GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. 6,0002 – 5,0002 = 36,000,000 – 25,000,000 = 11,000,000 Since 3,0002 = 9,000,000, and 4,0002 = 16,000,000, the correct answer must be between 3,0002 and 4,0002.


GMAT sequences and patterns note: Factoring The number 5,995 is a multiple of 5 just under 6,000, which equals 5(1,200). The number 4,796 is a multiple of 4 just under 4,800, which equals 4(1,200).


GMAT sequences and patterns note: This “coincidence” suggests factoring. 5,995 = 6,000 – 5 = 5(1,200 – 1) = 5(1,199) 4,796 = 4,800 – 4 = 4(1,200 – 1) = 4(1,199) The shared factor of 1,199 means that simplification is possible.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Begin by simplifying the equation given in the question.


GMAT algebra note: At this stage, you might be tempted to divide both sides of the equation by xy, in order to arrive at the simpler equation y2 + x2 = 2xy. However, that would be a mistake—never divide both sides of an equation by an unknown (in this case, xy), unless you are certain that the unknown cannot equal zero.


GMAT algebra note: (Division by zero is undefined, and can lead to nonsensical results.) So rather than divide by xy, subtract 2x2y2 from both sides to group all terms on one side of the equals sign. This last line implies that either xy = 0 or y – x = 0.


GMAT algebra note: In other words, either xy = 0 or y = x. If y > x, then it is impossible that y = x.


GMAT algebra note: If x < 0, then it is possible that xy = 0 (i.e., if y = 0) or that y = x (i.e., y is negative, too). If y = x = any negative number, then there are infinitely many solutions for xy.


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


GMAT coordinate geometry note: Is it possible for a wooden block in the shape of a rectangular solid measuring l by w by h centimeter to pass through a square hole with sides of length 4 centimeter? The volume of the wooden block is 16 cm3.(1) L > w > h > 1(2) 52.


GMAT coordinate geometry note: Three of the four vertices of a rectangle in the xy-coordinate plane are (–5, 1), (–4, 4), and (8, 0). In the coordinate plane, circle C has radius r and its center is at point (x, y).


GMAT coordinate geometry note: Is at least 50% of the area of circle C contained in a single quadrant of the coordinate plane?


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A group of men and women gathered to compete in a marathon.


GMAT statistics note: Each competitor was weighed before the competition. The average weight of the women was 120 pounds, and the average weight of the men was greater than 120 pounds.


GMAT statistics note: If the average weight of the entire group was w pounds, what percent of the competitors were women? The average weight of the men was 150 pounds.(1) The difference between w and the average weight of the women was twice the difference between w and the average weight of the men.


GMAT statistics note: What is the average (arithmetic mean) of these numbers: 12. A set of 5 numbers has an average (arithmetic mean) of 50.


GMAT statistics note: The greatest element in the set is 5 greater than 3 times the least.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the median of the set equals the mean, what is the greatest possible value in the set?


GMAT statistics note: 5, 2, 4, m, 9, 5 For the list of numbers above, what is the median? The median is an integer.(1).


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Set M contains seven consecutive integers, and set N contains three values chosen from set M.


GMAT statistics note: Is the standard deviation of set N greater than the standard deviation of set M? Set N contains the median of set M.(1) The range of set M and set N are equal.(2) 60.


GMAT statistics note: Four different children have jelly beans:. Aaron has 5, Bianca has 7, Callie has 8, and Dante has 11.


GMAT statistics note: How many jelly beans must Dante give to Aaron to ensure that each child has within 1 jelly bean of all the other children?


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


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Start by considering the different types of blocks that could or could not fit through the hole.


GMAT word problem note: If all dimensions of the block are less than 4 centimeters (cm), the block can definitely fit through the hole. If exactly one of the three dimensions is greater than 4 cm, the block can still definitely fit through the hole.


GMAT word problem note: To do so, rotate the block so that the two shorter dimensions are aligned with the sides of the hole.


GMAT quant note: However, if at least two of the dimensions of the block are greater than 4 cm, further investigation is required. It may be possible to fit the block through the hole by rotating it to take advantage of the longest dimension of the hole, which is the diagonal of but if the dimensions of the block are too great, this may be impossible.


GMAT quant note: The block can have any dimensions such that Lwh = 16. However, a block with dimensions 8, 8, and 0.25 cannot fit through the hole.


GMAT quant note: Since the answer can be Yes or No, this statement is insufficient. A block with dimensions 2, 3, and 4 can fit through the hole, but a block with dimensions 100, 101, and 102 cannot.


GMAT quant note: Lwh = 16 and h is greater than 1, so Lw is less than 16. Therefore, L and w cannot both be greater than 4.


GMAT quant note: Since h is the least dimension, it must also be less than 4. Your GMAT scratch pad has a grid.


GMAT geometry note: Complete the rectangle with the dashed lines shown. The fourth point must be located approximately where the bigger dot is drawn.


GMAT geometry note: Alternatively, compute the location of the fourth point, using the fact that the short sides have the same slope. The known short side connects the points (−5, 1) and (−4, 4).


GMAT geometry note: In other words, the bottom le corner is 1 to the le and 3 down from the top le corner. The question asks whether at least half of the circle is contained in one quadrant.


GMAT geometry note: However, it does not specify a specific quadrant. If at least half of the circle is contained in any one of the four quadrants, the answer to the question will be Yes.


GMAT geometry note: Note that if circle C lies in exactly one or two quadrants, then the answer to the question must be Yes, because at least half of the circle must be in.


GMAT geometry note: If x = 2, y = 1, and r = 0.5, then the answer to the question is Yes, since the entire circle is contained in the first quadrant. However, if x = 2, y = 1, and r is much greater than 2, then approximately a quarter of the circle is contained in each quadrant and the answer to the question is No.


GMAT geometry note: Since the x coordinate is greater than the radius of the circle, no part of the circle can be in the second or third quadrants.


GMAT geometry note: Track mean, median, range, distribution, and how operations change spread. Therefore, at least half of the circle must be in the first quadrant or at least half of the circle must be in the fourth quadrant.


GMAT geometry note: The answer to the question is definitely Yes, and this statement is sufficient. The question stem indicates that the average weight of the women was 120 and the average weight of the men was greater than 120.


GMAT geometry note: In order to determine what percent of the competitors were women, you would need to know more about the weight of the men and you’d also need to know something about the relative number of women vs. If there were not an equal number of each, then this is a weighted average question.


GMAT geometry note: This does not provide the weight of the men or even the value of w, but it does indicate where the weighted average falls between the two values.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the weighted average is twice as far from the women’s end of the line, then the men are responsible for of the total weight and the women are responsible for of the total weight.


GMAT statistics note: Therefore, , or approximately of the competitors were women. You do not need to calculate this figure.


GMAT statistics note: The simple average formula applies to this problem.


GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. However, the chance of computational error is high on a problem with this many terms of such great value.


GMAT word problem note: Try grouping the similar terms. Group A: 12, 13, 14 (equidistant terms with an average of 13, the middle term) Group B: 510, 520, 530 (equidistant terms with an average of 520, the middle term) Group C: 1,115.


GMAT word problem note: 1,125 (equidistant terms with an average of 1,120, the middle term) Since each group of terms consists of three values (and are therefore equally weighted in the set of nine terms), the average of all nine original terms is the average of the respective averages of Groups A, B, and C.


GMAT word problem note: Average = You can make that division easier by breaking 1,653 into lesser numbers that are divisible by 3: 1,653 = 1,500 + 150 + 3. Divide each separately by 3 to get 500 + 50 + 1 = 551.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Two techniques will help you efficiently interpret the information given in the question.


GMAT statistics note: First, draw a number line with five dots representing the five numbers in the set. Second, label these numbers A, B, C, D, and E, with the understanding that A ≤ B ≤ C ≤ D ≤ E.


GMAT statistics note: A + B + C + D + E = 250 (The set of five numbers has an average of 50.) E = 5 + 3A (The greatest element is five greater than three times the least element in the set.) C = 50 (The median of the set equals the mean.) You’re asked to maximize E. Arrange the dots on the number line such that you obey the constraints, yet also note where you have some flexibility.


GMAT statistics note: Point D can be anywhere on the line from point C to point E. Given the information from the question stem, you can maximize E by minimizing D.


GMAT geometry note: Similarly, point B can be anywhere on the line from point A to point C. Maximize E by minimizing B, so make B = A.


GMAT geometry note: Therefore, E = 5 + 3A = 5 + 3(29) = 5 + 87 = 92. Many elements in this triangle could vary.


GMAT geometry note: You don’t even know the placement of B relative to AC, so the triangle itself might stretch. Even for a fixed triangle, DE could slide up or down, so various lengths are possible for DE, as shown here.


GMAT geometry note: You don’t know the lengths of any sides of the triangle. The side that most affects the length of DE is AC, so stretch that side.


GMAT geometry note: Stretching the triangle out to the right stretches DE. AC must be 14 and DE must be parallel to AC and halfway between AC and B, in order to maintain BE = EC.


GMAT geometry note: Even though vertex B is free to move, DE will always be the average of the width of the triangle at AC (14) and the width at B (0). Thus, DE must be 7, no matter how the picture shi s.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. To find the median of a set of numbers, line them up in order of value.


GMAT statistics note: The question of interest is, “Where is m relative to the other values?” This set has six values, an even number of terms, so the median is the average of the two middle terms. These are the three scenarios shown above.


GMAT statistics note: If m ≤ 4, then the two middle terms are 4 and 5, and the median is If 4 < m < 5, then the two middle terms are m and 5, and the median is If m ≥ 5, then the two middle terms are 5 and 5, and the median is 5. For the case where 4 < m < 5, the median is Thus, if the median is an integer, it must be 5.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Standard deviation is a measure of the “spread” of a group of numbers.


