GMAT Quantitative Problems - Incorrect
A circle is drawn on a coordinate plane. If a line is drawn through the origin and the center of that circle, is the line's slope less than 1? (1) No point on the circle has a negative x-coordinate. (2) The circle intersects the x-axis at two different positive coordinates.
(1) NOT SUFFICIENT: If no coordinates are negative, then the circle must be located entirely in quadrants 1 or 4 (to the right of the y-axis). Test some cases; to make your life easier, consider small circles. If a small circle is centered at a point whose y-coordinate is greater than its x-coordinate, as in the left-hand diagram below, then the slope of the line will be greater than 1; if the circle is centered at a point whose x-coordinate is greater than its y-coordinate, as in the right-hand diagram, then the slope of the line will be less than 1. (2) NOT SUFFICIENT: Test some cases. For instance, place the circle with its center on the x-axis, so that the line coincides with the x-axis and thus has slope 0: Alternatively, place the circle mostly above the x-axis, as in the below example. In this case, the slope is greater than 1: (1) AND (2) SUFFICIENT: Using both statements, the circle must lie entirely to the right of the y-axis and it must cross the x-axis at two points. Test some cases. For instance, it is still possible to have a slope of 0, as shown during the examination of statement 2 above—so there is at least one case where the slope is less than 1.Are there any cases where the slope is greater than 1? The circle's center would have to lie in the first quadrant (above the x-axis). To satisfy both statements, the circle must have two x-intercepts, and must either be tangent to the y-axis or not intersect the y-axis at all. Consider these cases visually, designating the center of the circle as ( a, b):In the first case—in which the circle is tangent to the y-axis—the value of a is equal to the circle's radius, but the value of b is less than the radius. That is, a > b so the line's slope b/a must be less than 1. In the second case, a is greater than the circle's radius, while b is still less than that radius. Therefore, the slope b/a is also less than 1 in this case. The slope has to be less than 1 in all cases satisfying both statements. The correct answer is (C).
A special deck of cards has 3 each of 8 different cards. The deck has been shuffled so that the cards are randomly distributed. If 3 nonmatching cards are dealt, what is the probability that dealing 2 more cards will result in at least one matching pair of the same cards with the original 3 dealt cards or 3 of the same card?
Answer: 1/2 nalyze the Question Three different cards are dealt from a deck that has 3 cards each of 8 types of cards. Thus, the total number of cards in the deck is 24. The cards are randomly distributed. State the Task Determine the probability that dealing two more cards will create either a matching pair with the original three cards dealt or three of a kind. Approach Strategically This is a multi-step probability problem. Since the answer will be either one outcome or another, individual outcome probabilities will be added. After 3 cards are dealt, there will be 21 remaining. Of these, 2 + 2 + 2 = 6 would match one of these cards, so the probability that the next card dealt will create a pair is 621=27621=27. The outcome of the second card does not matter in this case since the result would be either one pair, two pairs, or three of a kind; the draw is already a "success," and regardless of which card is drawn second, the result is still a "success." The probability of not succeeding with the first card is 1−27=571-27=57. If the first card does not produce a pair, the probability that the next card will make a pair with one of the original three dealt cards is 620=310620=310. So, the probability of not succeeding on the first card but getting the desired result on the second is 57×310=157057×310=1570. So, the desired result can occur either of two ways, with the first card or with the second card. The probability of success on the first card is, as determined above, 2727. The probability of success on the second card is 15701570. The total probability of obtaining the desired result one way or the other is 27+1570=2070+1570=3570=1227+1570=2070+1570=3570=12. Choice (E) is correct. Confirm Your Answer Check your calculations, making sure you accounted for the drawing of cards without replacement in determining the denominator of each probability. Also, make sure you multiplied when calculating the probability that one event and another occurs and added when calculating the probability that one event or another occurs. TAKEAWAY: Break multi-step probability questions into their component pieces.
Target Test Prep - Work Problem
How to solve: For X: x hours = 1080 --> Xrate = 1080/x For Y: x + 20 hours = 1080 --> Yrate = 1080/x+20 We know that X's rate is 20 percent more than Y's rate, hence we can say: (1080/x) = 1.2 * (1080/x+ 20) 1/x = 1.2(1/x+20) x = 100 hours to make 1080 light bulbs. Therefore, for machine X, we have 1080/100 bulbs per hour = 10.8. Hence, the machine can make 10.8h lightbulbs in h hours.
Target Test Prep - Coordinate Geometry Problem
In this case we want to move right 2 units and up 3 units. From A. If we try each combination, we'll get each areas and see that 9 is only one that doesn't work.
How many positive three-digit integers have an odd digit in both the tens and units place?
There are 5 odd digits: 1, 3, 5, 7, and 9. For each of the 5 possible odd tens digits there are 5 possible odd ones digits, for a total of 5 × 5 or 25 combinations where both the tens and ones digits are odd. There are no restrictions on the hundreds' digit so it could be any of the 9 digits 1 through 9, for a total of 25 × 9 or 225 three-digit numbers that fit the given conditions.
