Performing a quick search on the internet about the definition "
Quantitative Comparison Question" (now on QCQ) the word more frequently comes up is
bizarre.
The format of a QCQ adheres to the following definition:
Quote:
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:
(A) Quantity A is greater.
(B) Quantity B is greater.
(C) The two quantities are equal.
(D) The relationship cannot be determined from the information given.
A symbol that appears more than once in a question has the same meaning throughout the question.
Quantity A is greater means that for any and every possible case, Quantity A will always be greater than Quantity B.
Quantity B is greater means that for any and every possible case, Quantity B will always be greater than Quantity A.
The two quantities are equal means that for any and every possible case, the two quantities will always be equal.
The relationship cannot be determined from the information given: This will be true when more than one relationship stated above is true.
This format is unique to the GRE exam. There is no any rumor about or any corroboration of the following assumption I am going to make: in 2011 this peculiar question format is the answer by ETS to Data Sufficiency format the GMAT use to test the students' logic ability.
Little digression: if you imagine being an analyst just in front of three or four computer monitors, let say at the NYSE, and your complex task boils down to the comparison of two values/variables/quantities/mortgages or whatever they be, your goal is basically to compare these. For your brain, even though is so powerful, is almost impossible to process millions of data at the same time; for this specific reason, we have invented huge mainframes to perform the calculation. But, after all, they are machines, nothing more than silicon chips in an array supported by hard drives to stock information to recall at the right momentum. Is the operator/user/mankind that will be decided which data will be processed based on our knowledge or experience or fantasy or all in a beautiful mash-up.
QCQ, actually, is a representation of this process on a small scale: quantity A is greater than B, quantity B is greater than quantity A, they are equal or I cannot say which is which. Akin, in Data Sufficiency question we have to evaluate: A is sufficient, B as well, both are needed, no one is used to reach our decision, every piece of information is good on its own to reach a conclusion.
All this to say, QCQ must be treated with the right strategy and attitude towards, otherwise your score will be under the par, considering that they account for roughly 40% of the total questions in each of the two quantitative sections of the test AND they are the first 7/8 questions in a row since the beginning of the quant section. My grandmother said:
who begins well is a goal of the work.
#1 -
Down to the businessIn the general part of "
All you Need to Know about Quantitative Reasoning" we explained that you must have a strategy to tackle every question during the test, in the allotted time, but the question is: how do I tackle QCQ ?? which is the strategy that I must follow to perform well and get a high score?
I have read and read constantly tons of articles, books, quick guides, everything is possible to conceive to improve both on the quant and verbal side of the test and to learn always something possible new. At the end of the day, the strategy that most of the teachers/tutors/companies out there suggest is a classic methodology top-down, typical of well-established literature of the case. Nothing wrong with that.
At this point, the question is: why do a lot of students fail ?? Once again, read very carefully the general part of this guide. Here, we are going to show the right strategy for QCQ, a bullet-proof strategy for the maximum score.
We have to think of our strategy on a three levels
- Micro-level strategy
This strategy entails that you do know as cold every possible way to manipulate your knowledge of algebra/geometry/number theory/factorials/absolute value/standard deviation/probability/combinatorics/ and so forth. We are not here to explain these concepts per se. All you need to know about is contained in the Math_book that you find attached in the general part, at the bottom. Download it and study thoroughly. The concepts explained for the GMAT are the same encompassed during the GRE exam. No worries about, guys.
The thing that must be clear in this stage is that the concepts have to be crystal clear in your mind. Let me give you an example:
Quote:
Quantity A |
Quantity B |
\(\sqrt{96} < x \sqrt{6}\) |
\(\frac{x}{\sqrt{6}}\) \(< \sqrt{6}\) |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
The question sounds a little bit on the tough side, isn't it ?
However, if you do know as cold the rules to solve the roots like a second skin, this question can be solved with a 30 seconds approach:
Quantity A: \(\frac{\sqrt{96}}{\sqrt{6}} < x\) \(= \sqrt{\frac{96}{6}}\) \(= \sqrt{16} < x = 4 < x\) or \(x > 4\)
Quantity B: \(x < \sqrt{6} * \sqrt{6} = \sqrt{36} = 6\) or \(x < 6\).