GMAT statistics note: If set M contains the numbers {1, 2, 3, 4, 5, 6, 7}, then 4 is the median. Set N could be {3, 4, 5}, which has a lesser standard deviation than set M because the two sets have the same average but N is not as spread out as M.


GMAT statistics note: Alternatively, N could be {1, 4, 7}, which has a greater standard deviation than set M because the two sets still have the same average, but N is now more spread out than M. (It doesn’t have additional values that are closer to the average.) (2) SUFFICIENT:.


GMAT statistics note: If set M contains the numbers {1, 2, 3, 4, 5, 6, 7}, and set N has the same range, then N must contain 1 and 7. If set N contains {1, 4, 7}, then N has a greater standard deviation than M.


GMAT statistics note: If set N contains {1, 2, 7}, the standard deviation actually increases even more (because the numbers are no longer evenly distributed}. No matter what combination you try, the standard deviation of N has to be greater than the standard deviation of M.


GMAT statistics note: Conceptually, the transfer of jelly beans from Dante to Aaron reduces the range of the number of jelly beans held by individual children. The constraint is that the final distribution represents a range of just 1 jelly bean, a condition Bian.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Aaron and Dante must end up with a number of jelly beans that is either 7 or 8.


GMAT algebra note: If either Aaron or Dante has a number of jelly beans other than 7 or 8, he will differ too much from either Bianca’s or Callie’s number. You can count out the necessary change on the number line above or you can write out the algebra.


GMAT algebra note: The solution to both equations is x = 3:. Add 3 to Aaron to get 8, and subtract 3 from Dante to get 8.


GMAT algebra note: The resulting number of jelly beans is A = 8, B = 7, C = 8, and D = 8.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. At least one side of rectangle R has an integer length.(1) The diagonals of rectangle R have integer lengths.(2) 62.


GMAT fractions, decimals, and percents note: A group of friends charters a boat for $540 and each person contributes equally to the cost. They determine that if they can get three more of their friends to join them, every person in the group will pay $9 less.


GMAT fractions, decimals, and percents note: If they find three more friends to join them, what is the total number of people renting the boat? In Smithtown, the ratio of right-handed people to le -handed people is 3 to 1 and the ratio of men to women is 3 to 2.


GMAT fractions, decimals, and percents note: If the number of right-handed men is maximized, then what percent of all the people in Smithtown are le -handed women?


GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. The sum of the interior angle measures for any n-sided polygon equals 180(n − 2).


GMAT fractions, decimals, and percents note: If Polygon A has interior angle measures that correspond to a set of consecutive integers, and if the median angle measure for Polygon A is 140°, what is the least angle measure in the polygon? When the positive integer x is divided by the positive integer y, the quotient is 2 and the remainder is z.


GMAT fractions, decimals, and percents note: When x is divided by the positive integer a, the quotient is 3 and the remainder is b. The ratio of y to a is less than 3 to 2.(1) The ratio of y to a is greater than 2 to 3.(2).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If a and b are odd integers, a ∆ b represents the product of all odd integers between a and b,inclusive.


GMAT number properties note: If y is the least prime factor of (3 ∆ 47) + 2, which of the following must be true? Set S is the set of all prime integers between 0 and 20.


GMAT number properties note: If three numbers are chosen randomly from set S, and no number is chosen more than once, what is the probability that the sum of all three numbers is odd?


GMAT geometry note: Sets A and B each consist of three terms selected from the first five prime integers. No term appears more than once within a set, but any integer may be a term in both sets.


GMAT geometry note: If the average of the terms in set A is 4 and the product of the terms in set B is divisible by 22, how many terms are shared by both sets? If Q is the sum of the areas of squares SAND and SURF and C is the area of the circle, then the fraction is.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set S consists of n consecutive positive integers.


GMAT number properties note: If n > 3, what is the value of n? The number of multiples of 2 contained in set S is equal to the number of multiples of 3 contained in set S.


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


GMAT geometry note: A rectangle is a square if and only if its length equals its width. If the length is 1 and the width is 2, the rectangle is not a square.


GMAT geometry note: If the length and width are both 2, the rectangle is a square. Since R is a rectangle, both of its diagonals have the same length.


GMAT geometry note: It is only necessary to examine one of the diagonals. Since the diagonal and two of the sides form a right triangle, the length of the diagonal is always equal to The diagonal of the rectangle below has an integer length, but the rectangle is not a square.


GMAT fractions, decimals, and percents note: However, in the rectangle below, the diagonal has an integer length and the rectangle is a square. The answer could be Yes or No, so this statement is insufficient.


GMAT fractions, decimals, and percents note: According to the statements, the rectangle has at least one integer side, and it also has integer diagonals. In a square, the legs and hypotenuse must be in the ratio .


GMAT fractions, decimals, and percents note: To be a square, a rectangle must have a diagonal that is times as long as its sides. However, is an irrational number, so if x is an integer, then is not an integer (and vice versa).


GMAT fractions, decimals, and percents note: It is impossible for the rectangle described by these statements to be a square, and the answer to the question is definitely No. The statements together are sufficient.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Call the initial number of friends in the group f and the initial cost c.


GMAT algebra note: The problem allows you to write two equations. The question asks for f + 3.


GMAT algebra note: These equations can be solved algebraically, but the math is going to result in having to solve a quadratic. Try plugging them into the problem to find the correct answer.


GMAT algebra note: F + 3 f Orig Cost New Cost Are the costs $9 apart? More people need to join the group in order to bring the costs closer together.


GMAT algebra note: The answer must be between 9 and 18.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Check the math if you’re not sure, but do practice this technique enough that you know when you can actually tell what the answer must be (without doing the math).


GMAT number properties note: Use a double-set matrix to solve this problem. Thus, both x and y are integers.


GMAT number properties note: Moreover, the total number of people must be a multiple of 4 and of 5 in order for the given ratios to be possible. From this constraint, there are two ways to solve.


GMAT number properties note: Algebraic Solution Since the question specifies that the number of right-handed men be as great as possible, assume that all the men are right-handed. Of course, that means that none of the men are le -handed.


GMAT number properties note: Because each column in a double-set matrix must total, you can also fill in the number of le - handed women (the group you want).


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Smart Number Solution Since the total number of people in Smithtown must be a multiple of 20, set the total to 20 and determine the subtotals of men, women, le - handed, and right-handed based on the ratios given in the problem.


GMAT fractions, decimals, and percents note: Right-Handed Le -Handed Total Men 12 Women 8 Total 15 5 20.


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. To maximize the number of right-handed men, assign all the men to the “right-handed men” cell, and fill in the remaining cells.


GMAT number properties note: Right-Handed Le -Handed Total Men 12 0 12 Women 3 5 8 Total 15 5 20 Therefore, le -handed women represent of the population. If the median angle measure is 140° and the interior angle measures correspond to a set of consecutive integers, then the average angle measure must equal 140°.


GMAT number properties note: Since the sum of the angles must equal 180(n − 2), the average angle must equal.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Therefore, the polygon has nine sides and nine interior angles, and the measures of these angles are equal to a set of consecutive integers centered at 140.


GMAT number properties note: If has a quotient of 2 and a remainder of z, then x is z more than 2y. If has a quotient of 3 and a remainder of b, then x is b more than 3a.


GMAT number properties note: The question asks whether z > b, and the statements give information about . Simplify the question by replacing z and b with their equivalents and solving for the combination.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Note that the inequality sign flipped in the last step because of the division by −2.


GMAT number properties note: The variable a is a positive integer, so no additional flip is required for that manipulation. Rephrase the question as, “Is (1) SUFFICIENT:.


GMAT number properties note: This directly answers the rephrased question: “Yes, .” Therefore, z > b. The answer might be Yes if .


GMAT number properties note: This product is a very large odd number, as it is the product of only odd numbers and thus does not have 2 as a factor. Therefore, (3 ∆ 47) + 2 = Odd + Even = Odd, and (3 ∆ 47) + 2 does not have 2 as a factor either.


GMAT number properties note: Every odd prime number between 3 and 47 inclusive is a factor of (3 ∆ 47), since each of these primes is a component of the product. The sum will only be divisible by 3 if k is also divisible by 3.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Divide (3 ∆ 47) + k by 3, you are evaluating .


GMAT number properties note: Because is an integer, must also be an integer to yield an integer sum. In this problem, k = 2, which is not divisible by any of the odd primes between 3 and 47.


GMAT number properties note: Since (3 ∆ 47) IS divisible, but 2 is NOT divisible, the sum (3 ∆ 47) + 2 is NOT divisible by any of the odd primes between 3 and So (3 ∆ 47) + 2 is not divisible by any prime number less than or equal to 47. The least prime factor of (3 ∆ 47) + 2 must be greater than 47.


GMAT number properties note: Thus, the minimum possible prime factor is 53, since that is the least prime greater than 47. If set S is the set of all prime integers between 0 and 20, then S = {2, 3, 5, 7, 11, 13, 17, 19}.


GMAT number properties note: There are seven odd terms and one even term in set S. If the even term is among those selected, the sum will be even (E + O + O = E).


GMAT number properties note: The sum will be odd if all three terms selected are odd (O + O + O = O).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The first five prime integers are 2, 3, 5, 7, and 11.


GMAT number properties note: These are the only terms that can appear in sets A and B. There are some other restrictions on the sets:.


GMAT number properties note: The average of the terms in set A is 4, so the sum of the terms is (4) (3) = 12. There is only one way for three of the first five primes to sum to 12: 2 + 3 + 7.