Target Test Prep - Rate Problem
Why are neither sufficient? Depending on the difference in rates, the cat can either not be caught in 20 minutes or can be.
Target Test Prep - Rate Problem
Why is statement 1 correct? Because it gives the time and you already have the rate.
Target Test Prep - Rate Problem
Why is statement 1 sufficient? Statement 1 is sufficient because we know that y = x + 432. This obviously shows that y -x = 432. Remember, they're not asking what x and y are, just what the difference is.
A circle is drawn within the interior of a rectangle. Does the circle occupy more than one-half of the rectangle's area? (1) The rectangle's length is more than twice its width. (2) If the rectangle's length and width were each reduced by 25% and the circle unchanged, the circle would still fit into the interior of the new rectangle.
(1) SUFFICIENT: Let L and W stand for the length and width, respectively, of the original rectangle. From this statement, then, we have L > 2W. A circle has a constant diameter. If the entire circle is to fit within the rectangle, its diameter must be less than the shorter dimension of the rectangle, i.e., less than the rectangle's width. (Remember that the circle must fit both length-wise and width-wise into the rectangle!)The circle's diameter is less than W, or 2r < W; thus . The area of the circle is thus less than . The value of π is a little more than 3, so the area is a little more than . The rectangle's area is LW. Because L > 2W, this area is greater than (2W)(W) = 2W2.The circle's area is only a little more than , while the rectangle's area is greater than 2W2. The circle thus occupies less than half the rectangle's area. Statement 1 is sufficient. (2) SUFFICIENT: Let L and W stand for the length and width, respectively, of the original rectangle (whose area is LW). From this statement, then, the circle's diameter is smaller than both and (since the circle must fit into the interior both length-wise and width-wise). Because the diameter is twice as long as the radius, the radius must be smaller than either or . Use those two dimensions in place of the two r's in the area formula: The circle's area = π(less than )(less than ), or < . The value of π is a little more than 3, so . The circle's area, then, is about , or somewhat less than half the original rectangle's area of LW. Statement 2 is sufficient.
The integers n and t are positive and n > t > 1. How many different subgroups of t items can be formed from a group of n different items? (1) The number of different subgroups of n − t different items that can be formed from a group of n different items is 680. (2) nt = 51
Analyze the Question Stem: In this Value question we are selecting a subgroup of t items from n items, since order does not matter in a subgroup, this means that we are trying to find the number of combinations of selecting t items from n, or nCt. Evaluate the Statements: Statement (1): The number of ways to select a group of t items from n different items must be equal to the number of ways of selecting a group of (n − t) items from n different items. This is because when one selects a group of (n − t) items from n different items, one must decide which t items to not select. Thus, the number of ways to select a group of (n − t) items from n items is equal to the number of ways to decide which t items to not select from the n different items, and this is equal to the number of ways to select a group of t different items from n different items. So the number of ways to select t different items from n different items is 680. Statement (1) is Sufficient. We can eliminate choices (B), (C), and (E). Statement (2): The question stem says that n > t > 1 and here we have that nt = 51. The prime factorization of 51 is 3 × 17. The only factors of 51 other than 3 and 17 are 1 and 51. Since t > 1, we cannot have t = 1 and n = 51. Because n > t, we must have t = 3 and n = 17. We can now use the combinations formula to find the number of ways to choose 3 different items from 17 different items. We don't have to actually use the combinations formula to find the number of ways to choose 3 different items from 17 different items. Knowing that we could is enough. Statement (2) is Sufficient. Therefore, Answer Choice (D) is correct. Combined: Unnecessary.
Geometry Problem
Analyze the Question Stem: In this Value question, because OA, OB, and OC all radii of the circle, we have OA = OB = OC. Because OA = OB, triangle ABO is an isosceles triangle. Because OB = OC, triangle CBO is an isosceles triangle. Now let's consider the statements. Evaluate the Statements: Statement (1): Let's consider triangle ABO. Because sides OA and OB of triangle ABO are equal, the angles opposite these sides are equal. So OBA, which is opposite side OA, must be equal to OAB, which is opposite side OB. We know that OAB is 28 degrees, so OBA must also be 28 degrees. Now let's consider triangle BCO. DOC is an exterior angle of triangle BCO. So the measure of DOC is equal to the sum of the measures of OBC and OCB. Since sides OB and OC of triangle OBC are equal, the angles opposite these sides are equal. So OCB, which is opposite side OB, is equal to OBC, which is opposite side OC. Thus, DOC is equal to twice OBC. Since DOC is 40 degrees, OBC is of 40 degrees, which is 20 degrees. We now know that ABO is 28 degrees and OBC is 20 degrees. Then ABC = ABO + OBC. So the degree measure of ABC is 28 + 20 = 48. Statement (1) is Sufficient. We can eliminate choices (B), (C), and (E). Statement (2): AOD is an exterior angle of triangle ABO. So AOD = ABO + BAO. Since ABO and BAO and are opposite the equal sides OA and OB which are radii, ABO = BAO. Then AOD = ABO + BAO = 2(ABO). Thus, the measure of ABO is of the measure of AOD. Similarly, because COD is an exterior angle of isosceles triangle CBO with equal sides OB and OC, the measure of OBC is of the measure of DOC. Since ABO = (AOD) and OBC = (DOC): ABO + OBC = (AOD) + (DOC) = (AOD + DOC) Now ABO + OBC = ABC, and AOD + DOC = AOC. So ABC = (AOC). Since Statement (2) says that the degree measure of AOC is 96, the degree measure of ABC is of 96, which is 48.
