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QC Questions & Inequalities
Now, we can start talking about the advanced quantitative comparison strategies. So, the first three lessons introducing the quantitative comparison question format appear at the end of the General Math Strategy lessons, way at the beginning of the math content module. Here, I’ll assume that you’re familiar with those basics already.
In other words, you have the basic idea of what a QC question is and what’s being asked in this question format. I’ll also assume that you’re familiar with the content of the math lessons, all because the course is coming at the end of all the content, I assume that you watch most, or all of those lessons already, and you’re familiar with that content.
So, now we can start talking about more advanced strategies. You may remember the strategy I called matching operations in that section. The last video of that section. Here, we can discuss that idea of matching operations a little more formally. Essentially, quantitative comparison questions consist of two quantities and an unknown relation between them.
So in that box is an unknown relation. And this relation could be a greater than relation, in which case the answer’s A. It could be a less than relation, in which case the answer’s B. It could be an equals relation, in which case the answer is C. Or it could be some combination, sometimes one sometimes another, in which case the answer is D.
Our job on the QC question is to figure out this unknown relation between the two quantities. So it’s a little different from other math types. In other math types, we might be solving for a variable, here we’re solving for a relationship. Recall from the lessons in the algebra module that the relationships of inequalities are somewhat more restricted than equations.
In other words, we’re allowed to do things to both sides of an equation that we’re not always allowed to do to both sides of an ine, inequality. So we have to review. We’ve already talked about inequalities back in the algebra lesson, so this is just a review, what are we allowed to do on both sides of an inequality? We can always add any number to both sides of an inequality.
It doesn’t matter whether it’s a positive or a negative, we can add anything to both sides of an inequality. We can always subtract anything from both side of an inequality. Again, it doesn’t matter whether it positive or negative, we can add or subtract anything. Addition and subtraction always preserve the order of inequality, which is precisely why can add or subtract any number in both columns of a quantitative comparison.
We can multiply or divide both sides of an inequality by the same positive number. Multiplication or division by a positive factor still preserves the order of inequalities. This is why we can multiply or divide by positive numbers in a quantitative comparison. Recall that multiplication or division by a negative number reverses the order of inequality.
So for example, it’s a true statement that 7 is greater than 3. If I have $7, I have more money than if I have $3. But if I multiply both by a negative, well now the relationship switch, switches up. I am better off if I am $3 in debt than I am if I’m in $7 in debt. $7 in debt, I have quote, unquote less money than if I were merely $3 in debt. So negative 7 is less than negative 3.
So when we multiply by a negative, the order of the inequality is reversed. That’s why we can’t multiply or divide each quantity of a QC by a negative number. It could change the unknown relationship. Similarly, we can never multiply or divide each quantity by a variable if we do not know the sign of the number represented by the variable because a variable can hold any number.
It could hold a positive. It could hold a negative. So that’s the danger of multiplying or dividing by a variable. Now if we’re guarantee for some reason, if the, the nature of the problem guarantees us that the variable is positive, then we could multiply or divide by it. So what we’re allowed to do, we can add or subtract any number.
We can multiply or divide by a positive number. Those are the operations that are always legal, 100% of the time in QC questions, as long as you do the same thing to both quantities. Keep in mind that this includes cross-multiplying, as long as all the fractions are positive. Cross-multiplying is a 100% legitimate move if you’re comparing two fractions in a quantitative comparison.
So here’s a practice problem. This should look familiar, because this problem has already appeared in the opening lessons on quantitative comparisons. But pause the video and think about this. So here we’ll show a different way of attacking this problem than we showed in the previous lessons.
Here, I’m simply gonna cross multiply. So we saw this already. We’re gonna cross multiply here. And we get these multiplications. So we don’t need a calculator for this. 35 times 3, let’s think about this, 35 plus 35 is 70.
Plus 35 again is 105. What is 13 times 8? Well, 13 times 10 is 80, 13 times 3 is 24, 80 plus 24 is 104. So we get a 105 and 104, quantity A is bigger. So this is the same answer we got last time. But, a very different approach this time.
Cross-multiplying is 100% legal. Again, we just have to make sure the fractions are positive, but then cross-multiplying is always a valid thing to do when we’re comparing two fractions. If we know that both quantities are positive and in many cases we do know that, then there are a few more moves that are perfectly legal.