GMAT number properties note: The product of the terms in set B is divisible by 22, so 2 and 11 are terms in set B. Set B is {2, 11, x}, where x can be 3, 5, or 7, but not 2 or 11 (no duplicates).


GMAT number properties note: Sets A and B share at least one term: the 2. If x is either 3 or 7, the sets will share two terms.


GMAT number properties note: If x is 5, the sets will only share one term. The rephrased question is, “Is x = 5?” A definite Yes or No answer leads to a definite value answer for the number of shared terms (i.e., Yes = one shared term, No = two shared terms).


GMAT number properties note: If the product of the terms in set B is not divisible by 5, x ≠ 5 and the answer to the rephrased question is a definite No. If the product of the terms in set B is divisible by 14, then 2 and 7 are terms in B.


GMAT number properties note: Therefore, x = 7 and the answer to the rephrased question is a definite No.


GMAT geometry note: The area of a square is equal to the length of one side squared. The question asks for the fraction Inscribed angle USA cuts off a diameter (UA) of the circle, so angle USA must be a right angle.


GMAT geometry note: Therefore, the triangle is a right triangle with hypotenuse UA. The Pythagorean theorem indicates that US2 + SA2 = UA2.


GMAT geometry note: US2 also represents the area of the smaller square. SA2 also represents the area of the larger square.


GMAT geometry note: The sum of the two equals the quantity Q mentioned in the question stem, so Q = UA2. UA is a diameter of the circle, so the quantity Q = (2r)2 = 4r2.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. The value of n must be 4 or greater.


GMAT quant note: If n is odd, it could equal 3, 5, 7, or so on. Test cases to figure out what is possible.


GMAT quant note: N Set S # Multiples 2 = # Multiples 3? 4 3, 4, 5, 6 Yes: 2 multiples of each 5 3, 4, 5, 6, 7 Yes: 2 multiples of each (1) AND (2) INSUFFICIENT:.


GMAT quant note: The case of n = 5 was already proven in the last step. Is there a way to have n = another odd number?


GMAT quant note: 5 3, 4, 5, 6, 7 Yes: 2 multiples of each 7 3, 4, 5, 6, 7, 8, 9 Yes: 3 multiples of each.


GMAT quant note: Since there are still at least two possible values for n, none of the information is sufficient to answer the question.


GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. If the average (arithmetic mean) daily energy usage was greater than the median daily energy usage, what is the least possible value for n?


GMAT word problem note: A painting crew painted 80 houses. They painted the first y houses at a rate of x houses per week.


GMAT word problem note: Then, more painters arrived and everyone worked together to paint the remaining houses at a rate of 1.25x houses per week. How many weeks did it take to paint all 80 houses, in terms of x and y?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If a2 = b2, what is the value of y?


GMAT fractions, decimals, and percents note: A herd of 33 sheep is sheltered in a barn with 7 stalls, each of which is labeled with a unique letter from A to G, inclusive. Is there at least one sheep in every stall?


GMAT fractions, decimals, and percents note: The ratio of the number of sheep in stall C to the number of sheep in stall E is 2 to 3.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The ratio of the number of sheep in stall E to the number of sheep in stall F is 5 to 2.


GMAT number properties note: If (s × 10q) − (t × 10r) = 10r, where q, r, s, and t are positive integers and q > r, then what is the units digit of t? What is the greatest prime factor of 21054 − 21352 + 214?


GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. A decimal is called a “shrinking number” if its value is between 0 and 1 and each digit to the right of the decimal is not less than the digit to its immediate right.


GMAT fractions, decimals, and percents note: For instance, 0.86553221 is a shrinking number. If x is a shrinking number, which of the following must be true?


GMAT fractions, decimals, and percents note: Each of the seven paintings in an art gallery has a different price. Is it possible to purchase at least three paintings for no more than $1,800 in total?


GMAT fractions, decimals, and percents note: The median price of the seven paintings is $550.(1) It is possible to purchase four paintings at the gallery for a total of $2,300.


GMAT coordinate geometry note: An (x, y) coordinate pair is to be chosen at random from the xy- plane.


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


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Here’s what you know about the arithmetic mean:.


GMAT statistics note: In contrast, the median depends on n, but not in a linear way. To find the median, you order the terms and pick the middle one, so try various n values (i.e., vary the number of 8’s in the list).


GMAT statistics note: This implies the eventual need to plug n values into the mean formula above, so draw a picture to help eliminate some answers first. The “low” and “high” grouping is a fast way to find the relationship between median and mean for this set.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the number of “lows” and “highs” are equal, the median is the average of the middle terms.


GMAT statistics note: That is, if n = 6, then the median = 9. When n = 6, the mean must be greater than 9.


GMAT statistics note: Pairs of 8 and 10 terms average to 9, but there is one “extra” 8. Pairs of 7 and 11 terms average to 9, but there is one “extra” 11.


GMAT statistics note: These “extra” terms differ from 9 by −1 and +2 respectively, for a total difference of +1. That positive difference implies that (mean > 9), or (mean > median).


GMAT statistics note: To prove that n = 5 is too low, you could take a more conventional approach. If n = 5, the number of terms is 3 + 5 + 5 + 4 = 17.


GMAT statistics note: Thus, the median is 10, while the mean is closer to 9. That is, if n = 5, then median > mean.


GMAT statistics note: If n = 6, then mean > median.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. This is a combined work problem, so use the work formula: rate × time = work.


GMAT word problem note: The work and rates are given, but you need to calculate time, so manipulate the formula: . There are 80 houses, y houses are painted at a rate of x houses per week, and the rate increases to 1.25x houses per week for the remaining 80 − y houses.


GMAT word problem note: To make the math easier, choose values such that x and 1.25x are integers (i.e., x is a multiple of 4) and y and 80 − y are divisible by x and 1.25x, respectively.


GMAT word problem note: Variable Value Units Total Houses 80 80 Houses Houses Painted at Slower Rate y 20 Houses Houses Painted at Faster Rate 80 − y 60 Houses Initial Rate x 4 Houses/Week Increased Rate 1.25x 5 Houses/Week The total painting time is.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If a2 = b2, then (x + y)2 = (x − y)2.


GMAT word problem note: Distribute both sides using the square of a sum and square of a difference special products, then simplify.


GMAT quant note: There are three basic scenarios. X y xy = 0 0 Any nonzero ✓ Any nonzero 0 ✓ (1) INSUFFICIENT:.


GMAT quant note: This statement indicates that x and y must be non- negative for their square roots to be real values. The statement also eliminates the last scenario, in which x = y = 0.


GMAT quant note: But y could still be 0 or any positive value. X y xy = 0 0 Any positive ✓ ✓ Any positive 0 ✓ ✓ (2) SUFFICIENT:.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Eliminates the last scenario.


GMAT fractions, decimals, and percents note: If x and y were both 0, would equal 0. It also eliminates the first scenario.


GMAT fractions, decimals, and percents note: Thus, you can conclude that y = 0. Set up a table and assign sheep to stalls.


GMAT fractions, decimals, and percents note: Stall A B C D E F G # of Sheep 2x 3x Since a fractional sheep is not possible in this problem, x must be a positive integer. With 23 sheep remaining, it is possible for each of the other stalls to.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Hold at least 1 sheep (a Yes answer).


GMAT number properties note: However, the 23 other sheep might all be in stall B, leaving stalls A, D, F, and G empty (a No answer). Set up a table and assign sheep to stalls.


GMAT number properties note: Stall A B C D E F G # of Sheep 5y 2y If y = 1, there are 5 sheep in E and 2 sheep in F. With 26 sheep remaining, it is possible for each of the other stalls to hold at least 1 sheep (a Yes answer).


GMAT number properties note: However, 13 sheep might be in both stalls A and B, leaving stalls C, D, and G empty (a No answer). The only possibility is x = 5 and y = 3, since higher multiples would require more than 33 sheep total.


GMAT number properties note: Thus, 31 sheep are allocated among these stalls as follows.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Stall A B C D E F G # of Sheep 10 15 6 Thirty-one of the 33 sheep are in three of the 7 pens.


GMAT sequences and patterns note: With only 2 sheep unaccounted for, there is no way to place at least 1 sheep in each of the remaining four pens (a definite No answer). Group the 10r terms together.


GMAT sequences and patterns note: Since q > r, the exponent on 10q − r is positive. Since s is a positive integer, s × 10q − r is a multiple of 10 and therefore ends in 0.


GMAT sequences and patterns note: Any multiple of 10 minus 1 yields an integer with a units digit of 9.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The complicated expression in the question stem leads to a disguised Positive/Negative problem.


GMAT algebra note: In both cases (positive or negative 3), the end result is 3. Thus, in general, will always result in a positive value, or |x|.


GMAT algebra note: Rephrase the original question using the absolute value symbol in place of the “square root of the square” symbols. Then try to make the right-hand side look more like the le.


GMAT algebra note: Becomes Is |y − 4| = −(y − 4)? Since the absolute value of y − 4 must be positive or zero, you can rephrase the question further.


GMAT algebra note: The absolute value |y − 3| can be interpreted as the distance between y and 3 on a number line.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If y × |y| > 0, then y × |y| is positive.


GMAT number properties note: This means y and |y| must have the same sign. The term |y| is non-negative, so y must be positive.


GMAT number properties note: A base of 2 is common to each term, and 10 is the smallest exponent appearing with that base. Factor 210 out from all of the terms in the expression.


GMAT number properties note: Clearly, 2 is a prime factor, but is it the greatest? Examine (54 − 2352 + 24) to determine whether it has a larger prime factor.


GMAT number properties note: The expression is of the form x2 − 2xy + y2, where:. Write the expression in factored form.


