GMAT: Data sufficiency 29

We're on problem 119. Is 2x minus 3y less than x squared? 2x minus 3y less than x squared they ask us. All right, let's see if we can figure it out. Statement 1, 2x minus 3y is equal to minus 2. So they're telling us that the left-hand side of this original question is equal to minus 2. So now this question, if we take statement 1 into consideration, it boils down to is minus 2 less than x squared? So first you might be saying, oh, I don't know. I don't know what x is. But think about it. Can x be a negative number? At worst case, x is 0. That's the worst case. If x was anything else, x squared is going to be a positive number. So we actually know, just from statement 1, that negative 2 has to be less than x squared, unless x was some type of weird, imaginary number or something. I think that doesn't occur on the GMAT. So statement 1 alone is sufficient. x squared is going to be at least 0 or some positive number larger than 0. So statement 1 is sufficient. Statement two, x is greater than 2 and y is greater than 0. So let me just try to come up with two contradictory cases using these conditions to see. Let's say if x was-- and they don't tell us that anything is an integer here. So if x was 2.1, x is equal to 2.1 and y is equal to-- I don't know-- y is greater than 0. So let's say that y is equal to 0.1. Let's see what happens here. So 2 times 2.1. You get 4.2 minus 3 times y. So minus 0.3 is less than x squared. It's what? It's like 4 point something. So it's greater than 2. This is interesting. Actually this is interesting. So actually, you don't have to try examples out. Let's just write the equation down. 2x minus 3y is less than x squared. So if y is greater than 0, which they tell us in statement 2, then the quantity on the left-hand side is always going to be 2x. It's always going to be less than 2x, right? Because this'll have some positive value. It might have an insignificantly small value. We could make it 0.000001. But it's still going to have some positive value. So no matter what y we choose, as long as it's greater than 0, 2x minus 3y will be greater than 2x. So we could say that if 2x is less than x squared for any x we pick, than 2x minus 3y is definitely going to be less than x squared. Why is that? Because minus 3y is going to subtract from the 2x, because we know that y is greater than 0. So let me ask you a question. Is 2x always going to be less than x squared if x is greater than 2? Sure. I mean whatever x is, on the left-hand side it's going to be 2 times-- pick a random number-- 2 times x. It could be 2.5. But on the other side, on the right-hand side, you're going to have 2.5 squared. So if you think about it, you're going to have 2 times a number. That's always going to be less than that number, which happens to be larger than 2 squared. The smallest possible is 2 times 2.00001. You could make a lot of 0's there. But I think you get the point. That's still going to be less than 2.0001 squared. Because you only have a 2 here instead of a 2.0001. So actually, statement 2 alone is sufficient. Hopefully that made sense. You just have to see oh, if y is greater than 0, then the 3y is definitely going to take away from the 2x. Then as long as x is greater than 2, 2x is always going to be less than x squared. If that makes sense to you, then you should realize that statement 2 alone is sufficient for this. That was interesting. Oh, no sorry. Either of them alone were sufficient. We did that in the first one, right? Statement 1 or statement 2, independently are sufficient to answer that question. 120. A report consisting of 2,600 words is divided into 23 paragraphs. So 2,600 words and it has 23 paragraphs. A two-paragraph preface is then added to the report. Is the average number of words per paragraph for all 25 paragraphs less than 120? So this is interesting. So the average number of words per paragraph for all 25 paragraphs less than 120. So if we knew the total number of words in the document we'd be set. We would just take the total words divided by the total paragraphs they have now. They have the 23 original plus the 2 from the preface, you have 25 total paragraphs. Now it told you the average words per paragrah. Now, we don't know the total words, because the total is going to be equal to the 2,600 words in the body of the document plus the words from the preface-- I'll call that words sub preface-- all of that divided by 25. This is the average. This is what we really have to figure out, to figure out if it's less than 120. That's what they want to know. So statement number 1. Each paragraph of the preface has more than 100 words. So I guess we could say that the preface, the words from the preface are greater than 200. That's another way of saying that the total is going to be greater than 2,800. Now if the total is greater than 2,800, what does that tell us about the total divided by-- excuse me. So then the total divided by 25 is going to be greater than 2,800 divided by 25. What's 2,800 divided by 25? Let's see, 25 goes into 2,800. 1, 25, 30, 1, 1 times 25, 25, 50, 12. So it tells us that the average is going to be greater than 112. So that doesn't help us. If it told us that the average was going to be greater than 120, we would be done. We would say oh, we have enough information to answer the question. Actually, the answer is no, that the average is going to be greater than 120. But here, this tells us the average is greater than 112. It could be 113 or it could be 121. So it doesn't answer our question of whether the average is less than 120. So statement one alone is not that useful. Statement number 2 tells us each paragraph of the preface has fewer than 100 words. So the total preface has less than 300 words. They say each paragraph of the preface, and there's 2 of them, has fewer than 150 words. So the total preface has less than 300 words. So we could do the same logic. So that means that the total is going to be less than what? The total is going to be less than the original number of words plus the preface. So it's going to be less than 2,600 plus 300. It's going to be less than 2,900. Then if you divide both sides by 25 to get the average, 25 goes into 2,900 really four more times. That's right. The average is going to be less than-- let me make sure I'm getting that right. So 4, 25 goes into 2,900. 1, 25, 40, 1, 25. Then we have 150. 25 goes into 150 six times. So we have the average is less than 1/16. So statement 2 alone is sufficient. Statement 2 tells us that the average of the words per paragraph is less than 116, which is definitely less than 120. So statement 2 alone is sufficient, and statement 1 doesn't help us much. I've run out of time. See you in the next video.