, subtract b2(x2 – y2) from both sides of the equation, then factor:.


GMAT algebra note: So either a2 – b2 = 0 (in which case a2 = b2), or x2 – y2 = 0 (in which case x2 = y2). One or both of these must be true.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x2 > y2, then x2 cannot equal y2.


GMAT number properties note: It is possible that a and b are equal, in which case a – b = 0, so the answer is 0. It is also possible that a and b are equal in absolute value but have opposite signs.


GMAT number properties note: In this case, the answer is a – b = –4. More than one answer is possible, so this statement is not sufficient.


GMAT number properties note: This statement only says that a and b are positive integers, but does not give further indication of what they could be. There are many possible answers for the value of a – b, so this statement is insufficient.


GMAT number properties note: Thus, either a = b or a = –b. Therefore, a = b, and the answer to the question is a – b = 0.


GMAT number properties note: The statements together are sufficient. The fact that x and k are both integers (and that x > k) significantly limits the possible values for x and k.


GMAT number properties note: The possible pairings are as follows: x k.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If the absolute value of k is prime, then only the second possibility works: x = 25 and k = −2.


GMAT number properties note: The second and third possibilities both make this statement true, so it isn’t possible to determine a single value for x.


GMAT quant note: From the Pythagorean theorem, a2 + b2 = c2. Use the Square of a Difference template.


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. Representing Objects with Pictures Rubber Band Geometry Baseline Calculations for Averages Number Line Techniques for Statistics Problems.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Still, visual thinking is a powerful tool.


GMAT word problem note: It can expand your comprehension of a topic. It may enable you to solve particular problem types more easily or “break through” on a difficult problem.


GMAT word problem note: In fact, visualization is the only realistic way to approach certain problems. So it’s worth trying your hand with visual approaches.


GMAT word problem note: In this chapter, we will discuss the following Visual Solution techniques:. Representing Objects with Pictures:.


GMAT word problem note: Drawing a good picture will make the problem-solving process easier and less error-prone.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Different “rubber band” scenarios according to those constraints and freedoms can o en help you solve the problem without doing any computation.


GMAT statistics note: Baseline Calculations for Averages:. Visual techniques can help you compute averages (both basic and weighted) and can also foster a better understanding of those calculations.


GMAT statistics note: Number Line Techniques for Statistics Problems:. You can solve a variety of common problems involving statistics by using a number line to visualize and manipulate the problem.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Sometimes it’s the only viable approach!


GMAT word problem note: Even if you are good at visualizing objects in your head, draw the picture anyway. It’s just too easy to make a mistake on many of these questions.


GMAT word problem note: Try-It #7-1 A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into -inch cubes, what fraction of the resulting cube faces are painted?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Total cubes = (4 inches × 2 cubes per inch) × (1 × 2) × (1 × 2) = 32 cubes Total cube faces = 32 cubes × 6 faces per cube = 192 faces total Now, consider the faces that were painted on the front and back of the dowel, the top and bottom of the dowel, and the ends of the dowel.


GMAT fractions, decimals, and percents note: Of course, there are other sides: the back, the bottom, and the other end. Now you can find the number of painted cubes:.


GMAT fractions, decimals, and percents note: Even if you can easily picture 3-D shapes and objects in your head, it is still better to draw a picture on your scrap paper.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The test will usually include among the incorrect answers numbers that you might get by losing track of your progress as you process the object in your mind.


GMAT algebra note: This kind of process can also help you with questions that deal with the relative size of different objects. Try-It #7-2 Bucket A has twice the capacity of Bucket B, and Bucket A has the capacity of Bucket C.


GMAT algebra note: Bucket B is full of water and Bucket C is half full of water. When the water from Bucket B is poured into Bucket C, Bucket C will be filled to what fraction of its capacity?


GMAT algebra note: You could attempt to solve this problem algebraically, but the equations get messy very quickly. Instead, try drawing buckets A, B, and C in correct proportion to one another.


GMAT algebra note: Then think through the problem.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Algebra Picture The algebra and the picture say the same thing, but the picture has several advantages:.


