GMAT: Data sufficiency 5

We are on problem number 22. And their question is straightforward enough. What is the value of the integer x? And they tells us it's an integer. x is equal to what? So the first statement, x is prime. Well, there are a lot of prime integers. There's actually an infinite number of them. So that alone doesn't tell us what x is. Two. Let's see, 31 is less than or equal to x, which is less than or equal to 37. Well, immediately, this is useless information because 31 and 37 are both prime numbers as far as I can tell, right? They're both prime numbers. So this means that x could be 31. It could be 37. Or let's see, are there any prime numbers in between? 32, 33, 34, 35. Well, there are no prime numbers in between, but it could be either 31 or 37, so it doesn't help us much even used in combination. I mean, this alone is definitely useless, because this is a range of six numbers. And even if we know x is prime, if we use this and this, all we know is that x is either 31 or 37, so the answer is E. Together, they're not sufficient, E Now, if this was a little bit different, if this was x is greater than 31 and less than or equal to 37, then we would know, oh, x has to be 37. Or if this wasn't here and this was there, then we would know that x is 31. But anyway, they didn't do that. They gave us the option, so and the answer is E. Problem 23. If P, Q and R are three distinct points, do line segments PQ and PR have the same length? So P, Q and R. They want to know if these two lines are the same length. This is line PQ and this is the line PR. So are those the same length? Let's see if we can figure this out. Statement one tells us P is the midpoint of line segment QR. OK, P, midpoint, so I've already drawn it wrong. Midpoint of QR. So let me redraw it, so all three points actually end up being on the same line, because P is the midpoint of QR. So this is the line. This is point Q. This is point R. So if P is the midpoint, P is right there, and that tells us that this distance is the same as this distance, which also tells us that QP, line segment QP is equal to PR. And that's what I think what they wanted us-- yeah, right. They want to say does PQ and PR have same length? This is PQ So PQ and PR definitely have the same length. So statement one is definitely true, because P is the midpoint. Let's see about statement two. I mean, statement one implies that they have to be on the same line; otherwise, P couldn't have been the midpoint. Let's see if statement two is interesting. Statement two: Q and R lie in the same circle with center P. Interesting. So this is saying a circle with center P, and Q and R both lie on this circle. So Q could be here. I'm just picking two random points. R could be here. So we want to know whether PQ is equal to PR, right? That was the question. Well, the definition of a circle is all points that are equal distant from P. So R has to be the same distance from P. It's going to be one radius of the circle away from P as Q is because they're both on the circle, and they're all equal distance from p. So PR is going to be equal to PQ. They're going to be equal to each other. So once again, two alone is sufficient. So the answer is D, each statement alone is sufficient. Problem 24. The problems are getting a little bit more interesting. Is the number x between 0.2 and 0.7? So if I just write it 0.2 is less than x, which less than 0.7, that's what they want to know. Statement one tells us that 560x is less than 280. You know you don't even have to solve this. This is a fairly simple equation. You can easily solve for x, and if you can solve for x, you can answer this question. So you actually don't have to go to the trouble. If you want to, you can then just divide both sides by 560, but that takes time, valuable time. You're taking the GMAT. And that is equal to 1/2, right? 28/56 is 1/2. And then you'd say, sure, x is between the two numbers. But you didn't have to do this. That's a waste of time. All you have to do say, oh, I can solve for x, so I can test that, whether that's true or not. Statement two says 700x is equal to 280. Same thing. You just look at it. You're like, oh, I could solve for x. If I can solve for x, I can tell you whether x is between those two numbers, and that's the end of it. And you say, oh, each of these independently are sufficient to answer this question. But if you wanted to prove it to yourself and see if this question is true or false, you can just say, well, x is equal to 228/700. That is equal to 2.28/7. That is equal to 0.4. I just divide the top and the bottom by 7, and you get 0.4. So once again, it is between it. You answered the question. But you remember, the question doesn't have to be affirmative. You just have to figure out if the data is enough to answer the question, and we didn't have to go through this and this. You could have just looked at it and said, oh, I can solve for x in either way and then just test