For example, with positive numbers, both squaring and taking the square root preserve the order of inequalities. So if we know that a and b are both positive and b is bigger than a, then it’s absolutely true that if we square, they keep the same order of the inequality or if we take a square root, we keep the, the same order of the inequality. So again, if you know that both quantities are positive, these are also legitimate moves to do on both quantities.
The operations of cubing and taking a cube root always preserve the order of inequality, regardless of the positive or negative sign of the numbers, but these are relatively unlikely to come into play on the GRE quantitative comparisons. It would be a real long shot that you’d see anything where you’d have to cube or take a cube root on the quantitative comparisons. Also, remember that any positive number is greater than any negative number.
If we determine that one quantity is positive and the other is negative then we do not need to be concerned with their actual values. If the signs of the are quantities are different, automatically, we are done. We can do the comparison. Here’s a practice problem.
Pause the video and then we’ll talk about this. Okay, so we have some square roots, so obviously it’s gonna make sense to square. We know both numbers are positive. So we can square it, and it will not change the relationship. Whatever relationship we have there, it will stay the same.
So I’m just gonna square both quantities, and of course, squaring a number with a radical. I square the 4 and separately, I square the, the square root of 3. I square the 5 separately, I square the square root of 2. And of course, 4 squared is 16, square root of 3 squared is 3, 5 squared is 25. Square root of 2 squared is 2, 16 times 3 is 48, 25 times 2 is 50.
Quantity of B is bigger. Here’s another practice problem. Pause the video and then we’ll talk about this. So this one would be a relatively nasty one to do as is.
We’d have to find a common denominator of 3 and 14, and on the other side, we’d have to find a common denominator of 6 and 7. That’s a little bit nasty, but notice the following. Notice that 3 and 6, 3 6 is a multiple of 3. And notice 14 and 7, 14 is a multiple of 7. Would it be convenient to gather the terms so that we had multiples of each other grouped with each other.
So I’m just gonna add the, I’m gonna add one-fourteenth to both sides. You got the one-fourteenth over on that side and then I’m gonna subtract one-sixth, five-sixth from both sides. That way, I get the 3 and the 6 denominator in the same place. I get the 7 and the 14 denominator in the same place. It makes things much easier.
Now, finding common denominators are incredibly easier. I can just multiply 4 over 3 times 2 over 2, I can multiply three-sevenths by 2 over 2, and then we get very easy subtraction. On the left, in quantity A, eight-sixth minus five-sixth is three-sixth which is one-half. In quantity B, six-fourteenths plus one-fourteenth is seven-fourteenths.
That’s also one-half. Magic, magic. These two quantities turn out to be equal. And so, the answer is C. Here’s another practice problem.
Pause the video and then we’ll talk about this. Okay, on this problem, the folks who reach for the calculator are really making a big mistake, and I know it’s awfully tempting. You see scary fractions like this, and you say, okay, let me, let me just do this on the calculator.
That’s a big mistake. That’s a huge time sink to figure out everything with a calculator. Whenever a problem looks like it will involve a long, ugly calculation, think again. The GRE is not interested in having us do long, ugly calculations. Look at the two fractions in Quantity A.
And in fact, notice this. Look at that fraction 10 over 23. Half of 23 is 11.5, so 10 over 23 is less than one-half. Half of 37 is 18 over 5, 18.5, so 19 over 37 is greater than one-half.
So what I have in quantity A is something less than one-half minus something greater than one-half. Well less than one-half minus greater than one-half, that has to be something negative. Quantity A has to be negative. Now quantity B would be particularly ugly to calculate but it’s positive plus positive.
Quantity B has to be positive. Any positive is greater than any negative. And so the answer has to be B. So notice that I completely avoided doing a long complicated calculation. All we have to do is the comparison. In general, if you question whether something is legal to do to both quantities in a quantitative comparison question, it comes down to whether that action would preserve the order of an inequality.
So having a good understanding of inequalities is essential to having a good understanding of the quantitative comparison questions. Thinking about the, the GRE quantitative comparison questions is an unknown relation problem. In other words, you’re not solving for a variable. You’re solving for a relationship.
That’s really the way a mathematician would think about these problems. And the more you can appreciate the way a mathematician looks at mathematics, the deeper your understanding will be. In a quantitative comparison question, we can always add any number to both quantities or subtract any number from both quantities. We can always multiply or divide both quantities by any positive number.
If we are guaranteed that both quantities are positive, then we can take a square root of both of them or square both of them. And as always, any positive number is greater than any negative number.
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