- The quantity in Column A is greater
- The quantity in Column B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
Many people dread choosing answer choice (D) on Quantitative Comparison (QC) Some feel it may be conceding defeat. Others think that the GRE is trying to trick them by making them pick (D). After all, they think, there must be some pattern that I’m not getting.
The truth is answer (D) comes up often. And to determine whether an answer cannot be determined is actually not too difficult.
#1 Determine a relationship
Say you find an instance, in which the answer is (A) the information in column A is greater. If that is the case, then the next step is to disprove that.
#2 Disprove that relationship
Meaning, see if you can come up with an instance, either through plugging in different variables, manipulating algebra, or manipulating a geometric figure, in which the answer is not (A). As soon you do that, you can stop. The answer is (D).
If you can’t disprove your answer, then it must be correct: it must be (A), (B) or (C).
1. \(-100 < x < 0\)
| Column A | Column B |
|---|---|
| \(x^{-4}\) | \(x^{-3}\) |
- The quantity in Column A is greater
- The quantity in Column B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
Explanations:
Question #1
After choosing a few numbers you should note something: that Column A will always be greater than Column B.
Why? Will, whenever, you have an even exponent, positive or negative, that exponent will always yield a positive number.
Odd exponents, on the other hand, give you a negative output if the base (the number below the exponent) is negative. Remember x has to be negative. So no matter what number you plug in Column A will always be positive, Column B negative. This is a definite not (D). The answer is (A).
hi chrish i put all the different values if i put 1 , they both are equal , if i put -1 B is greater . same way if i put 2 value is 0.0625 A is smaller and 0.125 B is greater . didn’t get that . pls help
Hi Lokesh,
Remember that we have an important limitation to which numbers we can choose. The question tells us that -100 < x < 0. This means that x must be a negative number. You are plugging in positive numbers, which will give you a different answer 🙂
Hi Chris,
I have a doubt related to quantitative comparison type questions. What option do we choose if both columns turn out to be infinity. For eg:
Column A Column B
The number of numbers from 1 to 2 The number of numbers from 1 to 5
D or C?
Hi Karan,
Great and interesting question! Honestly, you will not see a question like this on the GRE, in which you are comparing two quantities, each with the solution of “infinite.” That being said, the comparison of infinite to infinite is equal, so I’d go with (C). But again, you will not see something like this. Still, it’s a very interesting and awesome question! Have a great day! 🙂
Hi Chris,
In the above example
-100<x<0
Qty A : x^-4
Qty B : x^-3
Now multiply both sides by a positive number relation will not change. x^4 is a positive number.So
Qty A becomes : 1
Qty B becomes : x.
Now it is easy to plug -ve numbers for x.
In all definite ways, column A will be greater.
Can i approach this way or plugging numbers will be easy in test ?
I usually make mistakes while plugging -ve numbers .. Pls help
Hi Prasanth,
No, I do not recommend taking this approach. For example, imagine if you manipulated the problem by multiplying by x^5. So,
Qty A becomes: x
Qty B becomes: x^2
Now, in this case, Qty B is greater than Qty A, which is not true. It is best to approach this problem rephrasing this question.
Qty A: x^-4 = 1/(x^4)
Qty B: x^-3 = 1/(x^3)
Here you know that any negative number raised by an even exponent will be positive, and that same number raised by an odd exponent will be negative. This is how we know that column A is greater. I hope that helps! 😀
Hi Chris, regarding this problem :
-100<x<0
Qty A : x^-4
Qty B : x^-3
Prasanth multiplied both sides by a positive number (x^4 is a positive number),which will be at all times positive since (-100<x<0), and it is legal to multiply both sides of an inequality by a positive number..
But it wil not be true to multiply both sides by x^5 since (-100<x<0),so x^5 will give a negative number. And it will change the inequality..
i think Prasanth's approach is right. Does it ?
I would appreciate your explanation
thanks,,
Hi Fatima,
Prasanth’ approach is correct, in the sense that it really does work for this particular problem. Here, multiplying Quantity A and Quantity B by x to an even power will help simplify things and may help make them clearer.
The reason I’d still recommend against Prasanth’s approach is that it adds an extra layer of complexity that isn’t necessary. Under real testing conditions, every second counts, and simply recognizing the number properties at play here, without plugging in new numbers, is important. In addition under the pressure of test conditions, it becomes easier to make a mistake like choosing x^5 instead of x^4 without noticing that the inequality is reversed.
Thank you! Sir
In lesson intro to Quantitative comparison, there’s this example question
N is not an integer
6<N<10
Quantity A Quantity B
N 8
Here, the answer is mentioned as D) cannot be determined reason stated as it can be a decimal or a fraction, but integer itself means "Whole nos which can be -ve as well as +ve".
Hi Priyam 🙂
Thanks for your message 🙂 In that example, we’re told that 6 < N < 10. So, we know that N is a positive number between 6 and 10. However, we cannot determine whether N is greater than 8 using only the information given, that N is not an odd integer. N could be any positive number between 6 and 10 except 7, which is the only odd integer within this range. For example, N could equal 6.5 or 8.3. Neither of these numbers is an odd integer and therefore fits the description. Because N may be less than, equal to, or greater than 8, the answer to that example is (D).