GMAT algebra note: Based on this picture, you might pick a capacity of 1 for Bucket B, yielding a capacity of 2 for Bucket A and 6 for Bucket C. Bucket B would contain 1 unit of water and Bucket C, 3 units.


GMAT algebra note: It’s much easier to comprehend at a glance. It’s harder to mistake relative sizes (e.g., accidentally thinking A is smallest).


GMAT algebra note: You can easily represent both total capacity and amount of water visually. It prompts you to pursue the smartest, easiest solution: picking numbers for the capacities of the buckets.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Notice that the buckets are not labeled in alphabetical order, even though that would be easy to incorrectly assume.


GMAT quant note: The GMAT frequently adds little layers of disguise and complexity such as this to induce you to make a mistake. By drawing the buckets carefully, you minimize the chance that you will fall into a trap on a problem such as this one.


GMAT geometry note: We call these questions Rubber Band Geometry problems, because they simultaneously involve constraints and flexibility. Your job is to figure out what specifics in the problem are constrained and what specifics are flexible.


GMAT geometry note: In fact, it will always “rise” 2 units for every unit of “run.” However, you don’t know where the line will appear. The line is constrained in its slope, but it is flexible in that it can be moved up or down, right or le.


GMAT geometry note: You can draw many different lines with a slope of 2 (these lines will all be parallel, of course). If, however, the problem only specifies that a line must go through the point (4, 0) in the coordinate plane, then the line is “fixed” at that point.


GMAT geometry note: However, the slope of the line would now be flexible. You could draw many different lines with different slopes that run through that point.


GMAT geometry note: If you knew both of these specifications—that the line must have a slope of 2 and must run through the point (.


GMAT algebra note: Calculate the exact line that is being described. The slope of the line and a point that the line goes through specify the line precisely—there is no remaining flexibility for either the slope or the location of the line.


GMAT algebra note: In a DS problem, you could answer any such questions about this line without actually calculating the answer. Therefore for these types of problems, your goal is to figure out what combination of information “cements” the problem in place—in other words, what combination of information removes all of the remaining flexibility.


GMAT algebra note: No flexibility means sufficiency. And by using rubber band geometry thinking, you can o en do this without using any calculation or algebra at all.


GMAT algebra note: For each piece of information that you’re given in this type of problem, think about what is fixed and what is flexible. Try to draw multiple versions of each object (if possible), testing the boundaries of this flexibility.


GMAT algebra note: The following everyday objects may be useful as analogies in your thinking:.


GMAT quant note: Constrained Flexible Analogous Object(s) Mental Picture/Simplified Sketch A line passes through a specified point. Slope of the line Drinking straw = line Thumbtack = point Line free to spin about a point.


GMAT quant note: Two lines intersect at a specified point. Slope of the lines Drinking straw = line Thumbtack = point Both lines free to spin about a point.


GMAT quant note: The angle is free to change. A line passes through two points.


GMAT quant note: Nothing flexible Drinking straw = line Thumbtacks = points Two points pin down a line—no flexibility.


GMAT word problem note: Constrained Flexible Analogous Object(s) Mental Picture/Simplified Sketch Specified distance between two points Absolute or relative location of the points Thumbtacks = points Drinking straw = distance between points Fix one point temporarily. Line (straw) free to spin about one point, tracing the circle of possible locations of the other point.


GMAT word problem note: Slope of a line Location of the line, or points the line may pass through Drinking straw = line Line is free to “float around” but not rotate. Points are on a line (either in the coordinate plane or on a basic number line).


GMAT word problem note: Distance between the points Rubber band = stretched between points “Stretchy” distance between points.


GMAT geometry note: Constrained Flexible Analogous Object(s) Mental Picture/Simplified Sketch Points on a line are a specified distance apart.


GMAT geometry note: Order of the points (le -to- middle- to-right) Drinking straws = fixed lengths between points Manipulation will be determined by other stated constraints, but thinking of the points as the endpoints of rigid straws will ensure that you do not forget the distance constraint. Lines could be laid end- to-end, overlapping, separated, and flipped right-to-le.


