GMAT: Data sufficiency 6

We're now on problem number 28. And the question is, is x an integer? Statement one tells us that x over 2 is an integer. So if you think about it, x has to be an integer, because x is divisible by 2. This tells us that x is divisible by 2, because when I divide by 2 it's not 5.5 or something. I get an even-- I get an integer. So x has to be an integer. And if you want to use an example, well if x is divisible by 2, then x could be 6, which is clearly-- well anyway, that's almost too obvious. I don't think I have to explain this too much. But this alone is enough to know that x is an integer, because it's actually divisible by 2. Statement number two tells us that 2x is an integer. This doesn't give me a lot of information. Because if x is equal to 1/2, then 2x is equal to 1. So that 2x is an integer, but x is not an integer. But if x is equal to one, then 2x is equal to 2. And so 2x being an integer, doesn't necessarily tell me whether or not x is an integer. So this is useless, and this is useful. So the answer is one, that only statement number one is necessary. A. Problem 29. I should write smaller so I could save space. What is the value of x? Simple enough. The first one says, 2x plus 1 is equal to 0. Like I said before, you could just look at this and say, this is eighth grade algebra. I can solve for x. This is all I need. And you could solve for it if you want. You could say 2x is equal to minus 1, x is equal to minus 1/2. But that would be a waste of time, because they're not asking you what x actually equals, they're just ask you whether you could solve for it. Two says, this is interesting. x plus 1 squared is equal to x squared. Now you might be tempted to just take the square root of both sides. But that actually becomes complicated, because you could have the positive or negative square roots. You'd have to set up a bunch of different equations. The easiest thing to do actually would be to expand this. So what's x plus 1 squared? What's x plus 1 times x plus 1? So that's x squared plus 2x plus 1 is equal to x squared. And if you subtract x squared from both sides of this equation, you get 2x plus 1 is equal to 0. So you actually end up getting this again. So both of these pieces of information are equivalent. And each of them independently is enough to solve for x. So the answer is D. Each statement alone is sufficient. Problem number 30. What is the value of 1 over k plus 1 over r? And what information do they give us? So they tell us that k plus r is equal to 20. It's not obvious to me how to figure out from this, what 1 over k plus 1 over r is. And just to experiment, if we were to find a common denominator and add these together, what is another way of writing this expression? Well the common denominator would be kr. And 1 over k is the same thing as r over kr, right? You can cancel out the r's and get 1 over k plus-- and 1 over r is the same thing as k over kr. So this statement is equivalent to this statement. So we're trying to figure out what r plus k over k times r is. Statement one just gives us the top. So that by itself isn't enough. What does statement two give us? kr is equal to 64. So statement two gives us this information, right? Statement one gives us this information, up here. So we actually need both of them to figure out what this is equal to. So the answer is C. Both statements together are sufficient, but individually they're not that useful. Problem 31. If x is equal to one of the numbers, 1/4, 3/8, or 2/5, what is the value of x? So x is one of these, and we have to figure out which of they are. So statement one tells us that 1/4 is less than x, which is less than 1/2. So if x is greater than 1/4, statement one immediately tells us that x is not 1/4. Let's see. And x has to be less than 1/2. Well both of these numbers are less than 1/2, right? Statement one. Let me do this in a color. Statement one tells us that our answer is one of these two, right? That's what statement one tells us. Because both of those-- that's 0.4, that's less than 1/2. 3/8 is what? Actually, no. This is interesting. 3/8 is actually greater-- no, 3/8 is obviously less than 1/2, right? What am I thinking. 4/8 is exactly 1/2. So 3/8 is also less than 1/2. So statement one tells us it's one of those two choices. And statement two says that 1/3 is less than x which is less than 3/5. So this tells us that x has to be greater than 1/3. Let's see. 3/8 is greater than 1/3, right? Because 3/9 is 1/3. So 3/8 is greater than 1/3. 2/5 is also greater than 1/3, right? It's 0.4. So that's greater than 1/3. Let's see. Is 3/8 less than 3/5? Well, sure, right? 3/8 is definitely less than 3/5. And 2/5 is definitely less than 3/5. So it seems that statements two and one are actually giving you the same information. And when you get both of them, they actually don't clarify. They don't give you any clarity, whether x is 3/8 or 2/5. So they just leave you hanging. And so the answer is E. Together they're still not sufficient. Unless I missed something. Once again, remember, I'm just doing this in real time. I don't know if-- it's very possible that I make an error. Problem 32. Triangle PQR. Let me draw that. If PQ is equal to x, so this distance is equal to x. And QR is equal to x plus 2. And PR is equal to y, which of the three angles of triangle PQR has the greatest degree measure? So they want to know which of these angles -- so angle 1, 2, 3. You need to be able to figure out which of these angles has the greatest degree measure. And it might be intuition to you. But in order to figure out which angle has the greatest degree measure, you essentially have to figure out which angle is opposite the largest side. That's one way to think about it. And that should get you which is the largest degree measure. So if we could figure out whether x, x plus 2, or y is larger, then we'd be all set. So first of all, we know that x plus 2 is greater than x. So our answer, as far as the longest side, is either going to be y or x plus 2. And the shortest side is either going to be y or x. But anyway. What do they say? They say 1y is equal to x plus 3. So if y is equal to x plus 3, this is the largest side of the triangle, right? This is the largest side of the triangle. This is the second largest. And then this will be-- if I'm thinking about this right, this will be the largest angle. Let me think about that. I mean, you could go back to a little bit of our trigonometry, but this would be the largest angle. And let's see. So I think one alone is enough. It tells us this is the longest side. The triangle would look something like this. This is the longest side. This is the second longest side. And then this would be the shortest side. It would look something like that. And then angle three would be the largest angle. Let's see. Statement number two tells us that x is equal to 2. x is equal to 2 gives us no information about y. If we say that x is equal to 2, then we know that x plus 2 is equal to 4. And we have no information about y. So that really still doesn't help us. Because it could easily be a triangle like this. 2, and then the 4 could come back this way and be like that. In which case this would be the largest angle. Or y could be a really big distance. So that 4 could be like that, if y was a really large number. Let's see. If you have the combination of the two, well-- we know that one alone is sufficient. Two alone is not sufficient. And I'm just doing-- let me think about it a little bit. Let me make sure that I'm right, that one alone is sufficient. Yeah I'm pretty sure. I mean, I can't think of an example where I have to know that x is equal to 2. Yeah. So unless I'm wrong, the answer is a. Statement one alone is sufficient. I'll see you in the next video.