Bill and Sally see each other across a field. They are 500 feet apart. If at the same instant of time they start running toward each other in a direct line how far will Bill have traveled when they meet? (1) Bill ran at an average speed that was 50% greater than Sally's average speed. (2) Bill ran at an average speed 4 feet per second faster than Sally's average speed.
Analyze the Question Stem: The question stem tells us how far they were apart when they started, so this is a Value question. We must consider whether or not there is sufficiency to determine how far Bill traveled when they met. We know that the time each ran is the same because they start at the same time and stop when they reach each other. Evaluate the Statements: Statement (1): This statement provides a ratio of their respective speeds. When the time is the same, the distance traveled is directly related to the average speed (Distance = Rate × Time). Therefore, this statement says that Bill ran 50% farther than Sally. Since the total distance is known, the distance Bill ran can be calculated. Therefore, Statement (1) is Sufficient. We can eliminate choices (B), (C), and (E). Statement (2): This statement might appear to provide the same information as Statement (1) so it would be tempting to assume that this statement is also sufficient to answer the question. Upon more careful analysis, however, we can see that if we only know the difference in speeds, we cannot determine the ratio of speeds or the actual speeds. Therefore, the distance Bill ran cannot be uniquely determined. An example may illustrate this point. If Bill ran at 5 feet per second and Sally ran at 1 foot per second Bill would run 5 times as far as Sally, but if Bill ran at 8 feet per second and Sally ran at 4 feet per second Bill would run only twice as far as Sally. So, Statement (2) is Insufficient. Therefore, Answer Choice (A) is correct. Combined: Unnecessary
Is the positive integer z a prime number? (1) z and the square root of integer y have the same number of unique prime factors. (2) z and the perfect square y have the same number of unique factors.
Analyze the Question Stem: This is a Yes/No question. For sufficiency, a definite "yes" would show that the positive integer z is a prime number, or a definite "no" would show that the positive integer z is not a prime number. A prime number is an integer greater than 1 that has exactly two positive factors: 1 and itself. To know whether z is prime, we will need to learn how many factors it has: exactly 2 factors, and it is prime—any other number and it is not. We're also told about another term, y, which is the square of an integer (excluding 0). We expect the statements to relate z to y in a way that tells us how many factors z has. Evaluate the Statements: Statement (1): We are told that z and the square root of the integer y have the same number of unique prime factors. That unique number might or might not be one, so Statement (1) is Insufficient. (A prime number has two factors, but only one prime factor, as 1 is not a prime number.) Picking Numbers can illustrate this. If y = 9, then the square root of y is 3, which has one unique prime factor: 3. In this case, z would also have one unique prime factor, so z could equal 2, which is prime. However, if y = 36, then the square root is 6, which has two unique prime factors, 2 and 3. In this case, z would also have two unique prime factors, and z could then equal 6. Since z might or might not be a prime number, Statement (1) is Insufficient to answer the question with a definite "yes" or a definite "no." Eliminate choices (A) and (D). Statement (2): We are told that z and y have the same number of unique factors. We do not know what that number of unique factors is, but we only need to determine whether it must or must not be 2. Picking Numbers works well with this statement. Let's pick y = 4. In this case, y has three unique factors: 1, 2, and 4. This means that z must also have three unique factors and thus cannot be a prime number, which can have only two factors. Let's now consider a value for y that is not the square of a prime, in case that affects things—let's try y = 36. In this case, y would have 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36. z would also have 9 factors and thus would not be prime. There are two other perfect squares with peculiar properties: 0 and 1. For safety's sake, let's consider them. If y = 1, then y has only one factor: 1. This means that z has only one factor, and must, therefore, also equal 1—the only number with one factor;1 is not prime. And if y = 0, then it has no factors at all, as 0 is considered to have no factors. This means z must also equal 0—the only number with no factors—and would also not be prime. Since z cannot in any case be prime, the question is answered with a definite "no." Statement (2) is Sufficient. In fact, any non-zero perfect square will always have an odd number of unique factors. The reason for this is that factors of a number come in pairs. For a number that is not a perfect square, each factor has a unique partner, meaning there are two unique factors in each pair, and the total number of unique factors is even. However, for a perfect square, one pair of those factors consists of identical numbers, which means that there is only one unique factor for that pair, and the total number of unique factors is odd. If the number of factors is odd, it cannot equal 2. Since z cannot be prime, Statement (2) is Sufficient to answer the question with a definite "no." Eliminate choices (C) and (E). Therefore, Choice (B) is correct.
