استراتژی QC - شمارش اعداد
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Picking numbers. Now we can talk about picking numbers for the quantitative comparisons.
First I’ll 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, V-I-C for a variable in the answer choices.
So I’ll assume here that you’ve seen those lessons. If you haven’t 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 we’re gonna talk about quantitative comparisons and picking numbers.
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. It’s very important to understand the strengths and limitations of this strategy.
So here’s a practice problem. Pause the video and then we’ll talk about this. Okay. Here they’re very good to us. They tell us that n is an integer greater than 2.
So we know it’s an integer. It can’t be negative. It can’t be a fraction or a decimal, very good. So the most obvious choice if we’re 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, that’s 8. And of course, in Quantity B we get 3 squared, that’s 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. That’s 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, that’s 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 let’s pause that question for a minute. We’ll come back to that question, but here’s another practice problem.
Work on this. Pause the video, work on this, and then we’ll talk about this. Okay. So here, x is just a number. It could be positive, it could be negative, it could be whole, it could be a fraction or a decimal.
So, the gamut is open. It just has to satisfy that absolute value inequality. Well, let’s pick a few numbers. First of all, we’ll just pick a few easy numbers. Turns out that x equals 3, that works because, of course, 3 times 3 minus 1, that’s 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, that’s 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, minus 1 is negative 10.
Absolute value of that is positive 10 which is between 5 and 15. So that satisfies the inequality. That’s 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, quantity B is bigger.
Exactly what can we conclude from this? Hm. Well, we don’t 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 can’t be sure about that. Maybe quantity B is bigger all the time. Or maybe it’s 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, there’s another relationship. Maybe the quantities are equal or, or quantity A is bigger, something like that. So in other words, it’s possible that B is the answer.
It’s 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 that’s 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 that’s 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 didn’t 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 that’s one of the big advantages of plugging in. It’s very useful when you’re doing guessing. It helps you guess intelligently. Let’s 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 don’t need to tell you this, you don’t have an infinite amount of time on the GRE.
So you can’t 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, we’ll have to use some sort of logic or mathematical reasoning to know for certain what the answer is.
Let’s return to that problem and think about it. So now instead of purely picking numbers, I’m gonna think about some logic.
Here’s the problem again. And let’s think about this. Let’s 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. All right. So, that’s too small. So, I, I’d 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, I’d 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, I’ll 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, it’s important to consider different categories of numbers when we’re picking numbers. Positive, negative and zero.
Integers, fractions and decimals. The entire family of numbers. Don’t get stuck just picking numbers that you can count on your fingers. Don’t just get stuck using the positive integers only and forgetting about all the other numbers. That’s very important.
Keep in mind, also, there’s an art to picking numbers well. It’s 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.
So here’s a problem. Pause the video and then we’ll 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, we have to keep that in mind.
Certainly, if x equals 0, quantity A would be one-fifth minus one-tenth. We don’t need to calculate that. That’s something positive cuz it’s bigger minus smaller. And so that’s 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 a large negative value, say negative 1,000.
Either way, adding that little one-fifth, that’s not going to make much of a difference. We’re still going to have an absolute value of something that’s 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 it would still be greater than zero.
And, in fact. We’re not gonna demonstrate this, but I’ll 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, you’ll always get that A is bigger.
So does that mean that A is the answer? Hm. The question is then, when does the expression in quantity A have its minimum, its lowest value? Well, let’s 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, it’s 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 I’m 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 that’s 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, it’s 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.
Here is another problem. 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 let’s think about this. Let’s pick just 1, 2, and 3. So our set is 1, 2, 2, 3. The sum is 8. 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 let’s have this exact same set, but we’ll just change one of the numbers to an outlier. So, I’m 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 don’t actually have to perform the division. Clearly, whatever 108 divided by 4 is, that’s gonna be something much larger than 2.
So, the mean now is much larger than 2. The median is still 2. So, we’ve changed the relationship. And, because we’ve 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, you’re, you’re logical analysis of the problem, so in that sense, it might be helpful.
When picking numbers, remember to use different categories of numbers. Don’t just pick positive integers. Remember negative numbers. Remember fractions and decimals.
And, most importantly, remember that it’s not just about picking numbers haphazardly. You want to think about the logic in the mathematical situation and use that to guide your choice of numbers.
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