I hope this clears up your doubts 🙂
I have a doubt regarding this. In the above explanation, you have mentioned that “N could equal 6.5 or 8.3”. But, in the question it has been specified that N is an integer and 6.5 or 8.3 are not integers.
Hi Bhavna 🙂
In that example, the prompt states that “N is not an odd integer.” This is not the same as saying that N is an even integer. N could be an even integer, but N could also be a decimal or fraction, since decimals and fractions are not odd integers. I hope this clears things up 🙂
Thank you so much.. 🙂 I missed this point.
Hi Chris,
In your explanation to the 1st question, in the 2nd last sentence you wrote
“So no matter what number you plug in Column A will always be negative, Column B positive.” Isn’t this the exact opposite because Column A wiil be positive and column B will be negative?
I got confused the first time I read this.
Chris,
For these types of problems I have a trick that allows me to correctly deduce what the answer is. I wanted to get your thoughts on my method.
An example problem that I fabricated –
n is an integer
Which is greater?
COL A – n^3 + 6
COL B – n^2 + 5n -30
I would attack this problem in this way – I would choose 5 arbitrary numbers (0,-0.6, 0.6,
3,-3) and plug each in till i get an explicit answer. The only problem I see with my this is that it can get to be time consuming.
Let me know what you think!
PS – I love it when you answer in a witty way with your sophisticated words!
Hi Sammy,
You put me in a little bit of a tough spot, because it’s kind of difficult to be witty with Quantitative Comparison :).
Your method definitely works well. Picking numbers can help you determine which column is bigger as long as you make sure to include 0, 1, -1, 2, -2 (or 3, -3) and 1/2 and -1/2 (I think 0.6 is a little unwieldy :)).
But in general this is a very effective method and one that I too employ. Often, esp. with algebraic equations, you may want to manipulate the equations and see if you can simply them.
Sorry, I didn’t drop any sophisticated words, but I hope that helped :).
Thanks Chris, my only qualm would be that the method can be somewhat time consuming.
Oh, that’s right – you mentioned that…if you have a strong facility with numbers, that is you can quickly plug-in, then it shouldn’t be time consuming. Also you can come up with a tiny column grid:
A B
0
1
-1
2
-2
By the time you get the 2 you can usually know for certain what column it is.
Hope that helps :).
Hi Chris,
Would it be correct to say that if n is an integer, one need not test decimals that can be written as a fraction. For example, 0.6 = 6/10, which means that 0.6 is actually a fraction, aka an integer divided by another integer, or two integers. And the same thing for 0.5, since it equals 1/2. In fact, when it comes to “n is an integer” type questions, we should be able to exclude all decimals, since according to the “official GRE guide,” only real numbers are involved (e.g. we would not test 3.14 which is a decimal but can’t be written as a fraction. This would mean that every decimal number is a real number (which implies it can be written as a fraction) for GRE purposes, as far as quantitative comparison type questions are concerned. Great question by the way.
Yes, you are exactly right. I was speaking more broadly when giving that spread of numbers. One should always obey the constraints of the question, i.e. if x is a positive integer, you have to plug in accordingly.
Thanks for catching that :).
Yes you are right, it was an egregious error made on my part. I meant to say ‘a real number’
No problem :). Glad I could help.
hey chris i am your big fan of u can u give me links for practcicng gre full lenth tests of free costs
Hi Harsha 🙂
I’d recommend that you first take the free PowerPrep tests from ETS, if you haven’t already. Next, here’s a free test from Manhattan (their materials are great!): Manhattan Free GRE Practice Test. Lastly, if you want access to more practice tests, I’d recommend purchasing one of the Manhattan books–each book comes with a code to access 6 online tests.
Hope this helps 🙂
Hi Chris,
I have a question regarding a concept in Question #2: you state “Three equal sides equal three equal angles.” Is this a characteristic of all polynomials, i.e., can we say “x equal sides equal x equal angles” for any figure?
Best regards,
Denis
Hi Denis,
The measure of an angle corresponds to the length of the side opposite that angle, in a triangle. With quadrilaterals, or figures with more than four sides an angles, that relationship between which side corresponds to which angle is not as clear cut. However, we can say with certain that if there are x equal sides then there are x equal angles.
Hope that helps :).
Hi Chris,
I greatly appreciate the expeditious reply. Thanks for the help.
I’d say the answer for question 1 is (D). If you pick -1, then you get 1 for both column A and column B, since you have 1 over 1 raised to some positive power. If you pick, say, -2, then column A and column B are obviously different.
Hi Amal,
If you take any negative to an even exponent it will always yield a positive number. Therefore, if you plug in -1, you will get 1 for Col. A and -1 for Col B.
Hope that helps :).
Yeah you’re right. I forgot the negative sign on the coefficient. When you raise a negative coefficient to a negative odd power, the coefficient (which winds up in the denominator) is still negative. The coefficients are always negative, but the even power in column A yields a positive quantity.
No problem 🙂