GMAT number properties note: The prime factors of 21 are 3 and 7, so the largest prime factor of the original expression is 7.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Alternatively, if you did not see the quadratic template in (54 − 2352 + 24), you could also perform the computation and factor the result.


GMAT fractions, decimals, and percents note: According to the text, a “shrinking number” is a decimal between 0 and 1 in which each digit to the right of the decimal is no smaller than the digit to its immediate right. That is, in a shrinking number, each digit is either larger than the one to its right, or the same as the one to its right.


GMAT fractions, decimals, and percents note: As you go through the digits from le to right, the digits will either stay the same or get smaller. For instance, 0.5331 is a shrinking number, but 0.37654 is not a shrinking number.


GMAT fractions, decimals, and percents note: This Roman Numeral problem asks which of three options must be true. Test cases and eliminate any options that are not necessarily true.


GMAT fractions, decimals, and percents note: Because the digits do not get larger a er the decimal point.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Number is relatively easy to do math with, start by using x = 0.1 to test cases.


GMAT fractions, decimals, and percents note: However, this is not a shrinking number, since 9 is greater than 0. So statement I is not necessarily true.


GMAT fractions, decimals, and percents note: , which is a shrinking number. In fact, statement II must always be true.


GMAT fractions, decimals, and percents note: Adding 9 and then dividing by 10 is equivalent to inserting the digit 9 to the immediate right of the decimal. Since 9 is the largest possible digit, it will never be lower than the digit to its immediate right, so the decimal will still be a shrinking number.


GMAT fractions, decimals, and percents note: , which is not a shrinking number, since the digit two places to the right of the decimal is greater than the digit immediately to the right of the decimal. Statement III is not necessarily true.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. A gallery has seven paintings with different prices.


GMAT statistics note: Is it possible to purchase at least three of the paintings for no more than $1,800? This is possible only if the three least expensive paintings at the gallery have a total price of $1,800 or less.


GMAT statistics note: The question can be rephrased as, “Is the total price of the three least expensive paintings $1,800 or less?” (1) SUFFICIENT:. The median price of the seven paintings is $550.


GMAT statistics note: Since all of the paintings have different prices, the three least expensive paintings have prices below the median. Therefore, the total price of the three least expensive paintings is less than $550 + $550 + $550 = $1,650.


GMAT statistics note: The answer is definitely Yes, so this statement is sufficient. This statement implies that the four least expensive paintings have a total price of $2,300 or less.


GMAT statistics note: Jot down some possible cases. Case 1: $1, $2, $3, and $2,000.


GMAT statistics note: The three least expensive paintings can be purchased for $1,800, and the answer to the question is Yes. To find a No answer, try to maximize the lowest three prices out of the four.


GMAT statistics note: To do so, make the highest price as low as possible, or only slightly more than one-fourth of the total. Since one-fourth of $2,300 is $575, a good case to test is $573, $574, $576, and $577.


GMAT coordinate geometry note: The inequality y > |x| represents everything above the line (on either side of the y-axis)—that is, the shaded region. Since the equation y = |x| forms a 45-degree angle from the x-axis, there are 90 degrees above the line (on both sides of the y-axis).


GMAT coordinate geometry note: This represents one-fourth of the xy-plane. Therefore, if a random pair of (x, y) coordinates is chosen from the plane, the probability is that the point will fit the criterion y ≥ |x|.


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. If 3a + 3a − 2 = (90)(3b), what is b in terms of a?


GMAT number properties note: If xy ≠ 0, what is the value of y = 4 − x(1) x(x − 6y) = −9y2(2) 84. When one new integer is added to an existing list of six integers, does the median of the list change?


GMAT number properties note: The mean of the original six numbers is 50.(1) At least two of the numbers in the original list were 50.(2) 85. If abc + dcb = 598, then which of the following must be equivalent to a?


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. For nonzero integers a, b, c and d, is negative?


GMAT number properties note: Ad + bc = 0(1) abcd = −4(2) 87. Set A consists of 8 distinct prime numbers.


GMAT number properties note: If x is equal to the range of set A and y is equal to the median of set A, is the product xy even? The smallest integer in the set is 5.(1) The largest integer in the set is 101.(2) 88.


GMAT number properties note: If x3 < 16x, which of the following CANNOT be true?


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Is the two-digit positive integer n divisible by 3?


GMAT quant note: If the digits of n are reversed to produce the two-digit integer m, then m is divisible by 3. If the digits of n are reversed to produce the two-digit integer m, then m + n is divisible by 3.


GMAT quant note: Of all the sheets of paper in the tray of Printer A, 20% are removed and transferred to the tray of Printer B. A er the transfer, are there more sheets of paper in the tray of Printer B than in the tray of Printer A?


GMAT quant note: The transfer increases the number of sheets of paper in the tray of Printer B by more than 25%.


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


GMAT geometry note: The circumference of the circle is equal to so The area of the circle is equal to The area of a square is side2. The diagonal of this square is the diameter of the circle, which is equal to .


GMAT geometry note: The diagonal of a square is always , so the side of the square is equal to Therefore, side and the area of square ABCD is:. The shaded area is the area of the circle minus the area of the square = πx − 2x.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Because the problem never provides real values for a or b, you can choose smart numbers to solve.


GMAT algebra note: Choose something for a that will make both exponents on the le side positive. Plug a = 3 into the answers and look for −1.


GMAT algebra note: Alternatively, you can solve algebraically. Some manipulation is required to determine b in terms of a, but spend a minute thinking strategically about what manipulations to do.


GMAT algebra note: It would help to isolate terms with an a by factoring 3a out on the le side. This will let you compare 3a to 3b, and thus compare a to b, once you clean up the constant terms that are le over.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Ninety is also the sum of 81 and 9, both powers of 3 themselves.


GMAT algebra note: It is likely that some constant terms will cancel, as shown below. The constraint in the question stem indicates that neither x nor y equals zero.


GMAT algebra note: You can prove insufficiency by Testing Cases. If y = 1, then x = 3, in which case is 3.


GMAT algebra note: If y = 2, then x = 2, in which case is 1. Testing Cases might not be such a great idea for this statement because there are multiple instances of each variable.


GMAT algebra note: Begin by distributing the le -hand side of the equation.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Now, you’ve got a quadratic, so solve that way.


GMAT statistics note: You can test cases to try to prove or disprove the statements. If the original set is 50, 50, 50, 50, 50, 50, the mean and median were 50 originally.


GMAT statistics note: The new term may be any value and the median will not change. If, on the other hand, the original set is 0, 0, 0, 100, 100, 100, the mean and median were 50 originally.


GMAT statistics note: If the new term is 75, the median increases to 75. More generally for this set, if the new term is not equal to the original median of 50, the median will change.


GMAT statistics note: If the original set is 50, 50, 50, 50, 50, 50, the median was 50 originally. If, on the other hand, the original set is 0, 0, 0, 50, 50, 50, the median was 25 originally.


GMAT statistics note: If the new term is 40, the median increases to 40.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Generally for this set, if the new term is not equal to the original median of 25, the median will change.


GMAT statistics note: The mean must be 50 and at least two terms in the original set must be 50. If the original set is 50, 50, 50, 50, 50, 50, the mean and median were 50 originally.


GMAT statistics note: The new term may be any value and the median will not change. If the original set is 0, 0, 0, 50, 50, 200, the mean was 50 and the median was 25 originally.


GMAT statistics note: If the new term is greater than 25, the median increases. If the new term is less than 25, the median decreases.


GMAT statistics note: Set up the addition and extract several equations by summing each digit place individually. Note that both the ones digit and tens digit come from the sum c + b.


GMAT statistics note: Since the result is different in the ones digit (8) and the tens digit (9), a 1 must be carried from the ones to the tens digit. If c + b = 18, then both c and b must be 9 (the largest digit).


GMAT statistics note: Place an 8 in the ones digit of the sum and carry a 1 to the tens place.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Tens digit is thus 1 + b + c = 1 + 18 = 19.


GMAT number properties note: Next, place a 9 in the tens digit of the sum and carry a 1 to the hundreds place. In the hundreds place, the sum is 1 + a + d = 5.


GMAT number properties note: If an even number (0, 2, or 4) of the integers a, b, c, and d is negative, each pair of negatives will cancel, because (−1)(−1) = +1 and . This would yield a positive result for Thus, a way to rephrase the question is, “Among the integers a, b, c, and d, are an odd number (one or three) of them negative?” (1) SUFFICIENT:.


GMAT number properties note: This statement can be rephrased as ad = −bc. A d b c ad = −bc Odd number of negatives?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. A d b c ad = −bc Odd number of negatives?


GMAT number properties note: − − − + ✓ Yes + − − − ✓ Yes Though the table doesn’t list all possibilities, it lists enough to realize that, in order for the signs of ad and bc to be opposite one another, either one or three of the four integers must be negative. You might recognize that the (−1)(−1) = +1 property implies that abcd is only negative when there are non-paired negatives among the integers.


GMAT number properties note: That is, an odd number (one or three) of the integers a, b, c, and d must be negative. If not, you could list a few cases to see the pattern.


GMAT number properties note: A b c d abcd = negative Odd number of negatives? + + − + ✓ Yes + − + + ✓ Yes − + − − ✓ Yes − − + − ✓ Yes.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The product xy will be even if x is even, y is even, or both are even.


GMAT statistics note: The prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, etc. The smallest possible term in set A is 2, which is the only even prime.


GMAT statistics note: X = the range of set A = largest term − smallest term in set A. If the smallest term in set A is 2, then x = odd − even = odd.


GMAT statistics note: If the smallest term in set A is odd (i.e., not 2), then x = odd − odd = even. The median of set A is the average of the two middle terms, since the number of terms in the set is even.


