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GMAT: Data sufficiency 35

138-140, pg. 289. Created by Sal Khan.

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Video transcript

We're on problem 138. In a certain business production index, p is directly proportional to efficiency index e. So we could say that p is equal to some proportionality constant times e. Because they say p is directly proportional to e. Fair enough. Which is, in turn, directly proportional to investment index i. So e is proportional to i. So we could say it's equal to some proportionality constant times i. And actually, we could go even a step further and say, well, if p is proportional to e and e is proportional to i, we could also say that p is proportional to i. Directly proportional to i. So it might be some other constant times i. And you could prove that mathematically, right? You take this and substitute it for e and you get p is equal to k times e, e is equal to some constant times i, and so p is equal to kn times i. But k and n were both arbitrary constants, so we could just call that m. So this is the information that the problem gives us. p is proportional to e, e is proportional to i, so p is also proportional to i. So what are they asking us? What is p if i is equal to 70? So if we knew this proportional constent, we'd be all set. We could actually solve this problem. Maybe they're going to go through e or something, I don't know. So statement number one. e is equal to 0.5 when e is equal to 60. Well, we can definitely use this information to figure out what n is. But that by itself isn't going to help us, because we can figure out what n is and then once we know what n is, we can figure out what e is when i is 70. We'll put 70 here times whatever we figure out n is. And we'll get e. But without knowing what k is, that still won't help us. So statement one, by itself, if I'm right, isn't right. That won't help us. And I'll show it to you. Oh, this would be waste of time if you understood that logic. Because you can say 0.5 is equal to n times 60-- this is so we can solve for n-- so n is equal to 0.5 over 60. That's the same thing as 1/120. So then that equation boils down to e is equal to i over 120. 1/120 times i. So when i is equal to 70, e would be equal to 70/120, which is equal to 7/12. But that still doesn't help us. E is equal to 7/12, but we don't know what k is, right? When i is equal to 70, e is 7/12, so we still don't know what k is based on just the information in statement one. So as far as I can tell right now, one by itself-- not so useful. Statement two. p is equal to 2 when i is equal to 50. Well, this is useful. Because we already said, p is proportional to i. P is equal to some constant m times i. So we could use this information to solve for m. So p is equal to 2 when i is equal to 50. Divide both sides by 50, you get m is equal to 1/25. So the question was, what is p when i is equal to 70? So we could just say p is equal to m times i. Well, i now is 70. And whatever 70 divided by 25 is, that's p. So statement two alone was sufficient to solve this problem. And statement one didn't help us much. Next problem. 139. If x does not equal minus y-- that's interesting-- is x minus y over x plus y greater than 1. So why do they say x does not equal minus 1? I'm sorry. x does now equal minus y. Well, if x were equal to minus y, if this were minus y, then you'd have minus y plus y, the denominator would be 0, and you'd be undefined. So maybe that's why they put that out there. Let's see, I think we can simplify this statement. So we could multiply both sides by x plus y. You get x minus y is greater than x plus y. And so this holds true if what? Let's see, we could subtract x from both sides. Subtracting x from both sides you get-- just get rid of the x's-- you get minus y is greater than y. Let's see, we could add y to both sides of this equation. So the left side, if you add y here, you get 0. If you add a y on this side, you get 2y. And then you could divide both sides of this equation by 2. And you get 0 is greater than y, or y is less than 0. Either way. This statement is true if, and only if, this statement is true. So this is really all we need to test for. Is y less than 0? If we know that y is less than 0, we know that this is true. Or if we know that this is false, then we know this is false. So statement one. Statement one tells us x is greater than 0. Well, this is useless. This actually has no bearing on whether y is less than 0. x can be anything. It doesn't change this. So this by itself is useless. Statement two is-- well, there you go. y is less than 0. So statement two tells us this is true and this is true if, and only if, this is true. So therefore, statement two alone is sufficient to answer this question. Very seldom do you boil down the statement to actually one of the statements that they provide you. But that was interesting. Next problem. 140. In the rectangular coordinate system, are the points r,s and u,v equidistant from the origin. OK, so they're essentially saying, is the distance of the point r,s-- distance from the origin-- equal to the distance-- I'll call it d sub o-- equal to the distance of the origin of point u,v. And here, just to get the intuition of distance-- I always find it silly that they teach something called a distance formula in high schools, because it's really just the Pythagorean theorem. And by calling it something different and making you memorize a different formula, it'll just clutter your head. So this is the x-axis, this is the y-axis. The point r,s will be here. This is r. This is s. This is the point r,s. What's its distance from the origin? Well, its distance from origin is the length of this line right there. What's the length of that? Well, we can use Pythagorean theorem. The height right there is s. The base here is r. And if we call this distance, we use Pythagorean theorum. r squared plus s squared is equal to the distance squared. Or we can say that the distance is equal to the square root of r squared plus s squared. The distance to the origin. So this statement up here boils down to that the square root of r squared plus s squared needs to be equal to the square root of u squared plus v squared. Well, just to simplify things, let's just square both sides of this equation. The question they ask is, is r squared plus s squared equal to u squared plus v squared? Is this true? That's what they ask us. And that kind of simplifies things. Makes them more concrete. So let's see what the statements give us. Statement one. r plus s is equal to 1. Just off the cuff, I don't see where that's going to be actually useful. It's not like you can just square this. r plus s squared is r squared plus 2rs plus s squared. A lot of people make the mistake that thinking, oh, r plus s squared is r squared plus s squared. No, that's not true. You have to distribute all the terms and you end up with three terms. So that's not right. I don't see an r plus s anywhere up here. Let's try statement number two. Statement number two. u is equal to 1 minus r, and v is equal to 1 minus s. So this seems like it could be interesting. Because it allows us to, essentially, reduce this question, which is a question of four variables, and turn it into a question of two variables by substituting u and v with these things. So let's do that. Let's turn this statement into a statement of two variables. I'll switch colors just to ease the monotony. So the left-hand side is r squared plus s squared is equal to u squared. Well, now they're telling us that u squared is 1 minus r squared-- plus v squared. Well, v is 1 minus s. 1 minus s squared. Let's just keep simplifying. r squared plus s squared is equal to 1 minus 2, r plus r squared plus 1 minus 2s plus s squared. Let's see, we could subtract r squared from both sides. We can subtract s squared from both sides. And we're left with 0 is equal to 2 minus 2r minus 2s. And let's see, we can bring the r and the s terms over to the other side. So add 2r plus 2s to both sides. So bring them over, you get 2r plus 2s is equal to 2. Divide the whole thing by 2-- both sides. You get r plus s is equal to 1. Interesting. So when you apply these two constraints on our original question, the question gets reduced to this. If we know that statement two is correct, the question that the problem was asking gets reduced to this: It's saying, if we know that this happens, then the question is true if this is true. Well, just from statement two alone, we don't know that this is true. But, as you see, statement one tells us that that is true. So if you use both statements together, you know that this question is correct. This question is true. And that was actually pretty interesting and a little bit hairier than normal. Normally, you can identify immediately just by eyeballing it. But anyway, I'll see you in the next video.