How to Improve GMAT Quant from Q49 to a Perfect Q51 Updated on Aug 28th, 2020 Many people on the forum are not happy with their 79th percentile Q49 scores. These are good scores, but understandably some require a higher score to beat their competition. This guide is designed to be a comprehensive overview for serious test-takers (those who previously achieved Q49) who are looking to get to the elusive Q51. Important: You only have one shot at the Q51 on your test. You will face a handful of elite questions and you need to be ready to face them. Here is a detailed inventory of the tough Q51 question types, concepts, and tips on how to handle them. For an overview of what question types appear on the GMAT, see this post: https://gmatclub.com/forum/overview-of- ... 11809.html 1. Do I have to get ALL Quant questions right to Q51? Answer: You do not have to answer all questions correctly; however, you must reach the hard questions, which is accomplished by answering 15-20 questions correctly in succession. Since the GMAT Math exam is in CAT form (Computer Adaptive Test), the questions increase in difficulty as you correctly answer questions, and get easier when you answer easy questions incorrectly. Therefore, after answering the first 7 questions correctly, you can expect your score to be around 35-40. These first few questions will be fairly basic and straightforward. In fact, you will have to keep going for a while, as the questions that are representative of a score of 50-51 usually appear after question number 17. Therefore, it is crucial to get the first 17 questions right as otherwise, you will not be able to reach the challenging questions that provide the opportunity to attempt achieving Q50/51. 2. What Concepts/Topics Do I Focus on when Targetting Q51 Score? Overview: It is vital to note in preparation for GMAT Math that Integer Properties, Statistics, Probability, Inequality, and Absolute Value are the 5 most important topics (Key Topics) that account for 60% of the GMAT math questions, and 80-90% of 50-51 level questions. Among them, Integer Properties and Statistics are especially important. If you get every question right in the “Integer” section, your score can go up as high as the mid-40s, and if you get every question right in the “Integer and Statistics” sections, you can expect to achieve a score in the high 40s for the Math section. Thus, it is vital that you correctly answer every key question from these five topics to score a 51. Integer Properties and Statistics . One question format in particular that keeps reoccurring is a question with a combination of 2 or 3 concepts of “different positive integers, median, mean, range, absolute value, and inequality” (that is what we have seen on the test and what we continually hear from our students). Answering it correctly or incorrectly will be the deciding factor in whether you get the perfect score or not – no second chances. These can be both PS or DS questions, and they are especially challenging in the DS section, as they are designed to look deceivingly obvious or simple, for example, questions dealing with 3 variables but only 2 equations/data points. (Note: Math Revolution has a methodology that provides you the knowledge required to answer up to 80% of easy/medium DS questions without solving them by matching the number of variables and the number of equations). It is not easy to solve these types of questions in just two minutes and they are purposely designed this way. Here are 2 examples of the Statistics and Integer Questions: Statistics: There are 3 different positive integers. If their average (the arithmetic mean) is 8, what are their values? 1) The largest integer is twice the smallest integer. 2) One of them is 9. →The correct answer is A . Integer Properties: The smallest of 7 positive integers is 1. Is the range of them greater than 3? 1) The average (the arithmetic mean) of them is 4. 2) The median of them is 4. → The correct answer is A . Logic Questions – On the path to Q51, you must master answering Trick Questions or, as we call them, “Common Mistake Type” questions. If you answer these questions incorrectly, the highest you can achieve is Q49. Side Note: What is a “Common Mistake Type” Trap Question? There are 5 types of questions that GMAT Math test makers design to trap students. Here are 2 of the most commonly used on the DS: DS Common Mistake Type 4 (A): If answer choice C seems too attractive and obvious, go back and check choice A or B alone. Look at the example below. If m and n are integers, is mn even? (1) m = 5. (2) m and n are consecutive integers. → The correct answer is B . DS Common Mistake Type 4 (B) If answer choice A or B seems too attractive and obvious, consider answer choice D. Look at the example below. If a, b, c, and d are different positive odd numbers, what is the median of the numbers? (1) a + b + c + d = 16. (2) The product of 3 numbers of a, b, c, or d is 15. → The correct answer is D . TIP! Important to note that the GMAC likes to use DS questions as the deciding factor for their Q50/Q51 scores…. most likely because DS questions are unique to the GMAT and thus provide a good level “measuring stick” even for individuals with a strong quantitative background. Therefore, if you are looking to score Q51, you will need to pay especially careful attention to the DS questions and learn how to spot Common Mistake Type questions. 