GMAT statistics note: However, y could be either even (e.g., when the middle terms are 11 and 13) or odd (e.g., when the middle terms are 7 and 11). A useful rephrase of this question is, “Is either x or y even?” (1) SUFFICIENT:.


GMAT statistics note: If the smallest prime in the set is 5, x = even, and therefore xy is even. If the largest integer in the set is 101, the range of the set can be odd or even (e.g., 101 − 3 = 98 or 101 − 2 = 99).


GMAT statistics note: The median of the set can also be odd or even, as discussed. Therefore, xy can be either odd or even.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It may be tempting to simplify this way:.


GMAT algebra note: The first step of this solution is wrong because you can’t divide by x without knowing its sign. If x is negative, you would have to flip the sign.


GMAT algebra note: Positive x2 < 16 (don’t flip) |x| < 4 0 < x < 4 Negative x2 > 16 (flip) |x| > 4 x < −4 There are two ranges of solutions for x. The question asks for something that is definitely NOT true.


GMAT algebra note: Alternatively, there is a way to do algebra with inequalities containing variables. Instead of dividing by the variable, subtract it from both sides, then factor.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. In order for the product of three numbers to be negative, either all three need to be negative, or exactly one of the three needs to be negative.


GMAT number properties note: This cannot occur if x is greater than 4, since all three values would be positive. Before diving into the statements, remind yourself of the divisibility rules.


GMAT number properties note: One way to check for divisibility by 3 is to add the digits. If their sum is divisible by 3, then the number itself is divisible by 3.


GMAT number properties note: If m is divisible by 3, then the sum of the digits of m must be divisible by 3 as well. Since m has the same digits as n, the sum of the digits of n must also be a multiple of 3.


GMAT number properties note: Therefore, n itself is divisible by 3. This statement requires a bit more work because it’s not immediately clear what would happen when you add m and n.


GMAT number properties note: Because the question asks you to move digits around to different places (the tens place, the units place), rewrite the information using place-value notation, where a = the tens digit of n and b = the units digit of n.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. The statement indicates that the sum m + n is divisible by 3, so 11(a + b) must also be divisible by 3.


GMAT algebra note: Eleven itself is not divisible by 3, so the sum a + b must be divisible by 3. This sum represents the sum of the two digits of the number m as well as the sum of the two digits of the number n.


GMAT algebra note: If the sum of n’s two digits is divisible by 3, then n is also divisible by 3. You can solve this problem algebraically or by Testing Cases.


GMAT algebra note: If Printer A starts out with 100 sheets, then 20 sheets are moved to Printer B, and Printer A ends up with 80 sheets. If B started out with 75 sheets, then an increase of 20 sheets would be more than 25% of the starting number, and B would end up with 95 sheets.


GMAT algebra note: In this case, B has more sheets than A a er the transfer. If, instead, B started out with 40 sheets, then an increase of 20 sheets would be more than 25% of the starting number, and B would end up with 60 sheets.


GMAT algebra note: In this case, B has fewer sheets than A a er the transfer. If Printer A starts out with 100 sheets and 20 are moved to Printer B, then A ends up with 80 sheets.


GMAT algebra note: If B started out with 2,000 sheets, then an increase of 20 sheets would be less than 30% of the starting number, and B would end up.


GMAT algebra note: Percent problems usually relate part, whole, rate, and percent change. If B started out with 80 sheets, then a 20-sheet increase would still be less than 30% of the starting number, and again B would end up with more sheets than A.


GMAT algebra note: If the 20 sheets represent a less-than-30% increase, then use 30% as the limiting figure. If 20 represented exactly 30% of B’s sheets, then B would have to start with sheets (ignore the fact that this isn’t an integer).


GMAT algebra note: In this case, B would end up with sheets, which is still greater than A’s 80 sheets. A er the transfer, B can’t drop below A.


GMAT algebra note: Algebraically, use a for the original number of sheets in Printer A and use b for the original number of sheets in Printer B. A er the transfer, A has a – 0.2a = 0.8a sheets.


GMAT algebra note: Rephrase the question: “Is b + 0.2a > 0.8a?” Or, “Is b > 0.6a?” (1) INSUFFICIENT:. Translate the statement into algebra.


GMAT algebra note: Flip the statement around to compare it more easily to the rephrased question: b < 0.8a.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Therefore, or The decimal equivalent of is , so b is indeed always greater than 0.6a.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If |x| ≠ |y|, xy ≠ 0, , and , then is equal to which of the following?


GMAT word problem note: If a certain culture of bacteria increases by a constant factor of x every y minutes, how long will it take for the culture to increase to 10,000 times its original size?


GMAT number properties note: Percent problems usually relate part, whole, rate, and percent change. In two minutes, the culture will increase to 100 times its original size.


GMAT number properties note: A cylinder of height h is full of water. When all of the water is poured into an empty cylinder whose radius is 25% larger than that of the original cylinder, the new cylinder is full.


GMAT number properties note: The height of the new cylinder is what percent of h? If a and b are distinct positive integers, what is the units digit of 2a8b4a + b?


GMAT number properties note: B = 24 and a < 24(1) The greatest common factor of a and b is 12.(2) 95. Employees of a certain company are each to receive a unique seven-digit identification code consisting of the digits 0, 1, 2, 3, 4, 5, and 6 such that no digit is used more than once in any given code.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. In valid codes, the second digit in the code is exactly twice the first digit.


GMAT sets, probability, and counting note: How many valid codes are there? An n-sided die has sides labeled with the numbers 1 through n, inclusive, and has an equal probability of landing on any side when rolled.


GMAT sets, probability, and counting note: If the die is rolled twice, what is the probability of rolling 1 both times? If the die is rolled twice, the probability that the two rolls are different is 80%.


GMAT sets, probability, and counting note: If the die is rolled twice, the probability that the two rolls are equal is 97. If w, x, y, and z are positive integers and , what is the proper order, increasing from le to right, of the following quantities.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. What is the value of |a + b|?


GMAT number properties note: (a + b + c + d) (a + b − c − d) = 16(1) c + d = 3(2) 99. Set A consists of all the integers between 10 and 21, inclusive.


GMAT number properties note: Set B consists of all the integers between 10 and 50, inclusive. If x is a number chosen randomly from set A, y is a number chosen randomly from set B, and y has no factor z such that 1 < z < y, what is the probability that the product xy is divisible by 3?


GMAT quant note: At a birthday party, x children will be seated at two different tables. At the table with the birthday cake on it, exactly y children will be seated, including the birthday girl, Sally.


GMAT quant note: How many different groups of children may be seated at the birthday cake table?


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


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


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. Workout Set 10 Solutions 91.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This problem can be solved algebraically, but this path is not recommended.


GMAT algebra note: Now that the fractions have a common denominator of x, subtract one from the other. You are not alone if the algebraic solution was not obvious to you!


GMAT algebra note: Algebraic “false starts” are common in this type of problem. In addition, there are other, equally valid algebraic paths whose final forms would not match any of the answers.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It’s fast, easy, and correct.


GMAT algebra note: To understand the question stem, pick some numbers: the bacteria culture begins with an initial quantity of I = 100 and increases by a factor of x = 2 every y = 3 minutes. Construct a table to track the growth of the bacteria.


GMAT algebra note: Time (min.) Bacteria 3 100(2) = 100(2)1 6 100(2)(2) = 100(2)2 9 100(2)(2)(2) = 100(2)3 12 100(2)(2)(2)(2) = 100(2)4 3n = t 100(2)n n represents the number of growth periods, and where t is time in minutes.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This question asks how long it will take for the bacteria to grow to 10,000 times their original amount.


GMAT algebra note: In other words, “What is t when F = 10,000I?” The rephrased question is, “What is t when (1) SUFFICIENT:. Note that the yth root of x is equivalent to x to the power.


GMAT algebra note: This statement indicates that . If you plug this value into the equation, you can solve for t (though stop the calculation at the point that you can tell that you can solve for t).


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. The culture grows one-hundredfold in 2 minutes.


GMAT fractions, decimals, and percents note: In other words, the sample grows by a factor of 102. Since exponential growth is characterized by a constant factor of growth (i.e., by a factor of x every y minutes), in another 2 minutes, the culture will grow by another factor of 102.


GMAT fractions, decimals, and percents note: Therefore, a er a total of 4 minutes, the culture will have grown by a factor of 102 × 102 = 104, or 10,000. Also note that the problem mentions the volume of the cylinder (V = πr2h), so you’re going to need a radius.


GMAT fractions, decimals, and percents note: That radius needs to be increased by 25% later in the problem, so pick a small integer that is easy to increase by 25%, such as 4. The problem also contains two fractions, and .


GMAT fractions, decimals, and percents note: Since one number is already a multiple of 4, make the other number a multiple of 5.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. Now, solve the problem using these numbers.


GMAT quant note: the original cylinder is only full. it is only full.


GMAT quant note: The original height was 5, so the new height is , or 80%, of the original height. This problem contains a common trap seen in many difficult DS questions.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Combined statements would provide the values of both a and b, which could be plugged into the expression to answer the question.


GMAT sequences and patterns note: Remembering the units digit patterns for powers of 2 will help on this problem:. The units digits for powers of 2 is a repeating pattern of [2, 4, 8, 6].


GMAT sequences and patterns note: If you can determine the relationship of 3a + 5b to a multiple of 4 (i.e., where 23a + 5b is in the predictable four-term repeating pattern of units digits), you will be able to answer the question. This question can be rephrased as, “What is the remainder when 3a + 5b is divided by 4?”.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. If b = 24, then 5b is a multiple of 4.