GMAT geometry note: Triangle with a fixed area and a fixed base (and therefore a fixed height) Position of the third vertex along a line parallel to the base Straw = fixed base Rubber bands = other sides of the triangle Thumbtacks = endpoints of the base This is not intended to be an exhaustive list. The idea is to show you a new way of thinking through some difficult Geometry problems.


GMAT geometry note: Try-It #7-3 A circle in a coordinate plane has a center at point A and a diameter of 6. If points B and C also lie in the same coordinate plane, is point B inside the circle?


GMAT geometry note: The exact locations of points A, B, and C do not matter—only the relative locations of the points matter. Therefore, you can arbitrarily assign point A to a specific location (when possible, choose the origin of the coordinate plane) and draw a circle with a radius of 3 units around it.


GMAT geometry note: However, it does constrain point B to be exactly 2 units away from wherever point C is. You could imagine point C at the center of a circle of size 2, with point B somewhere on the The distance between point A and point C equals 2.(1) The distance between point B and point C equals 2.(2).


GMAT geometry note: (1) The distance between A and C is (2) The distance between B and C is circle around it (and this circle could pick up and move anywhere on the coordinate plane). Finally, combine these two statements to see that depending on where point C is drawn, point B may be inside the dotted circle, and it may not be.


GMAT geometry note: As long as you can achieve a visual proof of the answer, you don’t need to prove it algebraically. That’s what rubber band geometry is all about: testing scenarios for Geometry problems without the need to plug in numbers or use algebra.


GMAT geometry note: All you need is a visual environment that can be manipulated—one that preserves all key constraints and freedoms in the problem and allows you to see and test them.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Baseline Calculations for Averages BASIC AVERAGES Try-It #7-4 What is the average (arithmetic mean) of 387, 388, and 389?


GMAT statistics note: Without even calculating, you may be able to see that the average is 388. How did you arrive at that answer?


GMAT statistics note: It’s very unlikely that you calculated the average the classical way:. You probably noticed that the numbers are very close together and evenly spaced: 387 is 1 less than 388, and 389 is 1 greater than 388.


GMAT statistics note: Thus, the average must be 388—right in the middle. Whether you realize it or not, you’re using a relatively advanced technique to solve this problem: a baseline calculation.


GMAT statistics note: The baseline in this case is 388—the middle number. This concept can be applied to more difficult calculations of averages, making the calculation process much easier.


GMAT statistics note: Try-It #7-5 A consumer finds that five bags of popcorn contain 257, 261, 273, 280, and 259 corn kernels per bag, respectively. What is the average (arithmetic mean) number of corn kernels per bag of popcorn?


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. First, note that all of the bags have at least 257 kernels, so the average must be greater than 257.


GMAT statistics note: First, consider how much each term differs from 257. The biggest numbers rise the highest.


GMAT statistics note: A number equal to the baseline has no height. The height of the column thus represents the difference between the number and the baseline value:.


GMAT statistics note: Calculate the sum of the differences: 0 + 4 + 16 + 23 + 2 = 45. Divide by the number of terms: the average difference is 45 ÷ 5 = 9.


GMAT statistics note: Therefore, the average number of kernels per bag equals baseline + average difference: 257 + 9 = 266. Simply put, a baseline picture is a column chart.


GMAT statistics note: The columns don’t show the actual value of any number—rather, they show the difference between the baseline and the number. The baseline can be any convenient number.


GMAT statistics note: The smallest term in the set The largest term in the set The median term in the set A round number near the range of values.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. For sets with apparent symmetry, choosing a baseline in the middle is a good way not only to confirm the symmetry, but also to compute the average.


GMAT statistics note: In this scenario, represent numbers lower than the baseline with columns that drop below the baseline. As before, the size of the column represents the difference between the number and the baseline.


GMAT statistics note: Use trial and error to pick a possible average baseline, then adjust the drawing and calculations if necessary. Try-It #7-6 If a small business paid quarterly taxes last year of $10,079, $10,121, $10,112, and $10,088, what was the average (arithmetic mean) quarterly tax payment last year?


GMAT statistics note: In this case, some of the numbers are below $10,100 and others are above $10,100, so $10,100 is a natural first guess:. If the baseline is the average, then the sum of the differences from the baseline will be zero.