3. Analysis of Specific Question Types to Master for Q51 DS Tips: As mentioned above, Integer and Statistics are the main chapters to focus on DS. Besides the “Mistake Type”, one specific type of DS questions to master is Hidden Integer Questions. These questions often present prices of products, the number of fruit, coins, people, stamps, cars, etc as “hidden integers”. These are countable numbers that are not given to increase complexity. These are logic questions and you should solve them logically rather than calculation. It is possible to calculate and solve these questions that way but it takes much longer and a lengthy calculation is exactly the trap the test wants to catch you with – to waste your time and slow you down. Integer Properties: M bought several pencils. If each pencil cost either 23-cent or 21-cent, how many 23-cent pencils did M buy? 1) M bought a total of 6 pencils 2) The total price of all pencils M bought was 130 cents → The answer choice C is very attractive, but the correct answer is B . Pencil is a hidden integer, but if you don’t recognize this fact, you would say C is the answer. In addition, Statement 1 is actually not giving you any helpful information Among Common Mistake Type questions, there are many cases in which A or B easily seems like a correct answer – when, in fact, D is the correct answer. One or two questions of this type play a decisive role in leading to a perfect 51. Integer Properties: Numbers a and b are positive integers. If a^4 - b^4 is divided by 3, what is the remainder? 1) When a + b is divided by 3, the remainder is 0. 2) When a 2 + b 2 is divided by 3, the remainder is 2. → The correct answer is D . In addition, You should be comfortable tackling inequality problems such as “greater than” and “less than” questions if you want to get a perfect 51. You should practice these so that your mind clicks when asked to figure out the minimum value (if asked “greater than”) and the maximum value (if asked “less than”). Statistics: There are five homes. If the median price of a home is $200,000, is the range of all prices greater than $80,000? 1) The average price of the five homes is $240,000. 2) Three of the five homes have the same price. → C looks like a right answer choice at first. However, the correct answer is A . Of course we would be incomplete if we did not include the famous DS Probability questions that you are likely to see on your quest to Q51. Here is an example: Probability: There are only apples and oranges in a box. What is the probability that the fruit selected at random from the box is an apple? 1) There are 2 apples in the box 2) The ratio of the number of apples to the number of oranges is 1 to 4. → C looks like a right answer choice, but the correct answer choice is B . In case of Absolute Value questions, a question regarding the relationship among |a|, a, and -a will determine a perfect 51. Absolute value: If |r|>|s|, is r>s? 1) r>0 2) s>0 → C is an attractive answer choice but A is a correct answer. Problem Solving Tips Be prepared to see questions about finding Hidden Integers and problems on the so-called “Pigeonhole Principle”. You have a good chance of seeing one of them. We have provided the definition of the Hidden Integer questions above in the DS section. Tip: if you see integer questions just like the number of fruits, people, animals, stamps, and candies, it is a good chance you are facing this question type. The name of Pigeonhole principle comes from the old problem of having to calculate the number of cases of pigeons entering pigeon lofts. (it will more clear once you read the example below). Integer Properties: There are 10 red balls, 10 blue balls, and 10 white balls in a box. What is the maximum number of balls you can draw from the box and have fewer than 5 balls of the same color drawn? A. 4 B. 5 C. 6 D. 10 E. 12 → This question asks for the maximum number of balls you can have, so the correct answer should be the unluckiest case. If you draw 4 red balls, 4 blue balls and 4 white balls all together, there can be no more than 4 balls of the same color. Hence, the correct answer choice is E . Questions related to Murphy’s Law and Sally’s Law are the questions for a perfect 51 in PS. * (Integer) There are 5 locks and 5 keys and each of the 5 keys matches each of the 5 locks. What is the minimum and the maximum trial numbers of attempts needed to confirm that each of the 5 keys matches each of the 5 locks? A. 5,15 B. 4,15 C. 5,10 D. 4,10 E. 5,20 → D is the correct answer choice. Can you figure out how to solve it quickly? Good – remember this approach so you can refer to it on your test. 5. Conclusion In this way, a perfect 51 is not an outcome of luck; you have to concentrate to excel in Integer and Statistics parts to that end. Especially, DS questions often will be the determining factor of a perfect Q51, and, therefore, you should not be misled by Mistake types. Please pay special attention to the Mistake type questions. Please remember solving 37 questions in 75 minutes in the actual GMAT exam is a big challenge. You will need to manage time effectively to solve hard questions in 2 minutes or less per question. How can you reduce the time spent per question significantly? Since GMAT is a logical test, you should not question whether you can solve a problem or not, but you should practice reaching answers logically without having to solve problems. *** We are math experts and if you find any grammatical issues – that is because we spend all of our time focusing on math; sorry grammar. Note that the information herein is based on the knowledge and experience of Max Lee, who taught 30,000+ students and solved 100,000+ problems in the past 15 years. He discovered and analyzed types of GMAT Math questions after continuous and numerous interviews with students who have achieved a score of 49 to 51 in the actual GMAT exam. Thus, please note that the information is herein is based solely on the experience of Max Lee. Due to the nature of the test and lack of transparency about the algorithm this guide is a best-effort attempt based on the best information available. If you have any questions or different information, please post it here for the benefit of the community.

=> Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Visit https://www.mathrevolution.com/gmat/lesson for details. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary. The question xy + 1 > x + y is equivalent to (x - 1)(y - 1)>0 for the following reason. xy + 1 > x + y => xy + 1 – x – y > 0 => xy - x - y + 1 > 0 (rearranging the equation) => x(y - 1) - 1(y - 1) > 0 => (x - 1)(y - 1) > 0 => x > 1, y > 1 or x < 1, y < 1 Condition 1) Condition 1) tells us that 0 ≤ x < 1 and 0 ≤ y < 1 . This tells us that x < 1 , and y < 1 , which fits our modified condition. Therefore, the answer is unique, 'yes,' and the condition is sufficient. Condition 2) If x = -1 , and y = -1 , then xy + 1 = (-1)(-1) + 1 = 1 + 1 = 2, x + y = (-1) + (-1) = -2 . In this case, xy + 1 is greater than x + y and the answer is ‘yes’. If x = 2, and y = -3 , then xy + 1 = (2)(-3) + 1 = -6 + 1 = -5, x + y = 2 + (-3) = -1. In this case, xy + 1 is less than x + y and the answer is ‘no’. Since condition 2) does not yield a unique solution, it is not sufficient. Therefore, A is the answer. Answer: A

=> Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Since we have 3 variables ( x, y, z ) and 0 equations, E is most likely to be the answer. So, we should consider each condition on its own first. As condition 1) includes 3 equations, we should consider it first. Condition 1) Since xy ≥ 16, yz ≥ 16 , and zx ≥ 16 , we have (xyz)^2 ≥ 16^3 or xyz ≥ 64 . Condition 1) is sufficient. Condition 2) If x = 20, y = 20 and z = 24 , then xyz = 9600 ≥ 64 and the answer is ‘yes’. If x = 1, y = 1 and z = 62 , then xyz = 62 < 64 and the answer is ‘no’. Since condition 2) does not yield a unique answer, condition 2) is not sufficient. Therefore, A is the answer. Answer: A This is a CMT(Common Mistake Type) 4(A) question. If a question is from one of the key question areas and C should be the answer, CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

=> Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Since we have 1 variable ( n ) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first. Condition 1) (n-17)(n-21) = 0 is equivalent to the statement n = 17 or n =21 If n = 17 , then 17 = 19 – 2 is a difference of two prime numbers and the answer is ‘yes’. If n = 21 , then 21 = 23 – 2 is a difference of two prime numbers and the answer is ‘yes’. Since it gives a unique answer, condition 1) is sufficient. Condition 2) (n-15)(n-17) = 0 is equivalent to the statement n = 15 or n = 17 If n = 15 , then 15 = 17 – 2 is a difference of two prime numbers and the answer is ‘yes’. If n = 17 , then 17 = 19 – 2 is a difference of two prime numbers and the answer is ‘yes’. Since it gives a unique answer, condition 2) is sufficient. Therefore, D is the answer. Answer: D In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

=> Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. The question is equivalent to asking if x ≥ -1 as shown below: \sqrt{(x+1)^2}=x+1 => |x+1| = x+1 => x ≥ -1 Since we have 1 variable ( x ) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first. Condition 1) x(x-2) = 0 => x = 0 or x = 2 If x = 0 , then x ≥ -1 and the answer is ‘yes’. If x = 2 , then x ≥ -1 and the answer is ‘yes’. Since it gives a unique answer, condition 1) is sufficient. Condition 2) x(x+2) = 0 => x = 0 or x = -2 If x = 0 , then x ≥ -1 and the answer is ‘yes’. If x = -2 , then x < -1 and the answer is ‘no’. Since it does not give a unique answer, condition 2) is not sufficient. Therefore, A is the answer. Answer: A If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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31 Jan 2021, 04:44

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[GMAT math practice question]

(inequalities) What is the solution set of the inequality, $bx - (4a + b) < 0$?

1) $a$ is negative and $a = -b$.

2) The solution range of $ax - (a - 2b) > 0$ is $x < 3.$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
The inequality $bx - (4a + b) < 0$ is equivalent to $bx < (4a + b).$

Condition 1)
Since $a < 0$ and $a = - b$, we have $b = -a$ and $b$ is negative.
$bx < (4a + b)$ is equivalent to $x > \frac{(4a + b)}{b}$, (the direction of the inequality sign changes because we are dividing by a negative number). Then $x > \frac{(-4b + b)}{b}$ (by replacing a with $-b$), $x > \frac{(-3b)}{b},$ or $x > -3.$

The solution is $x > -3,$ so we get yes as the answer.
Since condition 1) yields a unique solution ‘yes’, it is sufficient.

Condition 2)
The inequality $ax - (a - 2b) > 0$ is equivalent to $ax > (a - 2b)$
.
Since its solution set is $x < 3$, we have $x < \frac{(a - 2b)}{a} = 3.$ So, $3a = a - 2b$ or $a = -b$ and a is negative. It is same as condition 1).

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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[GMAT math practice question]

(Inequalities) Is $xy + 1$ greater than $x + y$?

1) $0 ≤ x < 1$ and $0 ≤ y < 1$.

2) $x + y$ is negative.

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[GMAT math practice question]

(Algebra) What is the value of $\frac{x}{y}$?

1) $5x(5 + 2\sqrt{5}) - 5\sqrt{5}y(3 - 2\sqrt{5})$ is a rational number.

2) $x$ and $y$ are rational numbers and $xy ≠ 0.$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have $2$ variables ($x$ and $y$) and $0$ equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)

$5x(5 + 2\sqrt{5}) - 5\sqrt{5}y(3 - 2\sqrt{5}) $

$= 25x + 10√5x - 15√5y + 50y$ (multiplying through the brackets)

$= 25x + 50y + 10√5x - 15√5y$ (rearranging the terms)

$= 25(x + 2y) + 5√5(2x - 3y)$ (taking out common factors)

We must have $2x – 3y = 0$ in order for $5√5(2x-3y)$ to be zero (and therefore cancel out the square root or irrational number because condition $1$ states that the answer is a rational number) when $x$ and $y$ are rational numbers.

Thus, we have $2x = 3y$ or $\frac{x}{y} = \frac{3}{2}$.

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems that require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with a probability of 70%), and E is the answer (with a probability of 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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[GMAT math practice question]

(absolute value) Is $|a|<1$?

$1) a^2<1$
$2) \frac{1}{(1-a^2)}>0$

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[GMAT math practice question]

(inequality) If $x, y, z$ are positive integers, is $xyz ≥ 64$?

$1) xy ≥ yz ≥ zx ≥ 16$
$2) x+y+z=64$

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[GMAT math practice question]

(number properties) If $\frac{k}{mn}$, where $k, m$ and $n$ are positive integers, is a fraction in its lowest terms, is $\frac{k}{mn}$ a terminating decimal?

1) $\frac{1}{m}$ is a terminating decimal

2) $\frac{1}{n}$ is a terminating decimal

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[GMAT math practice question]

(absolute value) Is $x<y<z$?