At a certain health club, 30 percent of the members use both the pool and sauna, but 40 percent of the members who use the pool do not use the sauna. What percent of the members of the health club use the pool?
Analyze the Question: Percent is the keyword for the problem. The fact that our club members are described as using the pool (or not) and using the sauna (or not) with a possible overlap, should also suggest looking at this problem as an overlapping sets question. Identify the Task: We need to determine the total number of health club members and the total number that use the pool, or otherwise determine the proportion of pool users to total members. Approach Strategically: The percent formula is: Now we need to find the number of people who use the pool and the total number of people in the health club. Let's also remember this is a percentage problem with no initial values given. It's best to pick 100 for the total number of people in the health club. If there are 100 people in the club, then 30 of those members use both the pool and the sauna because 30% of 100 is 30. It might be easiest to organize all the information into a table similar to what we'd use in an overlapping sets question. The total number of people in the pool are: Substitute this number back into the Percentage equation from above. So the correct Answer Choice is (C). Confirm your Answer: Be careful that you are answering the specific question asked and double-check you haven't made any avoidable errors in computation.
X is a set that contains 7 different numbers. Y is a set containing 6 different numbers. All of the numbers in set Y are also members of set X. Which of the following statements CANNOT be true? A - The range of X is equal to the range of Y. B - The range of X is less than the range of Y. C - The median of X is equal to the median of Y. D - The mean of X is less than the mean of Y. E- The mean of X is equal to the mean of Y.
Analyze the Question: The stimulus defines two sets of numbers, X and Y; all of Y's six numbers are also in X, along with one other number. State the Task: The right answer will be the one statement in the answer choices that cannot be true. Since it may be hard to prove that something can never be true, it'll be helpful to approach this from the other direction: try to show that a statement can be true, and if so, eliminate it. Approach Strategically: Move through the choices one-by-one to find the one that can't be true. If necessary, pick easy values for these number sets and try out each of the choices. If the statement in an answer choice can ever be true, eliminate it. The range of a set is the difference between the lowest and the highest values in a set. If the number in X that's not in Y is not the highest or lowest value, then X and Y have the same highest and lowest values and thus the same range. (A) can be true, so eliminate it. If, however, that extra value in X is either the highest or lowest value in the set, that makes the range of X greater than that of Y. Therefore, the range of X is either the same or greater than the range of Y. There's no way for the range of X to be smaller than that of Y, so (B) is correct. For the record, here's why the other choices are wrong: Since Y has an even number of members, the median is the average of the middle two. If the two middle numbers were, for instance, 3 and 5, the median would be 4, and if 4 were the extra number in X, then it would be X's median as well. Thus, (C) can be true. If Y contains 6 positive numbers, and the extra number in X was, say, -500, then X's mean would clearly be less than that of Y, so (D) can be true. Similarly, if the extra value in X was equal to the mean of Y, then the mean of X would be equal to the mean of Y, so (E) can be true. Confirm Your Answer: If needed, pick full sets of values for X and Y to be sure of your answer. TAKEAWAY: Picking numbers is a great way to make abstract questions more concrete.
A B C Question
Analyze the Question: This is a Roman numeral question that gives inequalities in terms of a and b. State the Task: The right answer represents the statements that must be true. Approach Strategically: It may be helpful to split up the inequality and solve each part separately: 0 > 1 - > 1 and 1 - > -1 - > -2 < 2 So, is between 1 and 2. Now, evaluate the statements. Since each appears in the choices exactly twice, approach them in any order. Statement I: It has already been determined that > 1. The only way a fraction can be greater than 0 is if both the numerator and the denominator have the same sign. Since b is greater than 0, a must also be greater than 0. Therefore, this statement is false; eliminate (B) and (D). Statement II: Since > 1, the numerator of that fraction must be greater than the denominator. This can also be seen by multiplying both sides by b (which is positive) to get a > b. Statement II is therefore true, so eliminate (A). Statement III: Like , the square root of 2 lies between 1 and 2 as well. However, there's no way to know whether is greater or less than the square root of 2, so this statement doesn't have to be true. Only Statement II must be true, so the correct answer is (C). Confirm Your Answer: Make sure that you didn't forget to reverse any inequality signs when multiplying or dividing by a negative number. TAKEAWAY: When a question asks for a statement that must be true, pick something that's true all of the time, not some of the time.
The only people in each of rooms A and B are students, and each student in each of rooms A and B is either a junior or a senior. The ratio of the number of juniors to the number of seniors in room A is 4 to 5, the ratio of the number of juniors to the number of seniors in room B is 3 to 17, and the ratio of the total number of juniors in both rooms A and B to the total number of seniors in both rooms A and B is 5 to 7. What is the ratio of the total number of students in room A to the total number of students in room B?
