GRE Quantitative Comparison Tip 1 Dealing with Variables
Does Plugging in Work on GRE Quantitative Comparison?
Picking numbers. Now we can talk about picking numbers for the quantitative comparisons.
First Ill say, way back in the Algebra module, we discussed the strategy of picking numbers to solve
algebraic problems that had variables in the answer choices. You may remember we talked about those as VICs, V-I-Cs,
So Ill assume here that youve seen those lessons. If you havent seen them, it would really be helpful to watch those lessons and
understand that logic before approaching this lesson. This lesson kind of depends on your understanding the logic
from those lessons. So having said that, now were gonna talk about quantitative comparisons and
And of course, picking numbers is an approach that we can use in the QCs, especially quantitative comparisons that have variables involved.
Maybe variables in both answer choices or comparing a variable to a number with some kind of algebraic expression, that kind of thing.
Its very important to understand the strengths and limitations of this strategy.
Pause the video and then well talk about this. Okay.
Here theyre very good to us. They tell us that n is an integer greater than 2.
So we know its an integer. It cant be negative.
It cant be a fraction or a decimal, very good. So the most obvious choice if were gonna pick numbers,
the most obvious choice is 3. 3 is greater than 2.
So, what happens if we pick n equals 3? Well in Quantity A, we get 2 to the 3rd.
2 to the 3rd is 2 cubed, thats 8. And of course, in Quantity B we get 3 squared, thats 9.
So Quantity B is bigger if we pick n equals 3. Now, the next choice we could pick, a very obvious choice is n equals 4.
Now Quantity A, this is gonna be 2 to the 4th. Thats 2 times 2 times 2 times 2.
That turns out to be 16. Turns out to be 4 times 4 because each 4 is 2 times 2.
And in Quantity B, we get 4 squared, thats also 16. So that makes the quantities equal.
Right away, two different choices give us two different relationships. And that means, of course, the answer has to be D.
There, picking numbers worked out particularly well.
When we can pick two different values, and get two different relationships, we know right away that the answer is D.
Picking numbers can be an incredibly efficient way for determining the answer when the answer happens to be D.
But, of course, D is the answer on average, only about 25% of the time. What happens when D is not the answer?
So lets pause that question for a minute. Well come back to that question, but heres another practice problem.
Work on this. Pause the video, work on this, and then well talk about this.
So here, x is just a number. It could be positive, it could be negative, it could be whole,
So, the gamut is open. It just has to satisfy that absolute value inequality.
Well, lets pick a few numbers. First of all, well just pick a few easy numbers.
Turns out that x equals 3, that works because, of course, 3 times 3 minus 1, thats 8.
And absolute value of 8 is between 5 and 15, so this satisfies the inequality. So of course, the absolute value of 3 is 3 and 3 squared is 9, so
Quantity B is bigger. Another value that would work is 5.
And so, of course, this works because 3 times 5 minus 1, thats 14. Absolute value of 14 is between 5 and 15.
So this satisfies the restriction and, of course, when we plug this in, we get 5 and 25.
Again, Quantity B is bigger. We might also try a negative.
One negative value that works is negative 3. This satisfies the inequality because 3 times negative 3 is negative 9,
Absolute value of that is positive 10 which is between 5 and 15. So that satisfies the inequality.
Thats allowed. And we plug this in, again, we get 3 and 9.
So again, Quantity B is bigger. So, when we pick a few numbers for x, and plug these in to both quantities,
Well, we dont know for sure that the answer is B. Remember that B technically means that quantity
B is always bigger 100% of the time for every possible choice. So if we pick just a few numbers and
B happens to be bigger, does that mean that it always works?
We, we cant be sure about that. Maybe quantity B is bigger all the time.
Or maybe its bigger some of the time and we just got lucky with our guesses. Some of the time, B is bigger and some of the time, theres another relationship.
Maybe the quantities are equal or, or quantity A is bigger, something like that. So in other words, its possible that B is the answer.
Its also possible that D is the answer. But we definitively know that neither A nor C could be the answer.
It definitely cannot be true that quantity A is always bigger or that the quantities are always equal because we have examples where quantity B is bigger.
So thats really important to appreciate that even one choice eliminates two answers.