GMAT statistics note: Since the differences from the baseline do in fact sum to zero, $10,100 is indeed the average of this set.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Some sets may have many terms, but each of those terms has one of only two possible values.


GMAT statistics note: Rather than add each individual term together, simplify the calculations by using a weighted average calculation. For weighted averages, use visualization to advance your understanding of the math.


GMAT statistics note: Two real-life analogies can make it easier to remember how the relative weights of high and low values determine where the weighted average falls. Teeter-Totter Hanging Scale These pictures represent the values in a set as horizontal positions—le -to-right, as on a number line, not as vertical columns.


GMAT statistics note: (It may help to imagine the balance beam or lever marked off in equal units like a number line.) Each pin or weight corresponds to the presence of a value in a set. The X in each picture marks the equilibrium point—in other words, the weighted average.


GMAT statistics note: Both visual interpretations regard weighted averages as a kind of balancing act: the weighted average will be closer to the end of the range that has more weight. In the pictures, there are more instances of the le -hand side number in the set, so the X is relatively closer to the le -hand side.


GMAT statistics note: You can think of this point as the point where the weight would be “balanced” between the two sets. Try-It #7-7 A convenience store stocks soda in 12-ounce and.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Note that 22 is much closer to 24 than to 12.


GMAT statistics note: This implies that there will be many more 24-ounce bottles than 12-ounce bottles. Because the question asks about 12-ounce bottles, you could strategically eliminate any answer greater than or equal to Use your understanding of weighted averages as a balancing act to work backwards from the weighted average to the ratio of high-to-low terms:.


GMAT statistics note: The total range between the high and low values (24 and 12) is 12 units. Mark off the distance from each end to the average of 22.


GMAT statistics note: Because the weighted average is closer to 24, that side of the teeter-totter is assigned the greater weight of 10 out of 12. The other side, 12, is assigned the smaller weight of 2 out of 12.


GMAT statistics note: Therefore, the 24-ounce bottles constitute of the total number of bottles. The 12-ounce bottles constitute of the total number of bottles.


GMAT statistics note: When using this technique, it is important to remember that the weighted average is closer to (i.e., fewer units away from) the side that has greater weight, so that side should always be assigned the higher fraction. It is easy to reverse this logic accidentally when solving a weighted average problem with this technique, so be very careful!


GMAT statistics note: Just remember: the weighted average point will be closer to the side with more weight.


GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. The citrusade is more than 45% water.


GMAT word problem note: The lemonade is 50% water and the limeade is 40% water, so the lemonade must be more heavily weighted:. Therefore, the lemonade must make up more than 50% of the mixture.


GMAT word problem note: Next, the citrusade is more than 24% sugar. The lemonade is 20% sugar and the limeade is 32% sugar, so what else can you figure out about the relative weighting?


GMAT word problem note: Since lemonade is more heavily weighted (the mixture is to the right of 50/50), the weighting of the mixture is somewhere between 24% and the 50/50 weighting of the sugar, 26%. What weighting does the 24% figure represent?


GMAT word problem note: The weighting is at most lemonade, or . The weighting, then, must be between and lemonade and the rest limeade.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Number Line Techniques for Statistics Problems Several other common types of questions involving statistics can be solved with visualization.


GMAT statistics note: Specifically, using a number line can help simplify the work for many of these problems. MEDIAN RELATIVE TO MEAN Most questions involving the term median are really asking about the order of terms in a set: you line up the terms in a set in order of size, then select the middle term.


GMAT statistics note: By contrast, the average, or arithmetic mean, is the sum of all of the terms divided by the number of terms. It can be visualized as the balancing point of all the terms laid out on the number line, as in the discussion of the balancing point for weighted averages in the previous section.


GMAT statistics note: This technique is similar to rubber band geometry discussed earlier in this chapter, except this technique applies to problems involving sets rather than problems involving the coordinate plane. In this technique, you must place fixed terms in order from least to greatest (as you would on a number line), then move variable terms around according to the constraints.


GMAT statistics note: By doing so, you can visualize what impact these changes hav.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The median depends on where n falls relative to −2 and 4: below −2, between −2 and 4, or above 4.