Analyze the Question: This is a ratios question. Our situation is students, classified as juniors or seniors, in two different rooms, A and B. We are given the ratios of juniors to seniors in each room and the ratio of total juniors to total seniors. The question asks for the ratio of all students (juniors and seniors) in A to all students in B. Identify the Task: This is a complicated ratios question, but ratios always give us easily translatable equations, so if we keep our variables organized and use those equations, we'll get to an answer. Approach Strategically: Picking Numbers is often useful on ratio questions with unknown values, but it will not help us here, as the only unknown involved is the answer. As a first step, let's assign a couple of variables. Let's make x the "weight" of one unit in the ratio for room A. In other words, the number of juniors in room A is 4x, the number of seniors in room A is 5x, and the total number of students in room A is 4x + 5x = 9x. Similarly, let's make y the weight of one unit in room B, so room B has 3y juniors, 17y seniors, and 3y + 17y = 20y total students. With these variables, we can now reframe our answer: since we are looking for the ratio of the total number of students in room A to the total number of students in room B, we want to find the value of . The question stem also gives us a numerical value for the ratio of all juniors to all seniors (in both rooms), so let's incorporate that into an equation. We know the ratio of the total number of juniors in both rooms to the total number of seniors in both rooms is 5 to 7. The total number of juniors students in rooms A and B is 4x + 3y. The total number of seniors in rooms A and B is 5x + 17y. Since the ratio of the total number of juniors in rooms A and B to the total number of seniors in rooms A and B is 5 to 7, we can write the equation . Now, we'll manipulate this equation to help us find . Since we only have one equation and two variables, we won't be able to solve for x or y, but we can cross-multiply and simplify to put one in terms of the other, which will allow us to find the ratio between them. 7(4x + 3y = 5(5x + 17y 28x + 21y = 25x + 85y 3x = 64y Now that we have x in terms of y, we can substitute into the expression to cancel out the ys and turn it into a numerical expression, which should match with an answer choice. When: Thankfully, this matches with Answer Choice (D), which is correct. Confirm your Answer: There's a lot of math in this question, no two ways about it. The fact that your calculations yielded a value in an answer choice should be reassuring, but take a quick glance back to your noteboard to make sure you didn't make any avoidable mistakes before moving on.
Yes/No Data sufficiency Questions
Analyze the question stem This Yes/No question asks whether n2 > n3. Squared values must always be positive, but cubed values may be positive or negative, depending on the sign of the base. Evaluate the statements Statement (1) indicates that n2 < 1, which means that n must be a fraction between 0 and 1 or 0. This is not enough to provide a clear relationship with n3, so this is insufficient. Eliminate (A) and (D). Statement (2): Here n can still be a fraction between 0 and 1, or it can be a negative number, or 0. Because this is not enough to provide a clear relationship between n2 and n3, this is also insufficient. Eliminate (B). Taking the statements in combination, n can still be a fraction, or it could equal 0. Therefore, together the statements are insufficient. (E) is correct. TAKEAWAY: Knowing how powers affect their bases and vice versa is key to many exponents questions.
Problem on squares
Analyze the question stem This Yes/No question states that x2x2 is a positive integer and asks whether x is a positive integer. Until further information narrows down the possibilities, x could be a positive or negative integer or a radical. Evaluate the statements Statement (1) says that √x2x2 is an integer. You already know from the stem that x2x2 is an integer, so this statement tells you that x2x2 is a perfect square. However, you don't know whether it is the square of a negative integer or a positive integer. In other words, you don't know whether x is positive or negative. Eliminate (A) and (D). Statement (2), √x2=xx2=x, means that x is either a positive integer or a radical. For example, if x = 5, then this statement is √52=552=5. But if x equal √55, then √(√5)2=√552=5. This statement, too, is insufficient, so eliminate (B) and proceed to evaluate the statements together. Statement (1) limits x to a negative integer or a positive integer. Statement (2) limits x to a positive integer or a radical. Only "positive integer" is common to both, so together the statements are sufficient to answer the question. (C) is correct. TAKEAWAY: Since the GMAT only deals with real numbers, the value of a square root radical (a value under the √ sign) is always positive.
If x is a three-digit number in which the hundreds digit is 1 greater than the tens digit and 2 greater than the units digit, then what is the value of x ? 1. When x is multiplied by 3, the tens digit of the result is 2. 2. When x is multiplied by 4, the units digit of the result is 2.
Analyze the question stem This is a Value question asking for the value of x, which is a three-digit number. The hundreds digit is 1 greater than the tens digit and 2 greater than the units digit. In other words, the digits are consecutive in decreasing order. There are seven such numbers (987, 876, 765, 654, 543, 432, and 321). For sufficiency, the statements have to limit x to just one of those seven values. Evaluate the statements Statement (1) says the result when x is multiplied by 3 has a tens digit of 2. If the tens digit of x is 4, then the tens digit when multiplied by 3 will be 2: 543 × 3 = 1,629. However, due to carrying, the tens digit of the result is also 2 when x is 876: 876 × 3 = 2,628. With two possible values of x, this is insufficient. Eliminate (A) and (D). Statement (2) says the result when x is multiplied by 4 has a units digit of 2. Any number with a units digit of 3 or 8 will produce a number with a units digit of 2 when multiplied by 4. However, there is one value of x with a units digit of 3 (543) and none with a units digit of 8. Thus, the only value of x that works is 543 (543 × 4 = 2,172). That makes (B) correct. TAKEAWAY: Don't do more math than you need to. When evaluating these statements, you only have to worry about their effect on a particular digit, not the entire product.