Now if we were running out of time on the Quant section, and we needed to guess, even a simple plug-in would limit the possible answers,
it would cut the possible answers from four down to two. And that enormously increases your odds of guessing.
So thats very, very important. Also, if it were a problem where you just looked at it and
you knew you had no idea how to solve it and you didnt want to waste time on it, again, you could pick one number, find that one relationship.
It would limit your choices. And then you could guess and move on.
So thats one of the big advantages of plugging in. Its very useful when youre doing guessing.
It helps you guess intelligently. Lets assume we are not in a rush, and
have the regular amount of time to spend on the problem.
We really, actually wanna solve it, not just guess. Suppose every value of x we plug in, quantity B is bigger than quantity A.
How many such values of x would we have to try before we could be sure? Well, unfortunately, infinity.
And of course, I dont need to tell you this, you dont have an infinite amount of time on the GRE.
So you cant plug in an infinite number of choices. This is a problem of the plug-in method.
See, picking numbers alone cannot be used to determine definitively that A or B or C is the answer.
Picking numbers can certainly eliminate some choices and may give us a sense of the relationship between the quantities.
Sometimes picking a few numbers will sort of trigger your number sense and help you see the logic, and so in that sense, it can be helpful.
But ultimately, well have to use some sort of logic or mathematical reasoning to know for certain what the answer is.
Lets return to that problem and think about it. So now instead of purely picking numbers, Im gonna think about some logic.
Heres the problem again. And lets think about this.
Lets think about that absolute value restriction at the top, what the numbers could be.
Every value of x that satisfies the given inequality, has an absolute value greater than 1.
First of all, think about positive numbers. If I plug in positive 1, I get 3 minus 1 is 2.
So, I, Id need a bigger positive value. So, it could be a positive value bigger than 1.
Or if I plug in negative value, so if I plugged in negative 1, Id get negative 3 minus 1, negative 4, absolute value positive 4.
That would still not work, so I would need a negative value that is less than negative 1.
That is to say, a negative number with an absolute value greater than 1, so either way the positive numbers or
the negative numbers have absolute values greater than 1.
Well when a number greater than 1, any number on the number line greater than 1 is squared, it becomes bigger, so, of course, quantity B would be bigger.
Remember if we square 1, we get 1, if we square a fraction between 1 and 0, they get smaller.
But numbers bigger than 1, even a decimal like 1.01, when we square it, it gets bigger.
Similarly, when a negative number less than negative 1 is squared, the square
is a positive number greater than the absolute value of the original negative. So this kind of makes sense, that if I square say, negative 1.5,
Ill get a positive number but that positive number, 1.5 squared, is still going to be bigger than 1.5.
In other words, the square is still gonna be bigger than the absolute value of the number.
For all these numbers, because the absolute value is greater than 1, the square is always greater.
And so, using logic, we can determine that B is the answer. To answer that, we had to analyze it with a little bit of logic.
Picking numbers alone was not enough to definitively determine an answer. Picking numbers game is a start, but
we had to finish the job by thinking about the problem, doing mathematical thinking. Keep in mind, its important to consider different categories of numbers
when were picking numbers. Positive, negative and zero.
Integers, fractions and decimals. The entire family of numbers.
Dont get stuck just picking numbers that you can count on your fingers. Dont just get stuck using the positive integers only and
forgetting about all the other numbers. Thats very important.
Keep in mind, also, theres an art to picking numbers well. Its not just about making haphazard choices.
Number sense can guide you in making good choices when you pick numbers. And this is very important.
Again, it should not be haphazard. You should be thinking about the logic of the problem itself and
what numbers would be the most significant to plug in.
Pause the video and then well talk about this.
Okay, so we have no restriction on x, so x could be anything. It, it might be an integer, it might be a fraction or a decimal,
Certainly, if x equals 0, quantity A would be one-fifth minus one-tenth. We dont need to calculate that.
Thats something positive cuz its bigger minus smaller. And so thats positive, so it would be bigger than 0.
So there, Quantity A is bigger. If we had a large positive value, say 1,000 or
Either way, adding that little one-fifth, thats not going to make much of a difference.