GMAT statistics note: One approach to this question is to think through the potential answers for all possible n values in the likely range (you can glance at the statements and other values in the list to get a sense of the relevant range) and draw out the scenarios on a number line. Try numbers around the relevant numbers, including a smaller and larger one at the far ends of the range.


GMAT statistics note: Note that, by definition, all numbers in a set are different, so n cannot be −2 or 4:. This requires a fair amount of up-front work, but evaluating the statements is fast as a result.


GMAT statistics note: By contrast, if n = 10, the mean would equal 4 and the median would still equal 4. By contrast, if n = −8, the mean would equal −2 and the median would still equal −2.


GMAT statistics note: Taken together, however, any number in the range of 2 < n < 3 would feature a median greater than the mean.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Notice from this problem that as you move the variable terms, the mean always changes when the value of the variable terms change, but the median typically changes in jumps.


GMAT statistics note: The median can get stuck while the number you’re changing doesn’t affect which number is in the middle. CHANGES IN STANDARD DEVIATION The GMAT will rarely (if ever) ask you to calculate the standard deviation of a list of numbers.


GMAT statistics note: However, the exam will expect you to have some intuition about standard deviations. One way in which the GMAT might test your intuitive knowledge of standard deviations is by changing numbers within a list and asking you what the impact on standard deviation would be.


GMAT statistics note: The relationship is relatively straightforward:. You might also see the term variance, which is also a measure of the spread of numbers in a set or list.


GMAT statistics note: Variance and standard deviation indicate the same information. A variance of 0 indicates that all of the numbers are identical, as does a standard deviation of 0.


GMAT statistics note: The larger the variance, or standard deviation, the more the numbers are spread out. (Variance and standard deviation are never negative.) Try-It #7-10 Last Year 9 9.5 10 10 11 11 11 11 11 12.5 13 13 This Year 9 x 10 10 11 11 11 11 11 y 13 13 Moving terms away from the mean increases t.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The monthly sales (in thousands of $) at a certain restaurant for the past two years are given in the chart above.


GMAT statistics note: If the standard deviation of the monthly sales is greater this year than last year, which of the following are possible values for x and y? Except for x and y, the two lists of monthly sales numbers are identical, so focus exclusively on those terms that changed: 9.5 and 12.5 from last year were replaced by x and y this year.


GMAT statistics note: If this year’s standard deviation is greater, then this year’s numbers must be more spread out from the mean than last year’s. The numbers are close enough together to indicate that the average should be somewhere around 11.


GMAT statistics note: Visually, here are the interesting terms from last year:. This problem does not require actual computation of the standard deviation using the new x and y values.


GMAT statistics note: The math would be too complex to complete in two minutes. Instead, determine visually which x and y values increase the standard deviation: the pair of x and y values that are farther from the mean than are 9.5 and 12.5 will increase the standard deviation.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Only in this case is the deviation greater.


GMAT statistics note: The GMAT may also test you on standard deviations by adding numbers to a list. When new terms are added, the GMAT will o en ask you to compare the old list to the new list, or to compare various options for the new list, or to do both.


GMAT statistics note: You must have a technique to evaluate the standard deviation of different lists relative to one another. Again, pictures make for great comparison tools!


GMAT statistics note: Try-It #7-11 A list of 12 test scores has an average (arithmetic mean) of 500 and a standard deviation of 50. Which of the following lists of additional test scores, when combined with the original list of 12 test scores, must result in a combined list with a standard deviation less than 50?


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. It is not generally true that all of the terms in a list are within 1 standard deviation of the mean.


GMAT statistics note: However, standard deviation is a measure of the spread of the terms of a list, so you could represent the original list of scores this way:. The oval spans ±1 standard deviation from the mean, where many of the scores will likely be.


GMAT statistics note: This simplification is acceptable as long as you represent all of the other lists the same way so that you can compare the relative effects of the new test scores systematically.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. In general, these are the rules for adding a single term to a list:.


GMAT statistics note: Note that mathematically this is a slight oversimplification, but for the purpose of adding terms to a list of numbers on the GMAT, you can accept this simplification as true. FLOATING TERMS IN A SET On GMAT Statistics problems involving elements (i.e., terms) in a list, you can usually focus your attention on a single term or two.