Question regarding series
Analyze the question stem This is a Value question. The function defining terms in a series is sn = sn-1 + 2. In other words, after the first term, each term is the previous term plus 2. The question asks for the standard deviation of the series. Since this is a finite series, there must be a first and a last term. Because this is a Data Sufficiency question, you will not actually have to calculate the standard deviation; sufficiency consists of having the information needed to calculate the standard deviation, which is based on the difference between each term and the mean of the terms. Evaluate the statements Statement (1): Knowing that the range is 22 and that the terms are 2 apart means that you could determine the number of terms in the series: . So, even without knowing the mean, you could determine the difference between each value and the mean. For example, if the numbers are {1, 3, 5, ... , 23}, the mean is , and the difference between the first term and the mean is 12 − 1 = 11. And if the numbers are {101, 103, 105, ... , 123}, the mean is , and the difference between the first term and the mean is still 11. The same process holds for finding the difference between each term and the mean for all the terms. This is enough information to calculate the standard deviation, so this statement is sufficient. Eliminate (B), (C), and (E). Statement (2): Knowing the median without knowing the number of terms only tells you that there are as many terms greater than 19 as there are less than 19. The question stem tells you these terms are evenly spaced, meaning that the mean equals the median. However, you still don't know how many there are and so cannot calculate the distance of each term from the mean. Statement (2) by itself is insufficient. The correct choice is (A). TAKEAWAY: Asking for standard deviation in a Data Sufficiency question is equivalent to asking whether you can identify the difference between the mean and every value in a group of numbers.
Question regarding Absolute Value
Analyze the question stem This is a Yes/No question. A statement is sufficient if it shows whether x is positive. The inequality states that the absolute value of the difference between x and another number is less than 1. That is, x and a are less than 1 apart on the number line—their range is less than 1—but you do not know whether x is greater or a is greater. For example, if x = 0, then -1 < a < 1. Evaluate the statements To figure out what statement (1), a < -x, means in combination with the information from the question stem, try picking some numbers. Remember to pick numbers that keep the difference between x and y less than 1. Because the focus is on determining whether x is positive, play around with numbers that are close to zero. If a = 0, then x could equal a negative number between 0 and -1 such as -0.5; 0 < 0.5. Can x be positive? Try a = -0.5. Now any value for x between -1.5 and 0.5 works; for example, if x = 0.4, -0.5 < -0.4. This statement is insufficient. Eliminate (A) and (D). Statement (2) says that a is between the square roots of and . Without trying to figure out exactly what those values are, you can estimate them: is between and , or between and . And is between and . Thus, a is definitely a value between 0.2 and 0.5. Combined with the information from the question stem, the possible values of x include negative values and zero as well as positive values. Insufficient. Eliminate (B). Now evaluate the two statements together: According to statement (2), a is between 0.2 and 0.5 (the range is actually tighter than that), and according to statement (1), a < -x. If a were set equal to 0.2, x would have to be in the range -0.8 < x < -0.2. If a were set equal to 0.5, there would be no values of x that work, so try a = 0.49. Now -0.51 < x < -0.49. The only values for x that could fulfill all the conditions are negative values, and the answer to the question is no. The statements combined are sufficient, making (C) correct. TAKEAWAY: When working with an inequality, remember that the solution is a range of values.
Overlapping sets question.
Analyze the question stem This overlapping sets question can be approached with this formula: Total = # in Set 1 + # in Set 2 − # in both + # in neither Here the equation becomes: 100 = 45 + 20 − both + neither Since the question asks how many drivers responded "Smooth" for both cars, sufficiency will require being able to determine the "neither" category. Evaluate the statements Statement (1) explicitly provides the number of people who did not respond "Smooth" and is therefore sufficient. Eliminate (B), (C), and (E). Statement (2) speaks only to the "Rough" category and does not include the "Not Sure" responses. Since statement (2) is insufficient, eliminate (D) and choose (A). TAKEAWAY: Memorize the overlapping sets formula. Taking the time to set it up saves time evaluating the statements.
Exponents problem
Answer = 50. The GMAT will not require you to calculate such extremely complicated expressions as lie in this question stem. Look for an easier way. Identify the Task: The numerator and the denominator of the fraction both match up with Classic Quadratic forms. Once the Classic Quadratic substitutions are made, the arithmetic should be much easier. Approach Strategically: The Classic Quadratics are: x2+2xy+y2=(x+y)2 x2−y2=(x+y)(x−y) x2−2xy+y2=(x−y)2 The second and first of these Classic Quadratics can be used to factor the numerator and the denominator here, respectively. After factoring, the addition and subtraction within the parentheses is more manageable. 4822+2(482)(118)+11823062−2942 =(482+118)2(306+294)(306−294) =6002600×12 Reducing the fractions leads to further simplification, leading to an answer: 6002600×12 =600×600600×12 =60012 =50 Answer Choice (A) is correct. Confirm your Answer: Since the math worked out so elegantly in the end, it is doubtful that an error was made in addition. It is, however, always a good idea to check.