GMAT sets, probability, and counting note: However, a could be any integer less than 24. Possible remainders when 3a is divided by 4 are 0, 1, 2, or 3.


GMAT sets, probability, and counting note: If the greatest common factor of a and b is 12, then 12 must be a factor of both variables. That is, both a and b are multiples of 12 and thus also multiples of 4.


GMAT sets, probability, and counting note: As a result, 3a and 5b will be multiples of 4 as well, so the remainder will be 0 when 3a + 5b is divided by 4. Valid codes must have a second digit that is exactly twice the first digit.


GMAT sets, probability, and counting note: There are three ways to do this with the available digits. Scenario A: 12XXXXX Scenario B: 24XXXXX Scenario C: 36XXXXX For each of these basic scenarios, there are 5!


GMAT sets, probability, and counting note: Ways to shuffle the remaining five numbers (represented by X’s above). Thus, the total number of valid codes is 3 × 5!


GMAT sets, probability, and counting note: When an n-sided die is rolled once, the probability of rolling any particular number (such as 1) is equal to . Therefore, the probability of.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Rolling 1 both times is equal to .


GMAT sets, probability, and counting note: Rephrase the question as, “What is the value of n?” (1) SUFFICIENT:. The probability that the two rolls are different is 80%, or When rolling an n-sided die twice, the first roll can have any value from 1 to n, inclusive.


GMAT sets, probability, and counting note: Regardless of the value of the first roll, the probability is that the second roll will have that same value. Therefore, the overall probability of rolling the same value twice is , and the probability of rolling two different values is Therefore, according to this statement, .


GMAT sets, probability, and counting note: The value of n is 5, and this statement is sufficient. The probability that two rolls of an n-sided die will have the same value is , regardless of the value of n.


GMAT sets, probability, and counting note: This statement would be true for any value of n, so the value of n cannot be determined and this statement is insufficient.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. It would require a lot of tricky work to solve this algebraically, so test cases instead.


GMAT algebra note: Make sure to pick values for the unknowns such that holds true. Before plugging those values in for the quantities, check the answers.


GMAT algebra note: All of them begin with 1 and , so don’t bother to test . Now, place in ascending order:.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The question asks for the absolute value of a + b, so try to manipulate the statements to isolate that combination of variables, a + b.


GMAT algebra note: This provides information about c and d, and the relationship between them, but no information about a or b. Manipulate the equation to group (a + b) and (c + d) terms.


GMAT algebra note: Note that this is of the form (x + y)(x − y), where x = (a + b) and y = (c + d). This is the difference of squares special product, (x + y)(x − y) = x2 − y2.


GMAT algebra note: Use this to transform this expression. This is not enough to determine the value of (a + b)2.


GMAT algebra note: You can substitute for c + d and solve. The solution is shown below, but note that you don’t have to do that math:.


GMAT algebra note: The question asked for the absolute value of a + b, so the fact that the equation has (a + b)2 doesn’t matter.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If y has no factor z such that 1 < z < y, then y must be prime.


GMAT number properties note: Because y is selected from set B, it is a prime number between 10 and 50, inclusive. The only prime number that is divisible by 3 is 3, so y is definitely not divisible by 3.


GMAT number properties note: Thus, xy is only divisible by 3 if x itself is divisible by 3. Rephrase the question: “What is the probability that a multiple of 3 will be chosen randomly from set A?” There are 21 − 10 + 1 = 12 terms in set A.


GMAT number properties note: Of these, 4 terms (12, 15, 18, and 21) are divisible by 3. 6 has a factor 2 such that 1 < 2 < 6: 6 is NOT prime.


GMAT number properties note: 15 has a factor 5 such that 1 < 5 < 15: 15 is NOT prime. 3 has NO factor between 1 and 3: 3 IS prime.


GMAT number properties note: 7 has NO factor between 1 and 7: 7 IS prime.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. There are 8 children at the party, and 5 will sit at the table with the cake.


GMAT statistics note: Sally must sit at the birthday cake table, so pick 5 − 1 = 4 of the other 8 − 1 = 7 children to sit at that table with her. How many different ways can you choose 4 from a group of 7?


GMAT statistics note: Set up an anagram grid, where Y means “at the cake table” and N means “at the other table.” A B C D E F G Y Y Y Y N N N Now, you can calculate the number of possible groups:. It may be the case that you will find the unsimplified factorial form, rather than 35.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. As an alternative to testing all five choices, you could use a hybrid approach to determine the formula using variables.


GMAT algebra note: The number of possible groups was , but remember that this formula took Sally into account.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Substitute these variable expressions in place of the numbers.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. A casino pays players with chips that are either turquoise or violet colored.


GMAT word problem note: If each turquoise-colored chip is worth t dollars and each violet-colored chip is worth v dollars, where t and v are integers, what is the combined value of four turquoise-colored chips and two violet-colored chips? The combined value of six turquoise-colored chips and three violet-colored chips is $42.


GMAT word problem note: The combined value of five turquoise-colored chips and seven violet-colored chips is $53. Jonas and Amanda stand at opposite ends of a straight road and start running toward each other at the same moment.


GMAT word problem note: Their rates are randomly selected in advance so that Jonas runs at a constant rate of 3, 4, 5, or 6 miles per hour and Amanda runs at a constant rate of 4, 5, 6, or 7 miles per hour. What is the probability that Jonas has traveled farther than Amanda by the time they meet?


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. If p is a positive integer, is p2 divisible by 96?


GMAT quant note: P is a multiple of 8.(1) p2 is a multiple of 12.(2).


GMAT geometry note: Parallel sides AB and CD are 7 inches apart and 6 and 8 inches long, respectively. What is the radius of the circle in inches?


GMAT geometry note: If 9y + 3b = 10(3b), then 2y is equal to which of the following?


GMAT geometry note: The radius of a circle is r yards. Is the area of the circle at least r square yards?


GMAT geometry note: (1 yard = 3 feet) The diameter of the circle is more than 2 feet.(1) If the radius of the same circle is f feet, the area of the circle is more than 2f square feet. For how many values of x from 1 to 300, inclusive, is the sum of the integers from 1 to x, inclusive, divisible by 3?


GMAT geometry note: If x and y are integers and |xy| = 8, what is the value of |x + y|? X and y are both divisible by 2.(1).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set X consists of exactly four distinct integers that are greater than 1.


GMAT number properties note: For each integer in the set, all of that integer’s unique prime factors are also in the set. How many of the distinct integers in set X are prime?


GMAT number properties note: The product of the integers in set X is divisible by 36.(1) The product of the integers in set X is divisible by 60.(2) 110. If p = (22)(3x) and r = (22)(3y), where x and y are prime numbers and x ≠ y, which of the following represents the least common multiple of p and r?


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


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Workout Set 11 Solutions 01.


GMAT number properties note: The question asks for the value of 4t + 2v, where t and v represent the values of the turquoise and violet chips, respectively. Note that the question asks for a combination of variables, or a combo.


GMAT number properties note: It may not be necessary to be able to solve for the individual values of t and v. Translate and simplify the statement.


GMAT number properties note: You can’t solve for t and v, so this statement might look sufficient—but remember what the question is asking! You need to find the value of 4t + 2v.


GMAT number properties note: Multiply the equation by 2: 4t + 2v = 28. There isn’t a way to simplify this one, but notice all those prime numbers.


GMAT number properties note: Primes tend to minimize the number of allowable scenarios, especially when the question also specifies that the variables have to be integers. See what combinations of integers would actually work here.


GMAT number properties note: The 5t term could be 5, 10, 15, 20, and so on. This term can contribute only numbers that end in 5 or 0.


GMAT number properties note: If that’s the case, the 7v term must have a units digit of either 3 or 8. List out the possibilities for 7v, and try only the ones that end in 3 or 8.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 7v: 7, 14, 21, 28, 35, 42, 49 Only one ends in 3 or 8!


GMAT word problem note: If 7v = 28, then v = 4 and the 5t term must equal 25, so t = 5. This one scenario works and it is the only possible scenario for this equation, given that v and t must be integers.


GMAT word problem note: If Jonas and Amanda run at the same rate, they will meet each other exactly in the middle. Jonas will only run farther than Amanda if Jonas’ rate is greater than Amanda’s.


GMAT word problem note: In math terms, Distance = rate × time, and since Jonas and Amanda run for the same time, their relative distances depend solely on their relative rates. Rephrase the question as, “What is the probability that Jonas ran faster than Amanda?” There are four possible rates for Jonas (3, 4, 5, and 6) and four for Amanda (4, 5, 6, and 7).


GMAT word problem note: In total, there are (4)(4) = 16 possible rate scenarios. Of Jonas’ four possible rates, only two (5 and 6) are greater than some of Amanda’s possible rates (4 and 5).


GMAT word problem note: List the three rate scenarios that result in a faster speed (greater distance) for Jonas. Jonas ran 5 miles per hour (mph), and Amanda ran 4 mph.


GMAT word problem note: Jonas ran 6 mph, and Amanda ran 4 mph.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The prime factorization of 96 is (2)(2)(2)(2)(2)(3) = (25)(31).


GMAT number properties note: In order for p2 to be divisible by 96, p2 would have to have the prime factors 2531 in its prime box. The rephrased question is therefore, “Does p2 have at least five 2’s and one 3 in its prime box?” (1) INSUFFICIENT:.


GMAT number properties note: If p is a multiple of 8 = (2)(2)(2), p has 23 in its prime box. Therefore, p2 has (23)2 = 26 in its prime box, and thus has the required five 2’s.


GMAT number properties note: However, it is uncertain whether p2 has at least one 3 in its prime box. If p2 is a multiple of 12 = (2)(2)(3), p2 has two 2’s and one 3 in its prime box.