$1) |x+2|<y<z+2$

$2) |x-2|<y<z-2$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have $3$ variables ($x, y$ and $z$) and $0$ equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Now, by condition 1),$x < x + 2 ≤ | x + 2 | < y < z + 2,$ and $x < y$.
By condition 2), $|x-2|<y<z-2 < z,$ and so $y < z$.
Thus, both conditions together are sufficient since they yield $x < y < z$, and the unique answer is ‘yes.’

Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If $x = 1, y = 4$, and $z = 7$, then the answer is ‘yes’.
If $x = 1, y = 4,$ and $z = 3,$ then the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
If $x = 1, y = 4,$ and $z = 7$, then the answer is ‘yes’.
If $x = 5, y = 4,$ and$z = 7,$ then the answer is ‘no’.
Condition 2) is not sufficient since it doesn’t yield a unique answer.

Therefore, C is the answer.
Answer: C

Normally, in problems that require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with a probability of 70%), and E is the answer (with a probability of 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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[GMAT math practice question]

(function) Is $x + y < -1$?

1) $|x|>x$ and $|y|>y$

2) $x^2+y^2>1$

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[Math Revolution GMAT math practice question]

(number properties) Can $n$ be expressed as the difference of $2$ prime numbers?

$1) (n-17)(n-21) = 0$
$2) (n-15)(n-17)=0$

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[Math Revolution GMAT math practice question]

(absolute value) Is $\sqrt{(x+1)^2}=x+1$ ?

$1) x(x-2) = 0$
$2) x(x+2) = 0$

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[GMAT math practice question]

(Inequality) Which is greater between $(a + 2b)^2$ and $9ab$?

1) $1 < a < 2$

2) $\frac{1}{2} < b < 1$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The question is equivalent to the statement $(a - b)(a - 4b)$ is greater than or less than $0$ for the following reason:

$(a + 2b)^2 - 9ab > 0$

=> $a^2 + 4ab + 4b^2 – 9ab > 0$

=> $a^2 - 5ab + 4b^2 > 0$

=> $(a - b)(a - 4b) > 0$

Since we have $2$ variables ($a$ and $b$) and $0$ equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since $1 < a < 2$ and $\frac{1}{2} < b < 1$, we have $\frac{1}{2} < b < 1 < a < 2$ or $b < a.$

Since $1 < a < 2$ and $2 < 4b < 4$ (by multiplying the equation given in condition 2) by $4$), we have $1 < a < 2 < 4b < 4$ or $a < 4b.$

Then we have $a – b > 0,$ and $a – 4b < 0$ or $(a - b)(a - 4b) < 0.$

Thus, we have $(a + 2b)^2 - 9ab > 0$ and $(a + 2b)^2$ is greater than $9ab.$

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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[GMAT math practice question]

(Equation) What is the value of $x + 2y$?

1) The system of equations with $ax + by + c = 0$ and $bx + 2cy + 4a = 0 (abc ≠ 0)$ has infinitely many solutions.

2) $x + 2y$ is negative and $|x + 2y|$ is an even prime number.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Even though we have $2$ variables ($x$ and $y$) and $0$ equations, D is most likely the answer, since each condition has $2$ equations. So, we should consider each condition on its own first.

Condition 1)

Since $ax + by + c = 0$ and $bx + 2cy + 4a = 0 (abc ≠ 0)$ has infinitely many solutions, we have $\frac{a}{b} = \frac{b}{2c} = \frac{3}{4a}$. Assume $\frac{a}{b} = \frac{b}{2c} = \frac{3}{4a} = k.$

Then we have $b = ak, 2c = bk$ and $4a = ck$. When we multiply those equations together, we have $8abc = abdk^3$. Since $abc$ is not equal to $0$, we have $k^3 = 8$ or $k = 2.$

Then, $b = 2a, 2c = 2b, 4a = 2c.$ The second equation reduces to $c = b,$ so we have $b = c = 2a.$

Thus both equations in the system of equations are $x + 2y + 2 = 0.$

We have $x + 2y = -2.$

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since the unique even prime number is $2$, we have $|x + 2y| = 2$ which yields $x + 2y = 2$ or $x + 2y = -2.$

Since $x + 2y$ is negative, we have $x + 2y = -2.$

Since condition 2) yields a unique solution, it is sufficient

Therefore, D is the answer.
Answer: D

This question is a CMT 4 (B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4 (B) questions, D is most likely the answer.