Were still going to have an absolute value of something thats in the thousands, and
then subtracting a tiny fraction is not going to make a difference. So, quantity A would be a big positive number and
And, in fact. Were not gonna demonstrate this, but Ill just point out, for
any integer we plug in, quantity A is always bigger. I happened to design the problem so that for every integer value,
quantity A is bigger. So if you just plug in integers, youll always get that A is bigger.
Hm. The question is then, when does the expression in
quantity A have its minimum, its lowest value? Well, lets think about this.
We have an absolute value. What is the lowest value that an absolute value can have?
The lowest possible output of an absolute value is zero, when the input is zero. So what would make the input of that absolute value, zero?
Well, its x plus one-fifth. So if we picked x equals negative one-fifth,
then it would make the input of that absolute value zero.
So what happens if we plug in x equals negative one-fifth? Notice that Im not just picking this number randomly.
I thought about the logic of the situation and that guided me to a very particular choice of number to plug in.
When x equals negative one-fifth, the absolute value part becomes zero and so
then, all of quantity A just becomes negative one-tenth, which is less than zero.
So thats a choice that makes Quantity B bigger. Many values of x make Quantity A greater than zero, but at least one makes it less.
And so we can change the relationship with different values and that means the answer is D.
Picking integers in that problem made it look as if the relationship went one way, but picking other kinds of numbers.
In this case, a negative fraction, made the relationship go the other way. Very important to consider different cases of numbers.
Also notice that a certain amount of thought about the unique mathematical situation.
In this case, the nature of the absolute value guided our choice of numbers. That is the ideal for picking numbers.
Once again, I cannot stress this enough. When you pick numbers,
its not just about a haphazard randomly picking whatever numbers come to mind. The ideal is to think about the nature of the mathematical situation and
what numbers would be critical in that situation. What numbers really might have an,
an influence in changing the direction of the relationship.
Pause the video and then we will talk about this.
So this one is a little harder cuz we have three numbers to pick, but we can still do this.
So we have to compare the mean and the median. So this is a statistics question.
So first of all, just notice that Quantity B, the median of this set always equals that number B.
Because the two middle numbers of that set, B and B, they would average just a B. So that would always be the median.
So lets think about this. Lets pick just 1, 2, and 3.
8 divided by 4 is 2, so the mean is 2. The median is also 2.
For, so for this particular choice, the mean and the median are equal. And in general, mean and median are equal,
you may remember when we have a nice symmetrical distribution. Well, when the distribution becomes asymmetrical, when we have an outlier,
remember that the median is not affected by the outlier, but means are pulled in the direction of the outlier.
So lets have this exact same set, but well just change one of the numbers to an outlier.
So, Im just gonna pick 1, 2, 2, and 103. Make that number ab, really absurdly large.
Now of course, the median is still 2. The median is not affected by the outlier.
The middle of that set is still 2. What about the mean?
Well now, the sum is 108. And, we dont actually have to perform the division.
Clearly, whatever 108 divided by 4 is, thats gonna be something much larger than 2.
So, the mean now is much larger than 2. The median is still 2.
So, weve changed the relationship. And, because weve changed the relationship,
that automatically means that D is the answer.
In summary, on GRE, quantitative comparison questions with variables, the strategy of picking numbers has strengths and weaknesses.
If picking numbers yields different relationship between the quantities, we definitively know that the answer is D, and once again,
when the answer happens to be D. Picking numbers is often the most efficient way to get there.
If you suspect the answer is D, picking numbers is an excellent route to take. If the answer is A, B or C, we cannot use picking numbers alone to ascertain that.
Picking numbers leads us to a definitive answer only 25% of the time. The rest of the time, we need to use logic and mathematical thinking.
I will say that picking numbers is very good, once again, for eliminating answer choices, if you have to guess.
Also, picking a couple numbers might guide you in thinking about the logic of the situation.
It might give you insight, might jumpstart, youre, youre logical analysis of the problem, so in that sense, it might be helpful.
When picking numbers, remember to use different categories of numbers. Dont just pick positive integers.
Remember negative numbers. Remember fractions and decimals.
And, most importantly, remember that its not just about picking numbers haphazardly.
You wanna think about the logic in the mathematical situation and use that to guide your choice of numbers.