GMAT number properties note: It is uncertain whether p2 has at least five 2’s total as there may or may not be three more 2’s in the prime box. Therefore, it is certain that p2 has at least five 2’s and one 3 in its prime box.


GMAT geometry note: In order for the trapezoid vertices to lie on the circle, the trapezoid must be symmetrical about the dotted line, which passes through the center of the circle. Draw this vertical and the radii to points B and C to create two right triangles, allowing you to use the Pythagorean theorem.


GMAT geometry note: In fact, you might play an educated hunch that the triangles are 3–4–5 common right triangles. If hypotenuse r is 5, then each triangle has a 3 and 4 side.


GMAT geometry note: The unknown vertical sides are thus 4 and 3, which sum to 7 as they must. Algebraically, set up the following equations from the picture.


GMAT geometry note: Setting the two equations for r2 equal.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Since (x + y)(x − y) = 7, (x − y) = 1.


GMAT algebra note: The radius of the circle is 5, because r2 = 32 + 42 = 25. First, simplify the given equation.


GMAT algebra note: First, translate the question stem:.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Note that the question is stated in yards, but the statements use feet, so convert the question stem to feet.


GMAT sequences and patterns note: The value of is approximately , or a little bit less than 1. In other words, or approximately .


GMAT sequences and patterns note: Therefore, f could be smaller than feet (just a bit smaller than 1) but it could also be larger. Start at x = 1, then x = 2, and so on, looking for a pattern.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X Sum of the integers from 1 to x, inclusive Is the sum divisible by 3?x Sum of the integers from 1 to x, inclusive Is the sum divisible by 3?


GMAT number properties note: 1 1 No 2 3 Yes 3 6 Yes 4 10 No 5 15 Yes 6 21 Yes 7 28 No It appears that the pattern is No, Yes, Yes, No, Yes, Yes, etc., repeating every three terms. Therefore, 2 out of every 3 values of x yield a sum that is divisible by 3.


GMAT number properties note: So 200 of the 300 values of x will yield a sum that is divisible by 3. If the two variables must be integers and the absolute value of their product is 8, then x and y have to represent either the factor pair (1, 8) or the factor pair (2, 4), in either order and with either sign (positive or negative).


GMAT number properties note: This narrows the possible number of cases considerably. List out the possible cases allowed by this statement.


GMAT number properties note: If x and y are both divisible by 2, then you’re dealing with the factor pair (2, 4). A er you try your first case, ask yourself what different case would be most likely to return a different answer to the question.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. In this case, just reversing the numbers (4, 2) won’t make a difference to the final question, so don’t list that case second.


GMAT quant note: Try one of the cases that allows a negative value. Now that you have two different values, you know the statement is not sufficient.


GMAT quant note: This statement indicates that the two values have the same sign: both positive or both negative. Again, list cases until you have two contradictory results (or until you’ve tried all cases and realized that there is only one possible answer).


GMAT quant note: (1) AND (2) SUFFICIENT: x y Both divisible by 2?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Only four possible cases are allowed: the variables must match the (2, 4) factor pair and the signs have to be the same.


GMAT number properties note: The chart above shows two of the four possible cases, but you could also reverse the order: (4, 2) and (−4, −2). Since you’re just adding at the end, though, the order of the two variables doesn’t matter.


GMAT number properties note: In all four cases, |x + y| = 6. Set X contains exactly four distinct integers.


GMAT number properties note: Distinct means the four integers must be different. Also, if a number is contained in the set, all of its unique prime factors must also be in the set.


GMAT number properties note: If a prime number, such as 3, is in the set, no additional numbers are required in the set: the only prime factor of 3 is 3 itself, which is already included. However, if the composite number 8 is in the set, its one unique prime factor, 2, must also be included.


GMAT number properties note: To better understand the question, jot down some four-integer sets that fit the description. Specifically, look for sets that have different numbers of primes.


GMAT number properties note: {2, 3, 5, 7} is a valid set that contains four primes. {2, 4, 8, 16} is a valid set that contains only one prime, since all of the terms are powers of 2.


GMAT number properties note: The question deals with primes and divisibility.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The prime factorization of 36 is 36 = 2 × 2 × 3 × 3.


GMAT number properties note: For the product of the four integers in the set to be divisible by 36, the four integers must include 2, 2, 3, and 3 as prime factors, in some combination. Therefore, based on the definition of the set, 2 and 3 must both be included in the set.


GMAT number properties note: The remaining two numbers must include at least one 2 and at least one 3 as prime factors between them. However, there are several different valid sets that fit this criterion.


GMAT number properties note: Case 1: {2, 3, 6, 9} has a product divisible by 36, since it includes at least two 2’s and at least two 3’s in its prime factors. Also, it is a valid set since it includes all of the unique prime factors of 6 (2 and 3) and all of the unique prime factors of 9 (only 3).


GMAT number properties note: The primes 2 and 3 must be included in the set. Include a third prime as well, such as 5.


GMAT number properties note: The final number in the set must include 2 and 3 as prime factors in order for the product of the whole set to be divisible by 36. Because the set might have either two primes or three primes, the statement is insufficient.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Each also has an additional prime factor that is not shared by the other.


GMAT number properties note: Draw overlapping circles in which to place the shared and non-shared prime factors of p and r. To find the least common multiple (LCM), multiply from le to right and include all the common factors in the product.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If y = |x − 1| and y = 3x + 3, then x must be between which of the following values?


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. If a is a positive integer, is a2 a multiple of 8?


GMAT word problem note: A3 is a multiple of 16.(1) (a + 4)2 is a multiple of 8. Between the two black lines, they will paint across the road a red line at each third of a mile, a white line at each fi h of a mile, and a blue line at each eighth of a mile.


GMAT word problem note: What is the smallest distance (in miles) between any of the painted lines on this stretch of highway?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set S consists of n consecutive integers, where n > 1.


GMAT number properties note: The sum of the integers in set S is divisible by 7.(1) The sum of the integers in set S is 14.(2) 116. A trapezoid is symmetrical about a vertical center line.


GMAT number properties note: If a circle is drawn such that it is tangent to exactly three points on the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle? The parallel sides of the trapezoid are 10 inches apart.(1) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Three boys are ages 4, 6, and 7, respectively.


GMAT sets, probability, and counting note: Three girls are ages 5, 8, and 9, respectively. If two of the boys and two of the girls are randomly selected and the sum of the selected children’s ages is z, what is the difference between the probability that z is even and the probability that z is odd?


GMAT sets, probability, and counting note: If two distinct integers x and y are randomly selected from the integers between 1 and 10, inclusive, what is the probability that is a positive odd integer?


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Positive integers a and b are less than or equal to 9.


GMAT sequences and patterns note: If a and b are assembled into the six-digit number ababab, which of the following must be a factor of ababab? The two-digit positive integer s is the sum of the two-digit positive integers m and n.


GMAT sequences and patterns note: Is the units digit of s less than the units digit of m? The units digit of s is less than the units digit of n.(1).


GMAT quant note: The tens digit of s is not equal to the sum of the tens digits of m and n.


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


GMAT algebra note: Workout Set 12 Solutions 11. Parallel lines have the same slope.


GMAT algebra note: Work backwards from the answers by rearranging into slope-intercept form: form, where the slope is m. Since only one answer can be correct, only one answer will have a slope of.


GMAT algebra note: Stop when you find the one with the matching slope.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Set the two equations for y equal and algebraically solve |x − 1| = 3x + 3 for x.


GMAT number properties note: This requires two solutions: one for the case that x − 1 is positive, the other for the case that x − 1 is negative. X – 1 is positive: x – 1 is negative:.


GMAT number properties note: In the x − 1 is positive case, the answer is x = −2. In this case, x − 1 is not actually positive, so this is a false case.


GMAT number properties note: As a result, there is only one solution:. In order for a2 to be a multiple of 8, the prime factorization of a2 must include all of the prime factors of 8.


GMAT number properties note: That is, to be a multiple of 8, a2 would have to include 2, 2, and 2 among its prime factors. When an integer is squared, all of its prime factors are duplicated.


GMAT number properties note: Therefore, if a only includes a single 2 among its prime factors, a2 might only include two 2’s. In order to guarantee that a2 includes at least three 2’s, a must include at least two 2’s in its prime factors.


GMAT number properties note: The question translates to: “Does the prime factorization of a include at least two 2’s?” Or, “Is a divisible by 4?”.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The positive integer a, when cubed, is a multiple of 16.


GMAT algebra note: Odd integers, when cubed, will always yield an odd result, so a cannot be odd. What positive even values could a have?


GMAT algebra note: If a = 2, then a3 = 8, which is not a multiple of 16. However, if a = 4, then a3 = 64, which is a multiple of 16, as are 83 = (43)(23) and 123 = (43)(33).


GMAT algebra note: Therefore, this statement indicates that a is a multiple of 4. When a multiple of 4 is squared, the result is always a multiple of 8, so the answer is definitely Yes.


GMAT algebra note: Start by multiplying out the expression in the statement. (a + 4)2 is a multiple of 8.


GMAT algebra note: A2 + 8a + 16 is a multiple of 8. A2 + 8(a + 2) is a multiple of 8.


GMAT algebra note: The second part of the expression, 8(a + 2), is a multiple of 8 for any value of integer a. In order for the entire expression to be a multiple of 8, the first part of the expression must be a multiple of 8 as well.


GMAT algebra note: So according to this statement, a2 is a multiple of 8. The answer is a definite Yes, so this statement is sufficient.


GMAT algebra note: When comparing fractional pieces of a whole, find a common denominator. In this case, the one-mile stretch is divided into thirds, fi hs, and eighths.