Note: Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as condition 2).

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C or E.

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[GMAT math practice question]

(Statistics) A company has $2$ departments, department $A$ has $5$ employees, and department $B$ has $6$ employees. All employees of the company did some “push-ups”. What is the standard deviation of these $11$ employees?

1) The average and the standard deviation of $A$ are $7$ and $1$, respectively.
2) The average and the standard deviation of $B$ are $7$ and $3$, respectively.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have the standard deviations of two sets, we have 2 variables and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Assume A1, A2, … , A5 are the “push-up” numbers of the employees in department A and B1, B2, … , B6 are the “push-up” numbers of the employees in department B.
The combined average of departments A and B is
$\frac{( 5*7 + 6*7 ) }{ 11} = 11*\frac{7}{11} = \frac{77}{11} = 7.$

The variances of A and B are squares of the standard deviations of A and B, respectively.
The variance of A is { (A1-7)^2 + … + (A5-7)^2 } / 5 = 1 and we have { (A1-7)^2 + … + (A5-7)^2 = 5.
The variance of B is { (B1-7)^2 + … + (B6-7)^2 } / 6 = 1 and we have { (B1-7)^2 + … + (B6-7)^2 = 6.

The combined variance of sets A and B is
{ (A1-7)^2 + … + (A5-7)^2 + (B1-7)^2 + … + (B6-7)^2 } / 11
$= \frac{{ 5 + 6 } }{ 11} = \frac{11}{11} = 1$.
The standard deviation of sets A and B is the square root of the combined variance equal to 1.

Since both conditions together yield a unique solution, they are sufficient.

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since we don’t have any information about department B, it is not sufficient, obviously.

Condition 2)
Since we don’t have any information about department A, it is not sufficient, obviously.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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[GMAT math practice question]

(number properties) If p and q are prime numbers, what is the number of the different factors of p^2q^3?

1) pq=143
2) p and q are different

Answer: D

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since p and q are prime numbers, condition 1) tells us that p = 11 and q = 13, or p = 13 and q = 11. Therefore, since p and q are different prime numbers, the number of different factors of p^2q^3 is (2+1)(3+1) = 12. Condition 1) is sufficient since it yields a unique solution.

Condition 2)
Since condition 2) tells us that p and q are different prime numbers, the number of factors of p^2q^3 is (2+1)(3+1) = 12.
Condition 2) is sufficient since it yields a unique solution.

Therefore, D is the answer.

Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as to condition 2).

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[Math Revolution GMAT math practice question]

(statistics) If the average (arithmetic mean) of $5$ numbers is $20$, what is their standard deviation?

1) Their minimum is $20$.
2) Their maximum is $20$.

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[Math Revolution GMAT math practice question]

(set) If $|X|$ is the number of elements in set $X$, and $“∪”$ is the union and $“∩”$ is the intersection of $2$ sets, what is the value of $|A∩B|$?

$1) |A∪B|＝50$
$2) |B|=40$

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[Math Revolution GMAT math practice question]

(inequality) If $x$ is integer and $3|x|+x<4$, what is the value of $x$?

$1) x<0$
$2) x>-2$

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[GMAT math practice question]

(function) A function $f(x)$ satisfies $f(x)f(y)=f(x+y)+f(x-y)$ for every real numbers $x, y$. What is the value of $f(0)f(1)f(2)f(3)$?