A can be 5 and B can 4 so \(A > B\)
A can be 4.1 and B can be 5.9 so \(B > A\)
Moreover, nothing is stated if x is an integer or a fraction. As such, the answer is D.
As you can see the strategy or tip I suggest you is this, in a nutshell: study very well all the math rules tested during the GRE exam. You will solve always a tricky question like this in a snap.
- Meso-level strategy
GRE Quantitative Comparison Strategy, even though it seems like a monolith topic, can be divided into areas of content or techniques to apply, in which several kinds of skills are necessary to tackle every single question.
The areas are basically:
- Algebra - it accounts for a huge part of your skills for the purpose;
- Fractions;
- Geometry;
- Number Property;
- Word problem.
The main techniques are:
- Approximation;
- Picking Numbers;
- Matching Operations;
- Looking for Equality;
A bridgehead of the two classifications above, in operation, are:
- PEMDAS (the order of operations)
- Relationships between even and odd numbers
- Simplify and combine values containing exponents or square roots
- Simplify algebraic expressions
- Interpret graphs, charts, and tables
- Graph functions and lines on coordinate systems
- Probability and percentages
- Rules of geometry
The following definition is a perfect fit for how the strategy above in concrete is applied:
Quote:
Granularity - The level of detail considered in a model or decision making process. The greater the granularity, the deeper the level of detail. Granularity is usually used to characterize the scale or level of detail in a set of data.
Probably, it is the first time that some of you see this word and the consequent definition. However, it is the perfect definition of your strategy. It must be broken into small chunks to be able to handle every question and its aspect. The result: achieve the best score you can.
The following tips are designed to cascade and make up the bricks of your concrete strategy wall. However, keep in mind that my analogy of a wall has a twofold meaning: on one hand, your strategy must be just like a wall: solid, strong, consistent. The key concept here is consistency: any result cannot be obtained if you do not follow a consistent manner in what you do. Follow a study plan, training at least two hours per day of practice questions, dedicate at least one hour to learn or re-brush your theory notions. Every day from Monday to Friday. No pause. On the other hand, your strategy and the underlined tips to follow have to be flexible. Driving on a highway, you have to be able always to switch into the fast lane and overcome the traffic funnel, whenever is possible. Akin, if you hit a dead point, change your strategy to tackle the question or try something else to speed up the process. Always.
#1.
QCQ questions are relatively easier than other types of questions tested and require less time to solve. Therefore, you should allot approximately
45 seconds to 90 seconds to each QC question.
#2.
Memorize the 4 answer choices.
They are always the same. Not wasting time, having in mind the 4 answer choices, you will be able to focus completely on the question that pops out in front of you during the exam.
#3.
Focus on concepts more than on numbers.Never consider QCQ as intensive calculation questions. If you're doing a lot of number crunching you're probably overlooking whatever math concept is at the heart of the question. So try to minimize your calculation wherever is possible.
#4.
Do only as much work as you need in order to make the comparison.
This is a consequence from the previous tips: your task is to determine only which quantity is greater or if you cannot determine which is greater (i.e. the answer is D) — not how much greater one quantity is than the other.
#5.
Never select the fourth answer choice if the comparison involves only numbers (no variables).
This is a simple yet very important concept. If you know you can calculate a specific number value for each quantity, then even if you haven't determined which is greater you at least know that they can be compared.As a matter of fact, knowing that D is NOT the answer will increase your chances to pick the right one using a strategic guess. You have already eliminated 25% of possibilities to pick it wrong. i.e you do have 3 out of four chances to pick the correct one.
#6.
Trying to prove (D)Keep in mind that point #5, on the other hand, try to prove (D) when the two quantities contain variables. This is an important technique to apply, I do notice how in the upper level question very often D is the right answer, though a % in terms of the whole is quite difficult to assert, there is not a clear pattern.
Using this technique, use numbers from both sides of the number line. Employ \(-1,-\frac{1}{2},0,\frac{1}{2},1\). Moreover, do the following:
- Add or subtract to both quantities.