GMAT algebra note: The smallest common denominator of 3, 5, and 8 is 120.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. One-mile highway is divided into 120 equal increments, where will the red, white, and blue marks fall?


GMAT statistics note: Red (thirds): 40, 80 (out of 120 increments) White (fi hs): 24, 48, 72, 96 (out of 120 increments) Blue (eighths): 15, 30, 45, 60, 75, 90, 105 (out of 120 increments) The smallest distance between two marks is 75 − 72 = 3 or 48 − 45 = 3. Both statements provide information about the sum of the set, so rephrase with this in mind.


GMAT statistics note: For n = odd, the median is the middle term, an integer. For n = even, the median is the average of the two middle terms, a non-integer of the form “integer + 0.5.” You can determine n if you can determine both the median of set S and the sum of the integers in set S.


GMAT statistics note: Next, glance at the statements. Sum of consecutive set = (Median)(Number of terms) Sum of consecutive set = (Median)(n).


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Since n must be an integer, use divisibility rules to narrow down possible median values.


GMAT statistics note: Check some possible median and n values (remember that n > 1). There are at least two possible values for n.


GMAT statistics note: Set S can’t have an integer median if there are only two terms. Set S is {−1, 0, 1, 2, 3, 4, 5}, which has a sum of 14.


GMAT statistics note: Set S is {2, 3, 4, 5}, which has a sum of 14. There are four basic ways this picture could look as there are four sides of the trapezoid that could serve as the non-tangent side.


GMAT geometry note: If the circle is tangent to both of the parallel sides (Figure A or B), then the diameter must be 10. If the circle is tangent to only one of the parallel sides (Figure C or D), then the diameter is less than 10.


GMAT geometry note: Just knowing the length of the shorter parallel side is not enough to determine which of the basic figures above describes the correct situation. If Figure A or B represents the correct situation, the diameter of the circle is determined solely by the distance between the parallel sides.


GMAT geometry note: The diameter is independent of the length of the shorter parallel side. If Figure C or D describes the correct situation, then the diameter would depend on not only the 15-inch side but also the longer parallel side, which has an unknown length.


GMAT geometry note: In Figures C and D, the diameter must be less than 10 but also greater than the length of the smaller parallel side. If the length of the smaller parallel side is 15, then the diameter would have to be.


GMAT geometry note: Watch divisibility, factors, remainders, parity, and prime structure. Greater than 15 for Figures C and D, but this is impossible (since the diameter has to be less than 10).


GMAT geometry note: Therefore, Figure A or B represents the correct situation, and the diameter of the circle must equal 10. Alternatively, notice that Figure D could not represent the situation since the circle would have to have a diameter larger than 15 inches in order to be tangent to the short parallel side and the non-parallel sides of the trapezoid.


GMAT geometry note: The parallel sides of the trapezoid are only 10 inches apart, so the circle would be too large to be drawn entirely within the trapezoid as required. Similar logic explains why Figure C is also impossible when considering (1) and (2) together.


GMAT geometry note: Wherever possible, avoid computing. Use Odd & Even principles to reduce your computation.


GMAT geometry note: Boys Girls Sum z 4, 6 = Even 5, 8 = Odd E + O = O 4, 6 = Even 5, 9 = Even E + E = E 4, 6 = Even 8, 9 = Odd E + O = O 4, 7 = Odd 5, 8 = Odd O + O = E 4, 7 = Odd 5, 9 = Even O + E = O.


GMAT sets, probability, and counting note: Watch divisibility, factors, remainders, parity, and prime structure. Boys Girls Sum z 4, 7 = Odd 8, 9 = Odd O + O = E 6, 7 = Odd 5, 8 = Odd O + O = E 6, 7 = Odd 5, 9 = Even O + E = O 6, 7 = Odd 8, 9 = Odd O + O = E Of the nine scenarios listed, five yield an even z and four yield an odd z.


GMAT sets, probability, and counting note: This problem involves both probability and special quadratics. In order to find the probability that is a positive odd integer, you must find the number of values of x and y for which this is the case.


GMAT sets, probability, and counting note: The problem is really asking you to count the values of x and y for which is a positive odd integer. Because x2 — y2 is a special quadratic, the expression can be simplified to.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. This expression is an integer when (x — y)(x + y) is divisible by 4.


GMAT algebra note: Note that x – y and x + y are always either both even or both odd. Whenever x – y and x + y are both even, their product will be divisible by 4.


GMAT algebra note: In other words, when x and y are either both even or both odd, is an integer. But when x and y are different in parity (one even and the other odd), is not an integer.


GMAT algebra note: The expression also needs to be positive. This will occur whenever x is greater than y.


GMAT algebra note: Otherwise, x – y would not be positive and the entire expression would be negative or zero. Finally, when is an ODD integer?


GMAT algebra note: Test a few cases to determine when this occurs. Based on the reasoning above, you only need to test cases in which x is greater than y and in which x and y are either both even or both odd.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Even Even Even It appears that whenever x and y are both odd, will be even.


GMAT quant note: Therefore, x and y cannot both be odd. Test some cases in which x and y are both even: x y Odd or even?


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Odd Even Odd Even If x and y are both even, then whether the case is valid depends on whether x and y differ by a multiple of 4.


GMAT quant note: Therefore, whenever x – y is a multiple of is an even integer. In summary: x and y must both be even.


GMAT quant note: X – y must NOT be a multiple of 4.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. That leaves only a few cases to count: x y There are six valid cases.


GMAT number properties note: In total, there are (10)(9) = 90 ways to choose two distinct values for x and y from the integers between 1 and 10, inclusive. The sum of the digits of ababab is 3(a + b).


GMAT number properties note: This must be a multiple of 3. Below, you’ll find explanations for the other answers.


GMAT number properties note: An integer is divisible by 4 if its last two digits represent a two-digit number that is itself divisible by 4. It is uncertain whether the two-digit integer ab is divisible by 4.


GMAT number properties note: An integer is divisible by 5 if the last digit is 0 or 5. It is uncertain whether the positive integer b is 5.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. An integer is divisible by 6 if it is even and divisible by 3.


GMAT sequences and patterns note: Note that you can also use logic to eliminate this answer. If ababab were divisible by 6, it would also have to be divisible by 3, but that would lead to two correct answers!


GMAT sequences and patterns note: Try out a few randomly chosen numbers to help you understand the question. If m = 10 and n = 10, then s = 20 and the units digits of m and s are equal.


GMAT sequences and patterns note: If m = 11 and n = 12, then s = 23 and the units digit of s is greater than the units digit of m. If m = 19 and n = 11, then s = 30 and the units digit of s is less than the units digit of m.


GMAT sequences and patterns note: What’s going on with all of these cases? Basically, if the units digits of m and n are small enough, then the units digit of s will be equal to or greater than the units digit of m (and of n).


GMAT sequences and patterns note: On the other hand, if the units digits of m and n are large enough to cause you to carry over a 1 to the tens digit, then the units digit of s will end up being smaller than the units digit of m (and of n). If the units digit of s is definitely less than the units digits of one of the smaller numbers, then the carryover s.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Which case the units digits of both m and n must be larger than the units digit of s.


GMAT sequences and patterns note: There are only two ways in which the tens digit will not equal the tens digits of the two smaller numbers:. The tens digits of the two smaller numbers result in a number that needs to carry over into the hundreds digit.


GMAT sequences and patterns note: This is impossible for this problem because s is also a two-digit number. The units digits of the two smaller numbers result in a number that carries over into the tens digit.


GMAT sequences and patterns note: This must be what is happening in this case. If so, then the units digit of the larger number, s, must be smaller than the units digits of the two smaller numbers, m and n.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If x and y are positive integers such that x2 – y2 = 48, how many different values of y are possible?


GMAT number properties note: Is written in the form p × 10q, where p and q are integers, what is the maximum possible value of q? If x and y are integers, and x ≠ 0, what is the value of xy?


GMAT geometry note: The three sides of a triangle have lengths p, q, and r, each an integer. Is this triangle a right triangle?


GMAT geometry note: The perimeter of the triangle is an odd integer.(1) If the triangle’s area is doubled, the result is not an integer.(2) 125. The average (arithmetic mean) cost of three computer models is $900.


GMAT geometry note: If no two computers cost the same amount, does the most expensive model cost more than $1,000?


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The most expensive model costs 25% more than the model with the median cost.


GMAT statistics note: The most expensive model costs $210 more than the model with the median cost.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. K-numbers are positive integers with only 2’s as their digits.


GMAT number properties note: What is the K-weight of 600? If the reciprocals of two consecutive positive integers are added together, what is the sum in terms of the greater integer x?


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


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


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Workout Set 13 Solutions 21.


GMAT number properties note: The choices themselves, as well as how many in the question text, suggest that y could have more than one value. To begin, simplify the given information using algebra and logical reasoning.


GMAT number properties note: Since x and y are integers, both x – y and x + y are integers. The product of two integers is 48.


GMAT number properties note: One of those integers is equal to x + y and the other is equal to x – y. Since x2 – y2 is positive and x and y are both positive, x must be greater than y.


GMAT number properties note: Therefore, x + y and x – y are a pair of positive factors of 48, with x + y being the greater of the two factors. List the pairs of factors of 48 and solve for the values of x and y.


GMAT number properties note: Since the factors are y above and y below x, the value of x equals the average. Therefore, if for a particular case the average of (x – y) and (x + y) is not an integer, that case is not valid.


GMAT number properties note: In order for the average of the two terms to be an integer, the sum has to be even, so if the sum is odd, eliminate that possibility immediately. X – y x + y x y 1 48 n/a (sum not even) 3 16 n/a (sum not even).