$1) f(0)=2$

$2) f(1)=1$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

If $x = 1, y = 0$, then we have $f(1)f(0) = f(1) + f(1) = 2f(1).$

If $x = 1, y = 1,$ then we have $f(1)f(1) = f(2) + f(0)$

If $x = 2, y = 1$, then we have $f(2)f(1) = f(3) + f(1)$

Since we have $4$ variables $(f(0), f(1), f(2)$ and $f(3)$) and $3$ equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
If $f(1) = 1,$ then we have$f(2) = f(1)f(1) – f(0) = 1 – 2 = -1, f(3) = f(2)f(1) – f(1) = (-1)*1 – 1 = -2$ and $f(0)f(1)f(2)f(3) = 2*1*(-1)(-2) = 4.$

If $f(1) = 2$, then we have $f(2) = f(1)f(1) – f(0) = 4 – 2 = 2, f(3) = f(2)f(1) – f(1) = 2*1 – 2 = 0$ and $f(0)f(1)f(2)f(3) = 2*2*(2)*0 = 0.$

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since $f(1) = 1,$ we have $f(0) = f(1)f(0) = f(1) + f(1) = 2, f(2) = f(1)f(1) – f(0) = 1 – 2 = -1, f(3) = f(2)f(1) – f(1) = (-1)*1 – 1 = -2$ and $f(0)f(1)f(2)f(3) = 2*1*(-1)*.(-2) = 4.$

Condition 2) is sufficient since it yields a unique solution.

Therefore, B is the answer.
Answer: B

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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[GMAT math practice question]

(Function) $f(x)$ is a function. What is the value of the non-zero solution($s$) of $f(x) = f(-x)$?

1) $f(x)+2f(\frac{1}{x})=3x$

2) $x$ is an irrational number.

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since a function has many variables to determine, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) together give us:
When we substitute $x$ in the equation $f(x)+2f(\frac{1}{x})=3x$ by $\frac{1}{x},$ we have $f(\frac{1}{x}) + 2f(x) = \frac{3}{x}.$

When we subtract double the second equation from the first equation, we have:
$f(x) + 2f(\frac{1}{x})] – 2[f(\frac{1}{x}) + 2f(x)] = 3x – 2(\frac{3}{x})$

⇔ $f(x) + 2f(\frac{1}{x}) – 2f(\frac{1}{x}) - 4f(x) = 3x – \frac{6}{x}$

⇔ $-3f(x) = 3x – \frac{6}{x} $

⇔ $f(x) = \frac{2}{x} – x.$

$f(x) = f(-x)$ is equivalent to:

$\frac{2}{x} - x = \frac{-2}{x} + x$

⇔ $2x = \frac{4}{x}$

⇔ $x^2 = 2,$

Thus, we have $x = ± √2.$

Since condition 2) tells us x is an irrational number, the answer is not unique.
The conditions are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) and 2) together are not sufficient.

Therefore, E is the answer.
Answer: E

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.

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[GMAT math practice question]

(Algebra) What is the value of $2x + y$?

1) $\frac{(x - y)(y + 3)}{4(x - y)^2 +(y + 3)^2}= - \frac{1}{4}$

2) $x – y = -1$

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since we have $2$ variables ($x$ and $y$) and $0$ equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Assume $a = x – y$ and $b = y + 3.$

When we try to check both conditions together, we realize condition 1) alone is sufficient because condition 1) tells us $2x – y = -3$ for the following reason.

$\frac{(x-y)(y+3)}{4(x-y)^2+(y+3)^2}= - \frac{1}{4}$

⇔ $\frac{ab}{4a^2+b^2}=-\frac{1}{4}$ ($a = (x – y)$ and $b = (y + 3)$)

⇔ $4a^2+b^2=-4ab$ (cross multiplying)

⇔ $4a^2+4ab+b^2=0$ (adding $4ab$ to both sides)

⇔$(2a+b)^2=0$ (factoring)

⇔$2a+b=0$

⇔ $2(x - y) + (y + 3) = 0$ (substituting ($x – y$) and ($y + 3$) back in)

⇔ $2x – 2y + y + 3 = 0$ (multiplying $2$ through the first bracket)

⇔ $2x – y + 3 = 0$ (adding like terms)

Thus, we have $2x – y = -3$ from condition 1) alone.

Condition 2)
Condition 2) is not sufficient, obviously, since condition 2) does not yield a unique solution.

Therefore, A is the answer.
Answer: A

This question is an application of CMT 4(B): condition 2) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, we may assume condition 1) is sufficient. Since we can figure out condition 2) is not sufficient, we should be able to choose A.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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Current Tuck MBA students and Discover Tuck co-chairs share advice for prospective students and reflect on why they chose Tuck for their MBA.

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The Ultimate Q51 Guide [Expert Level]

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615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

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