- Multiply or divide both quantities by a positive number.
- Square or square root both quantities if they are positive.
- If a variable has no constraints, try to prove (D).
- When absolute values contain a variable, maximize the absolute value by making the expression inside as far away from zero as possible. Add positives to positives or add negatives to negatives.
#7.
Establish without rooms of error the type of Math being testedThis last statement is really important. At first glance, it is easy to identify a question for what it really is. To pinpoint. This is a common belief. However, as the most thing in life, they are misleading. ETS is a champ to misleads you and push you back among the rest of the pack. Your goal instead is to achieve the best score. Being in the 99 percentile. You have to stand out above the rest of the pack. A such, see a question what it is testing and that you are able to see much deeper what is the real gist of the question will permit you to obtain several benefits:
- seeing what a question is testing and more, you will reach a faster solution;
- the first point will make you gain time. After all, this is also a timed test. half of your battle is against time and everything is possible to NOT waste your time.
- gainging time, will permit you to spend a little bit more on the hardest question of the test.
- psichologically, solving a question faster, you will not have mental fatigue and stress.
Let me explain with a clear example what I am trying to convey to you.
Quote:
Quantity A |
Quantity B |
\(\frac{2^3 * 17 * 5^2}{60}\) |
\(\frac{255}{2}\) |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
A good approach and fast is the following, though classic:
Quote:
QTY A = \(\frac{2^3 * 17 * 5^2}{60} = \frac{8 * 17 * 25}{60} = \frac{170}{3}\)
As we compare we find the numerator of QTY A is less than the numerator of QTY B and
the denominator of QTY A is greater than the denominator of QTY B
Hence QTY A < QTY B
Option B
What type of considerations could we spend on the solution provided to the question above ? well, it is good and perfectly fit the case. However, at a closer look, it requires a certain amount of calculation, even though not intensive. Remember: GRE is not a test of intense calculation but logic.
Notice how the fraction is treated as a calculation:
\(2^3=8\); 17 is a prime number; and \(5^2=25\). The approach is perfectly fine. What if, do we think about it from the prime factorization standpoint instead of from the fraction comparison?
QA is \(\frac{2^3*17*5^2}{60}\)
QB the faction is multiplied per 30: \(\frac{255*30}{2*30}= \frac{255*30}{60}= \frac{5*51*3*2*5}{60}\). Now disregard the denominator because is equal between the two.
Now:
QA is \(2^3*17*5^2\) and
QB is \(5^2*51*3*2\)
\(5^2\) could be cross off and we left with \(2^3*17\) against \(51*3*2\). Clearly, the second quantity is greater. This approach costs you 30 seconds to operate, which is 10 less than the minimum time required for a QCQ question.
Conclusions:
- save time for the hardest questions;
- faster approach.
As you can see from the example above you have learned how you should handle a question from different angles, searching for different ways to solve it. The question was about fractions but at a closer look, it was also a number property question, even though not in a classical sense. However, this shows you how your strategy has to be concrete and at the same time flexible. Using an analogy is like having the ax and the sledgehammer at the same time, the two fundamental tools of a hard worker, to destroy the question in front of you in the less amount of time you do can. This leads us to the next
#8 Never select D as the answer when you have in one of the two quantities only numbers and NO variables.If you can calculate the two quantities, then D cannot be the right one because is one way or another you can compare two certain quantities. Even if you haven't determined which is greater you at least know that they can be compared. You can always reach a solution. You do have always enough information.
#9 Consider all the possible scenarios when you are dealing with variables.
Algebra or algebraic manipulations account for a considerable proportion of the test and in particular when we are dealing with QCQ. The way variables are presented under the form of a question lead you to two important considerations:
- pinpoint any clue that gives you a possible hint about the strategy to employ;
- from this: which types of skills you have to use to reach faster the right solution.
The
if........
then ........scenario is the most suitable this turn.