In a junior high school, the ratio of sixth graders to seventh graders is a:b, and the ratio of seventh graders to eighth graders is c:d, where a, b, c, and d are all positive. Which of the following is the ratio of sixth graders to eight graders? A - ab:cd B - ac:bd C - ad:bc D - bc:ad E - bd:ac
Since both of the given ratios include seventh graders, let's state both ratios so that the quantity representing seventh graders is the same in both ratios. The ratio of sixth graders to seventh graders is a to b and the ratio of seventh graders to eighth graders is c to d. In one ratio, seventh graders are represented by b while in the other ratio seventh graders are represented by c. We can have seventh graders represented by bc in both ratios if we multiply both members of the ratio a to b of sixth graders to seventh graders by c, and we multiply both members of the ratio c to d of seventh graders to eighth graders by b. Multiplying both members of the ratio a to b of sixth graders to seventh graders by c, we have the ratio ac to bc of sixth graders to seventh graders. Multiplying both members of the ratio c to d of seventh graders to eighth graders by b, we have the ratio bc to bd of seventh graders to eight graders. Since the ratio of sixth graders to seventh graders is ac to bc and the ratio of seventh graders to eighth graders is bc to bd, the ratio of sixth graders to eighth graders is ac to bd. The correct answer is (B).
Question regarding divisibility
This Yes/No question asks whether y3−yy3-y is divisible by 4, given that y is positive. Simplify the expression by factoring: y3−y=y(y2−1)y3-y=y(y2-1). Notice that the term in parentheses is the difference of two squares, so the complete factorization is y(y + 1)(y − 1). Evaluate the statements Statement (1) says that y2+yy2+y is divisible by 10, meaning that y2+yy2+y is some integer multiple of 10. Factoring this expression gives y(y + 1) = 10(i), where i is some integer. Now substitute 10(i) for y(y + 1) in the factored expression from the question stem. The question now becomes whether 10(i)(y − 1) is divisible by 4. Since 4's prime factorization is 2 × 2, a number is divisible by 4 if its own prime factorization includes at least two 2s. In this case, 10's prime factorization is 2 × 5, so that's one 2. But without knowing i or y, or at least whether either is even (thereby having 2 as a factor), there is no way to determine whether 10(i)(y − 1) is divisible by 4. Eliminate (A) and (D). Statement (2) states that y = 2k + 1. Since k is an integer, this means that 2k is even and y is odd. If y is odd, then (y − 1) and (y + 1) must be even, which means they both have 2 as a factor. Thus, the expression y(y + 1)(y − 1) has at least two 2s in its prime factorization, and since 2 × 2 = 4, the overall expression must have a factor of 4. Statement (2) is sufficient, and the correct answer is (B). TAKEAWAY: Prime factorization can be quite helpful on questions that ask about divisibility.
Question on A,B,C
This is a Value question seeking a definite value for an expression with three variables. It may help to rewrite this expression as a single fraction with one denominator: . Information providing values for a, b, and c or for their products as needed for the expression above will be sufficient. Evaluate the statements Statement (1) gives the sum of the variables but says nothing about the products. Don't fall for look-alike answer choices—the reciprocal of a + b + c is not the same as (for example, if a = 1, b = 2, and c = 3, 11+2+3≠11+12+1311+2+3≠11+12+13), and you can't manipulate this equation arithmetically to solve for . Eliminate (A) and (D). Statement (2) provides the denominator of the simplified fraction but nothing else. Eliminate (B) and evaluate the statements together. The statements combined still do not provide enough information to evaluate the expression. It may look as though you have three equations and three unknowns, but the stem contains only an expression, not an equation: it doesn't say that is equal to anything. (E) is the correct answer. TAKEAWAY: Restating the question stem in a different form, such as by giving fractions a common denominator in this case, can make it much easier to evaluate the statements.
For each month, the number of accounts, a, that a certain salesperson has contracted that month is directly proportional to his efficiency score, e, which is directly proportional to his commission rate, c. If c = 3.0, what is the value of a ? 1. If c = 4.0, e = 0.3. 2. If c = 6.0, a = 80.
This is a Value question. It will be helpful to first note that because a is directly proportional to e, which is in turn directly proportional to c, a is then directly proportional to c. To say that a is directly proportional to c is to say that there is a constant k such that ck = a, or, perhaps more simply, that there is a fixed ratio between a and c. A statement, or set of statements, will be sufficient if it determines that ratio. It could do this directly, or it could provide enough information about the relationship of a to e and c to e that e works as a "bridge" between the two values of interest. Evaluate the statements Statement (1): From this, the proportional relationship between e and c can be determined. However, nothing is said about the relationship of a to e, so there is no way to compute the value of a when c = 3.0. Insufficient. Eliminate (A) and (D). Statement (2): This gives the ratio between a and c. There's no need to actually calculate the value of a if c = 3.0; it's enough to know that it's possible. But for the record... Because a is directly proportional to c: ac=806.0=x3.0ac=806.0=x3.0. Since 6 is divided by 2 to get 3, divide 80 by 2 to find x = 40. This statement is sufficient, so eliminate (C) and (E). (B) is correct. TAKEAWAY: When the ratio between two terms is known and you have the value of one of the terms, you can find the value of the other term.