GMAT statistics note: These terms could be considered the floating terms—the terms that are unknown or not completely defined among a list of more clearly defined terms. As you approach a question of this type, try to rephrase the question quickly so that you focus on the unknown, or floating, terms rather than on the known terms.


GMAT statistics note: Try-It #7-12 List A contains 5 positive integers, and the average (arithmetic mean) of the integers in the list is 7. If the integers 6, 7, and 8 are in List A, what is the range of List A?


GMAT statistics note: Adding a new term more than 1 standard deviation from the mean generally increases the standard deviation. Adding a new term less than 1 standard deviation from the mea.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The average of all five integers in the list is 7.


GMAT number properties note: Three of the integers in the list are given (6, 7, and 8), and they all have an average of 7. Therefore, the floating terms in this problem must also have an average of 7.


GMAT number properties note: Assign x and y to represent these terms:. The rephrased question is thus, “Given that x + y = 14, what is either x or y?” Once you know one of the values, you can solve for the other and thereby determine the range of the list.


GMAT number properties note: If 3 is one of the unknown integers, the other must be 11. The range is thus 11 − 3 = 8.


GMAT number properties note: This statement might seem a little too vague to be sufficient, but by visually listing the possible pairs that add up to 14, you can rule out pairs that don’t fit the constraint from this statement:. Notice that the pairings represent the constraint x + y = 14.


GMAT number properties note: Visually, this means that x and y are balanced around 7. The largest term in List A is greater than 3 times and less than 4 times the size of the smallest term.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Only one pair of integers results in a ratio strictly between 3 and 4.


GMAT word problem note: The unknown terms must therefore be 3 and 11, and the range is 11 − 3 = 8. In this problem, the constraint x + y = 14 is a fixed sum.


GMAT word problem note: Another common constraint is a fixed difference, such as a − b = 2. A fixed difference can be represented visually as a fixed distance between a and b on the number line, with a to the right because it is larger.


GMAT word problem note: That distance could move le or right:. MAXIMIZING (OR MINIMIZING) ONE TERM Another visual technique for statistics involves maximizing (or minimizing) the value of a term in a set or list of numbers, subject to some constraints.


GMAT word problem note: Such problems will usually employ the word maximum or minimum. For these problems, you o en should maximize (or minimize) the term by minimizing (or maximizing) the other terms, because the constraints usually involve mathematical trade-offs.


GMAT word problem note: Try-It #7-13 8 is 1.33 times the size of 6 (the ratio is too low). 9 is 1.8 times the size of 5 (the ratio is too low).


GMAT word problem note: 10 is 2.5 times the size of 4 (the ratio is too low). 11 is 3.66 times the size of 3 (an acceptable ratio).


GMAT word problem note: 12 is 6 times the size of 2 (the ratio is too high).


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. In a certain lottery drawing, five balls are selected from a tumbler in which each ball is printed with a different two-digit positive integer.


GMAT statistics note: If the average (arithmetic mean) of the five numbers drawn is 56 and the median is 60, what is the greatest value that the lowest number selected could be? The goal is to maximize the value of the lowest-numbered ball.


GMAT statistics note: All balls contain a two- digit positive integer, and none of the balls have the same number. The problem provides enough information to calculate the sum and to lay out a visual listing of the numbers:.


GMAT statistics note: In order to maximize the value of the first (lowest) number in the set, what do you need to do to the other numbers? You’d want to minimize them.


GMAT statistics note: Select the smallest numbers that you can for the remaining slots:. In some problems, you might actually make different slots equal to each other (e.g., the three largest numbers could be 60, 60, and 60).


GMAT algebra note: Track mean, median, range, distribution, and how operations change spread. The five numbers must sum to 280, so you can set up an algebraic equation and solve.


GMAT algebra note: Can you think of a way to minimize the arithmetic needed to solve in that way? The three numerical values, 60, 61, and 62, are all larger than the average of 56.


GMAT algebra note: Specifically, they are +4, +5, and +6 away from that average. The other two numbers, then, need to make up for that overage of 4 + 5 + 6 = 15:.


