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QC Questions & Algebra
In this lesson, we will discuss some strategies to keep in mind for quantitative comparison questions that involve algebra. First of all, many of the important algebra patterns discussed in the Algebra module remain important here. So we’ll assume that you’re familiar with all the videos in the Algebra module. Where, there’s not a full review of algebra here, obviously, because it’s all covered there.
We’re just talking about how these things play out in the QC questions. So, the three big patterns to know. The square of a sum. Remembering that r plus s quantity squared, that’s not just r squared plus s squared. We have to FOIL out r plus s times r plus s.
It equals this pattern. Very similar pattern, square of a difference. And then probably the most important, the difference of two squares, r squared minus s squared, and how to factor that. All three of these are very important factors to know well, and these are ones that they love to test in the quantitative comparisons.
For example, here’s a practice problem. Pause the problem and then we’ll talk about this. Okay, remember that we can multiply both quantities by the same thing as long as we know that it’s positive. Well, a, b, and c, these are all lengths.
So obviously, each length is a positive number. We add two lengths, a plus c. That has to be positive. So we’ll multiply by that denominator, a plus c. So on the left, we just get b squared. And on the right, we get the difference of two squares pattern, c minus a times c plus a.
And of course, that’s gonna equal c squared minus a squared. Well, then just to get everything positive, we’ll add a squared to both columns. And then we get a squared plus b squared on the left, and c squared on the right. Now remember, a squared, a and b, these are the legs of our right triangle. c is the hypotenuse.
And of course, Mr. Pythagoras guarantees that for any right triangle, a squared plus b squared would have to equal c squared. That is the Pythagorean theorem. So, these two columns are always equal as long as the angle’s a right angle. And of course, the diagram does guarantee that it’s a right angle. So always equal, the answer is c.
Here’s another problem. Pause the video and then we’ll talk about this. Now, you might be tempted to plug in numbers here. And plugging in numbers could actually lead to some problems.
Instead, what I’m gonna say is, if you have algebraic expressions, try and simplify it much in the same way as if you were solving an equation. It’s very, it’s easiest if you can get x in one, one of the two quantities. Just have that equal x by itself. And so you’re comparing x to a number. So the first thing I’m gonna do is I’m gonna subtract x from both quantities.
And of course, whether x is positive or negative, I’m allowed to add or subtract it. That’s perfectly fine. So I subtract it. Then what I’m gonna do is subtract 2 from both sides. Then I’m gonna divide by 6.
So that gives me x in Quantity A and 13 over 6 in Quantity B. Those are the two things we’re comparing. So if x is greater than 2, which of these two quantities is bigger? Well, notice a few things. Notice that, for example, x could equal 13 over 6 because 13 over 6 itself is a number slightly bigger than 2.
Again, there’s no guarantee that x is an integer. So don’t add that restriction here. x could be any number, so x could be 13 over 6. And in that case, the two columns would be equal. If we picked 3, for example, and plug in, then Quantity A is 23, Quantity B is 18, so A is bigger.
So, two different numbers give us two different answers. In fact, if we picked a number that was less than 13 over 6 but larger than 2, say 2 and one-hundredths, something like that, then it would actually turn out that quantity B is bigger. So, we can get any kind of relationship that we want here, greater than, less than, or equal, so obviously there’s not a constant relationship.
The answer must be D. Here’s another practice problem. Pause the video and then we’ll talk about this. Okay, here also, it would be kind of a mistake to plug in numbers.
In other words, if you plug in numbers, you’ll always find that Quantity A is bigger, but plugging in a bunch of different numbers is not gonna help you too much. Instead, notice that that expression in Quantity A is very close to one of our algebraic patterns, the sum, the square of a sum. In particular, if we had x squared plus 16x plus 64, that would be a perfect square.
So let’s just rearrange it a bit, okay? So we’ll separate that 67 into 64 plus 3. And then that x squared plus 16x plus 64, that is the sum, that is the square of a sum. So that is x plus 8 squared. This is why it’s very important to recognize those patterns.
That’s a perfect square. So in other words, what we have here is x plus 8 squared plus 3. Now, putting it in that form makes some things very clear. When something is squared, it’s always positive or it could be zero. If x equaled negative 8, that would equal zero. When we add 3, it’s always greater than 0 because even when x equals negative 8, we’d get 0 squared, which is 0.
But then plus 3, that would still be 3. That would be greater than zero. So it’s either 3 plus 0 or 3 plus a positive number. That’s always greater than 0, so answer A is in fact always correct. Here’s a practice problem.
Pause the video and then we’ll talk about this. Okay, this is one that involves function notation. So don’t be intimidated. Sometimes they will throw function notation at you. They define the function notation.
And c is a number such that f of c is 0. So let’s think about this. f of c equals 0. So in other words, what that would mean is c squared minus 25 equals 0. Well, if we rearrange a little bit, that would mean that c squared equals 25, and we have to be very careful here.
What does c equal? Yes, c could be 5, but we have to keep in mind when we have, take the square root of a variable, we have to include the plus or minus sign. So c could be 5 or negative 5. Those are the two numbers which, when squared, equal positive 25. So if c is plus 5 or minus 5, well then, we have no way to determine whether Quantity A is bigger or smaller because if it’s 5, it’s bigger than 3.
If it’s negative 5, it’s smaller than 3. So, it could go either way. Notice that if you forgot to take, forgot to include the plus or minus sign, you would have gotten the wrong answer to this. This question is, in disguise is actually testing whether you remember that plus or minus sign.
Because there is a plus or minus sign, we can’t decide on the relationship, and so the answer is D. Here’s another practice problem. Pause the video and then we’ll talk about this. Sometimes, when you’re given one of these very complicated equations before the comparison, it helps to work a little bit with that equation and simplify it.
So the first thing I’m gonna do is I’m just gonna subtract w to get the fraction all by itself. And then notice that I have an m plus w in the denominator, and an m minus w on the other side, so that’s very suggestive. I notice if I multiplied those, I’d have the difference of two squared patterns. So multiply by that denominator.
And so, I just get k squared plus 1 on that side, and then I get the difference of two square patterns, m minus w times m plus w. And of course, that’s going to be m squared minus w squared. Then just to get the, the k and the w on the same side, I’m gonna add w squared to both sides. So now I have k squared plus w squared plus 1 equals m squared.
So this looks an awful lot like the, the two quantities we have here. And I’ll just point out, with any number on the number line, doesn’t matter whether it’s positive or negative, if we add 1, we make it bigger. So for any, say, call it a number, p plus 1 is always bigger than p. And it doesn’t matter whether p is positive or negative, doesn’t matter whether it’s an integer or fraction, that’s true for every single number on the number line.
So no matter what k squared plus w squared is, in fact we know it has to be a positive number of some kind, if we add 1, that’s something bigger. And when we add 1, that equals m, so this definitively means that m squared is 1 bigger than k squared plus w squared. And so, that Quantity B is always larger. So be familiar with these important patterns, the Square of a Sum, the Square of a Difference, and the Difference of Two Squares.
The folks who write the test really love these patterns. It may help to treat the two quantities as the two sides of an equation, and to isolate x. That’s often a good trick, especially if you have relatively simple linear expressions in both quantities. Remember the plus or minus sign when you take the square root of variable squared.
And always remember the basic rules of inequalities. As we’ve said above, understanding inequalities is essential for understanding the deep logic of GRE quantitative comparisons.
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