IF a variable | Then |
has a unique value (e.g. x=2) | solve for the value of the variable |
has a defined range (e.g. \(-\frac{1}{2} \leq x \leq \frac{1}{2}\)) | test the boundaries |
has a relationship with another variable (e.g. \(2r = s\)) | simplify the equation and make a direct comparison of the variables |
has no constraints | try to prove (D) |
has specific properties (e.g. y is positive and is an integer) | try to prove (D) |
#10 When quantity A or B is a number and the other is something else (e.g. an equation, an inequality, and so forth) use the quantity in the form of a pure number as benchmark#11 Don't rely on your sightIn the case you are dealing with a geometry figure NEVER compare quantities by visual estimation or measurement. Compare them instead based on your knowledge of mathematics and on the non-graphical data provided. You cannot assume that a figure is drawn to scale unless it is accompanied by a "the figure drawn the scale" usually at the bottom. That say, this holds for all possible QCQ you are going to solve. Instead, try to redraw the figure, keeping those aspects that are completely determined by the given information fixed but changing the aspects of the figure that are not determined.
What if the questions do not include a diagram.
- Establish what you need to know.
- Establish what you know.
- Establish what you don't know.
Look at this question. It fits perfectly our teaching intention:
Quote:
A cube of cheese is 3-inches high. The cheese is sliced twice.
Quantity A |
Quantity B |
Resulting surface area of all the slices of cheese |
90 square inches |
A. The quantity in Column A is greater
B. The quantity in Column B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
What about this question:
- you do not have any figure, so you have to draw it for the sake of purpose;
- for sure your figure will not draw the scale, so the figure you scratch on your paper could be by itself misleading. So, try to figure it out in several possible useful ways.
- Use B as a benchmark to reach the solution.
#12 Plugging numbersOne of the most simple yet powerful technique without a shadow of the doubt. Always consider a whole range of numbers: positive, negative, integers, fractions.
- Macro-level strategy
Quote:
Quantity A |
Quantity B |
PS |
SR |
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
Quote:
ExplanationFigure 1.From Figure 1, you know that PQR is a triangle and that point S is between points P and R, so PS is less than PR. and SR is less than PR. You are also given that PQ is equal to PR. However, this information is not sufficient to compare PS and SR. Furthermore, because the figure is not necessarily drawn to scale, you cannot determine the relative sizes of PS and SR visually from the figure, though they may appear to be equal. The position of S can vary along side PR anywhere between P and R. Below are two possible variations of Figure 1, each of which is drawn to be consistent with the information PQ is equal to PR.
Figure 2.Figure 3.Note that Quantity A is greater in Figure 2 and Quantity B is greater in Figure 3. Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.
#2 -
RemarksAs you can see from the diagram above and the question showed under the macro-level strategy, the strategy itself is NEVER carved in stone or in a way top-down/bottom up but ALWAY flexible and prone to changing/adjustments on the run. Multiform and versatile.
On top of that, after your tool-box is composed of all possible weapons provided by the strategy at the first two levels you will be able to tackle even the most difficult question. It is a holistic approach that you must have. When you will reach your way of reasoning as a second skin, then you are ready for the top score. always try to:
- being ready to change strategy when you are stuck;
- try to see the question from every possible angle to attack it as fast as you can without getting lost in a maze and not going in the wrong way, wasting your precious time;
- evaluate ALWAYS upfront a question for about 15/20 seconds before to start any kind of strategy: PEMDAS, plugging numbers and so on;
- when a question is difficult: skip it behind and move on to another question. back to it could be useful because you could have a different view to attack even after some minute:
- when nothing jumps out in your head: plugging number is the first strategy. At least, you try to eliminate some answer choice and you can still apply an educated guess;
- be proactive towards the question in front of you (very important point). Do not wait for the question will be solved by itself.
All the concepts exposed above are also explained,
beautifully I would say, during the course tenured by our lead board's tutor
GreenlightTestPrepI -
GRE Math - Intro to GRE Quantitative Comparison (QC)II -
QC Strategy - ApproximationIII -
QC Strategy - Matching OperationsIV -
QC Strategy - Plugging in NumbersV -
QC Strategy - Number SenseVI -
QC Strategy - Looking for EqualityVII -
QC Strategy - Miscellaneous----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Practice:
Quantitative Comparison Questions Hard -
Medium -
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