GMAT algebra note: The numbers x and x + 1 are also consecutive, so they need to be −8 and −7 away from the average of 56. The two remaining numbers are 48 and 49.


GMAT algebra note: Try-It #7-14 The average (arithmetic mean) of six numbers is 18 and the median of the six numbers is 16. What is the minimum possible value for the greatest number in the list?


GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. This time, the goal is to minimize the largest number:.


GMAT word problem note: In order to minimize the final term, you’d want to maximize all of the other terms. This time, though, the list contains an even number of terms, so you can’t just set the median to the middle number.


GMAT word problem note: The two middle numbers average to 16. The pair (16, 16) averages to 16, as does the pair (15, 17).


GMAT word problem note: The first pair, though, is better when the goal is to minimize the final term, since the terms to the right have to be equal to or greater than the terms to the le.


GMAT word problem note: In other words, when trying to minimize the final term, it’s true that you want to maximize the earlier terms, but you also have to think about how to do so in a way that doesn’t make the final term too large (since it has to be larger than the earlier terms). In this case, the first four terms are all 16:.


GMAT word problem note: In order to minimize the final term, set the last two terms equal to each other and solve algebraically:. Or you can use the over/under approach.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Those last two terms have to make up for the −8 on the other side, so each x must be 4 over the average, or 22.


GMAT statistics note: TAKING AND GIVING Another common scenario involves taking value from one term in a list and giving it to another term. The relative value of the terms in the list will change, leading to some interesting results.


GMAT statistics note: If an integer number of Keri’s points were taken from her and given to Luke, and the median score of the five players increased, how many points were transferred from Keri to Luke? The key to this problem is that by taking enough points from Keri and giving them to Luke, the median of the list can change.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Order the scores from low to high on a number line, and represent the change in Luke’s score with x:.


GMAT statistics note: In order for the median to change, Luke’s score must leap-frog those of Nora and Mia, pushing Mia into the bottom two scores and making Luke’s score the median. You don’t want to decrease Keri’s score so much that Luke and Mia surpass her, leaving Mia once again in the median score position.


GMAT statistics note: If 15 + x = 38, Luke would match the current median score. That is x = 23, and Keri’s new score would be 63 − 23 = 40.


GMAT statistics note: However, the median score would remain 38, with both Luke and Mia having that score. Therefore, x must be greater than 23.


GMAT statistics note: Luke’s new point value is 15 + 24 = 39. Keri’s new point value is 63 − 24 = 39.


GMAT statistics note: Both Nora and Mia are below Luke (and Keri), so the new median is 39:.


GMAT geometry note: Does a rectangular mirror have an area greater than 10 square centimeters? The perimeter of the mirror is 24 cm.(1) The diagonal of the mirror is less than 11 cm.(2) 2.


GMAT geometry note: A number line is numbered with the integers from 0 to 50 inclusive. An ant walks along the number line as follows:.


GMAT geometry note: First, it walks in the positive direction until it reaches a multiple of 5 that it hasn’t previously reached. Then, it walks in the negative direction for at least 1 unit, stopping when it reaches any multiple of 2.


GMAT geometry note: The ant starts at 0 on the number line and repeats this process until it first reaches the point marked 50. How many units does it travel in total?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If a, b, and c are positive, is On the number line, a is closer to b than it is to c.(1) b > c(2) 4.


GMAT number properties note: The length of one edge of a cube equals 4. What is the distance between the center of the cube and one of its vertices?


GMAT number properties note: A test is taken by 100 people and possible scores are the integers between 0 and 50, inclusive. For each of the following scenarios, determine whether the average (arithmetic mean) score would be greater than 30 (answer Yes, No, or Uncertain).


GMAT number properties note: More than 70 people scored 40 or higher.a) 75 people scored 40 or higher.b) Fewer than 10 people scored 50.c) No more than 2 test-takers scored any given score.d).


GMAT coordinate geometry note: As part of an experiment, a student repeatedly tests the temperature of a light bulb. The bar graph below displays the number of readings the student recorded at various temperatures, measured in degrees Fahrenheit.