it. Next, problem 25. If i and j are integers, is i plus j even? So statement number one is that i is less than 10. That tells me nothing. That to me seems kind of useless. Statement number two is i is equal to j. Well, that's interesting. So what's i plus j then going to be equal to? Well, that's the same thing as-- since i is equal to j, that's the same thing as j plus j, which equals 2j. Or you could say, if you substitute j for i, or if you substitute i for j, that's the same thing as i plus i, which is equal to 2i. In either case, you immediately see that either of these numbers are a multiple of two and therefore have to be even, because i and j are integers. So statement two alone is sufficient, and statement one is useless. Problem 26. n plus k is equal to m. What is the value of k? So we could solve for it, if we could answer this, because k is equal to m minus n, and that would solve our answer. Let's see if the statements give us any help. n is equal to 10. Well, that'll just tells us that k is equal to m minus 10. It still doesn't tell us what k is, because we don't know what m is. Statement number two. m plus 10 is equal to n. This is interesting. So what does this tell us? So let's see, if we can figure out from this statement what m minus n is, then we're in business. Let's see, let's subtract n from both sides. You get m minus n plus 10 is equal to 0. Subtract 10 from both sides, you get m minus n is equal to minus 10. Well, we know that k is equal to m minus n. m minus n is equal to minus 10, so this is equal to k. So we solved for k, using just statement two alone. Just statement two alone is sufficient. And statement one is-- I mean, it kind of gets us halfway, but you really don't need it. Statement two is all you need, so that is b. Let's see if we have time for one more. Yeah, sure, why not. They have drawn a triangle for us. I'll draw it quick and dirty. This is angle x. This is angle z, and this is angle y. And they want to know is this triangle equilateral, which means are all the sides equal to each other? So statement one tells us that x is equal to y. So we break out a little bit of our eighth or ninth grade geometry here. So if x is equal to y, if these angles are equal to each other, what that tells us is that these sides are equal to each other, the corresponding sides are the same. I could actually redraw this triangle like this. Maybe it'll jog those neurons from ninth grade. This is x. This is y, and this is z. And you could test it out, if you don't believe me. And we've gone over this in the geometry module. If x and y are equal to each other, then these sides have to be equal to each other. And you could just try to imagine drawing a triangle that has two base angles equal to each other and the sides are different. It would be impossible. The sides would have to be the same. This triangle would have to be at least isosceles and maybe equilateral. Isosceles is two sides being the same. But that by itself doesn't tell us that this is an equilateral triangle. In order for this to be equilateral, z has to be the same as x and y. They all have to be. And if they're all going to be the same, they all add up to 180, they all have to be 60 degrees. So you could have a situation where x and y are both 20 degrees. This is 20, this is 20, then they add up to 40. And then z would have to be 140 in which case, we would not have an equilateral triangle, so one alone is not sufficient. Let's see was statement two tells us. Statement two tells us z is equal to 60 degrees. So that by itself-- so first of all, we know if we know both of these pieces of information, we are definitely dealing with an equilateral triangle. Because think about it. If z is 60, and then we use this information that these two are equal to each other, what do we know? If z is 60, then these two added together have to-- so x plus y plus z are going to be equal to 180. If z is 60, then x plus y have to be equal to-- subtract z from both sides, or subtract 60 from both sides-- has to be equal to 120. And since these are equal to each other, x and y both would have to be equal to 60 degrees, right? So if you use both of this information, all of the angles are going to be 60 degrees and it's eqilateral. So we know that definitely in combination, we can figure it out. But what about just if z is equal to 60? Well, I can easily draw a triangle that disproves that. So if I have a 0-degree angle here-- in fact, it might be a 30-60-90 triangle like you learned in school, so it could be 60 degrees, 30 degrees, and 90 degrees. This could be z. And this definitely is not an equilateral triangle. And so if just by z being 60 degrees, in no way are you saying that this is definitely an equilateral triangle. You have to have the other condition that x is equal to y. And so the answer is, let's see, C, both statements together are sufficient, but neither statement alone is sufficient. The answer is C. See you in the next video.