John wants to have one dinner with 3 friends: Kathy, Mary, and Norman. Is there one night, this week, Monday to Friday, inclusive, on which John and his three friends are all available? (1) John and Kathy are only available from Tuesday through Friday nights. (2) Mary is only available on Monday and Tuesday nights, and Norman is available only on Wednesday night.
This is a Yes/No question that asks whether or not there is a night during which every one of four people are available. To answer yes, we must be able to identify at least one night for which we know the availability of each of the four people. However, in order to answer no, we need only identify two people who are not both available on any given night. Evaluate the Statements: Statement (1): mentions only John and Kathy, who are available on the same nights. However, we have no information on the availability of Norman or Mary, and John and Kathy are available on at least one night. Statement (1) is Insufficient. We can eliminate choices (A) and (D). Statement (2): Mary is only available on Monday and Tuesday, and Norman is available only on Wednesday. If there is no night on which both Mary and Norman are available, then there cannot be a night when John and all three of his friends are available. The answer to the question posed is always no, and so Statement (2) is Sufficient. Answer Choice (B) is correct. Combined: Unnecessary.
Number Properties Problem
This is a Yes/No question. Your task is to determine whether there is sufficient information to say that or . From the fact that xy is positive, you can deduce that x and y are either both positive or both negative. Evaluate the statements Statement (1) says that . Explore some possibilities by picking numbers. If x = 1 and y = 4, then √xy=√1×4=√4=2xy=1×4=4=2, which is less than 4, and the answer to the question is no. However, if x = -1 and y = -4, then √xy=√−1×−4=√4=2xy=-1×-4=4=2, which is greater than -4, and the answer is yes. Statement (1) is therefore insufficient. Eliminate (A) and (D). Statement (2) says that x − y < 0. Thus, x < y. This means that if x and y are positive, then the absolute value of y is greater than the absolute value of x, but if x and y are negative, then the relative absolute values are reversed. Experimenting with some numbers shows this. For example, if x = 1 and y = 4, then the statement is true, but if x = 4 and y = 1, the statement isn't true. On the other hand, if x and y are both negative, then the absolute value of x must be greater than the absolute value of y; for example, if x = -3 and y = -2, the inequality works, but if x = -2 and y = -3, the inequality doesn't work. Now plug some of these values into the question stem to see if it's possible to come up with a consistent answer. If x = 1 and y = 4, the answer to the question is yes; if x = -4 and y = -1, the answer to the question is no. Thus, Statement (2) is insufficient. Eliminate (B). Now evaluate the two statements together. When combined, the two statements say in effect "y is 4 times x (Statement (1)), and y is greater than x (Statement (2))." Thus, based on the evaluation of statement (2), x and y must be positive. Test some values in the question stem: Is ? No. Is ? Again, the answer is no. Further testing will reveal that any two permissible numbers will answer the question with a no, so the two statements together are sufficient and the answer is (C). TAKEAWAY: In questions involving unknown values in radicals, testing several values can determine whether statements are sufficient.
Target Test Prep - Rate Problem
Why is statement 2 sufficient? If william averaged 30 MPH from A to B, then the total speed can NEVER be 60 MPH. As an example, let's say the speed from B to A was 3000 MPH. Then we have: Rate = Total Distance/Time Which gives = x + x/(x/3000 + x/30) = 2x/(101x/3000) = 59.4 The same calculation applies if the speed from B to A is 6000000 mph. There will always be a limit at 60 mph.
Target Test Prep - Rate Problem
Why is statement two sufficient? Statement two is sufficienct because if Thomas traveled one direction from manchester to mayfield at 50 mph, and then traveled from mayfield to manchester and took zero seconds(which is impossible, but as a thought experiment), the total speed would be Average Speed = Distance/Time = x + x/(0 + x/50) = 2x/(x/50) = 100x/x = 100. Therefore, because we know that it will take thomas SOME amount of time to travel from mayfield to manchester, the total speed will be less than 100 mph.
When 4/7 is expressed as a decimal, what is the 38th digit to the right of the decimal point?
expressed as a decimal gives a result of 0.571428 repeating, which is a pattern that repeats every 6th term. To solve, we need to find where the 38th term would fit into this pattern. This can be done by solving for the remainder when 38 is divided by 6. If the remainder is zero, then we're looking for the sixth term in the sequence. Otherwise we're looking for the term that corresponds to the remainder. 38 divided by 6 = 6, remainder 2. The remainder of 2 means that we're looking for the second term in the pattern 0.571428. The second term is 7, so (B) is correct
Is angle DFE equal to 60?
https://gmatclub.com/forum/kaplan-geometry-ds-is-angle-dfe-60-degree-274936.html
