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GMAT: Data sufficiency 8

37-41, pgs. 280-281. Created by Sal Khan.

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Video transcript

Now on problem 37. And they say what is the value of xy minus yz. Statement one says, y is equal to 2. Well, that by itself doesn't help us. That just tells us 2x minus 2z. So if you don't know what x and z are you can't figure out what this whole thing is if you just know y is equal to 2. So that alone doesn't help. Statement two is x minus z is equal to 5. Now this is interesting because this expression, we can factor out the y. What happens if we factor y out of both of these expressions? I'm going to just rewrite it. xy minus zy-- I just switched the y and z-- that equals x minus z times y. Well, we can figure out x minus z from statement number two. x minus z, this is equal 5. And we know what y is equal to from statement one. So y is equal to 2. So we can definitely figure it out as long as we have both statements and we can show that the answer is y is equal to 2. So the answer is, 2 times 5 is 10. But we didn't have to figure it out. We just have to know that we could figure it out if we had both of these data points. One by themselves, you wouldn't be able to solve it. So the answer is C. Both statements together are sufficient but neither alone is. Problem 38. OK, they drew us a picture. Let me see if I can draw that same picture, looks like some type of device. So it looks like that and then they have the other end, and the other end looks like that. There are these handles, or something that looks like handles. I haven't read the problem yet. So they say will the first 10 volumes of a 20 volume encyclopedia fit upright in the book rack shown above? So this is a book rack. So I guess the books get stacked that way. And they labeled this right here, this dimension is x. And they're saying, will the first 10 volumes of a 20 volume encyclopedia fit upright in the book rack shown above? So essentially I'm going to put the first 10 in there. So the first statement they say, is that x is equal to 50 centimeters. Well, that doesn't help me because I don't know how big the first 10 volumes of the encyclopedia are. If each of them are, at least the first 10 are, less than 5 centimeters each or on average less than 5 centimeters, then maybe I could fit them. But one by itself doesn't help me. Two: 12 of the volumes have an average thickness of 5 centimeters. Well, that doesn't help me either because remember, they're saying will the first 10 volumes of a 20 volume encyclopedia? Maybe the 12 that have an average thickness of 5 centimeters, maybe those are the volumes 8 through 20 and maybe volumes 1 through 7 have an average thickness of 50 centimeters each or 5 million centimeters each. So even both of these conditions combined don't help me know if I can definitely fit the first 10 volumes of the 20 volume encyclopedia. So that is E. Both statements together are still not sufficient. Problem 39. A circular tub-- OK, so they've drawn this circular-looking tub. So the top looks like that. And then there's two sides. Let's me see how well I can construct what they've drawn. And the bottom looks something like that. And then they shade in a little area, a strip of this, like that. And they say a circular tub has a painted band around its circumference as show above. So this is the painted band around its circumference. What is the surface area of the painted band? And they tell us that the height of the painted band is x. So in order to essentially know the surface area, you'd have to know the height, which is x, times the circumference of the circle, which would be the length. So if you knew the height times circumference of the circle, you'd be able to know the surface area. So they tell us, the first statement, x is equal to 0.5, whatever, meters. So this is in meters. 0.5 meters, so that alone doesn't help us. We have to be able to figure out the circumference of this tub in order to really be able to figure out the surface area because the circumference times this height will be the service area. Point two: they say the height of the tub is one meter. So they're telling us that this is one meter. Well that doesn't help us. That still doesn't tell us how far around the tub goes. If they had given us the diameter or the radius or the circumference then we could've used our basic geometry to figure out the circumference. But they didn't. So either way-- it's a one meter height-- both of these combined still do not allow us to figure out the surface area of this green band. E again. Problem 40. What is the value of integer n? And n as an integer. So statement number one: they tell us that n times n plus 1 is equal to 6. Now we should already be able to figure this out because n is an integer, n plus 1 is an integer. So what are the factorizations of 6? You could have 1 times 6. But that doesn't fit n and n plus 1, right? This is n and n plus 5. You have 2 times 3, which work, right. If 2 is n, n plus 1 is 3 and they equal 6. So let me circle that. And then what other factorizations? You don't have any other-- these are all of the factors 6. So just looking at the first statement, you know that n has to be equal to 2. This is the only integer where this is true. Actually, let me take a step back. No, what if n is negative? Because I was assuming it's a positive integer but they didn't say it's a positive integer, so let me think about that. If I did minus 3 times minus 2, that also equals 6 and these are both integers. OK, so this isn't enough. Because in this case, n could be-- this would be n and this is would be n plus 1-- so n is either equal to 2 or n is equal to minus 3. That's a little tricky. The intuition is to just assume that n is positive but it doesn't have to be positive. So n could be 2 or minus 3 here. That's a tricky one. So that one by itself does not help us solve the problem. The second part of it, point two it says, 2 to the 2n is equal to 16. And whenever you have these in the SAT or the GMAT or anything, whenever you have a variable in the exponent, your goal really is to just get everything in the same base. So we could write the left-hand side the same. So that's 2 to the 2n. And how do we write 16 with the base 2? Well, 16 is just 2 to the fourth, right? And so we get 2n is equal to 4. n is equal to 2. So this statement alone is enough to figure it out. This statement alone is not. So the answer is B. Statement two alone is sufficient. 40 is B. And this was tricky because one, you think that 1n alone is sufficient but it is ambiguous because and n could be minus 3. 41. OK. They've drawn a bunch of, if I see this right, a bunch of-- two lines. Let's see if I can draw this. So a line and then there's a bunch of circles. One circle, two circles they have, and then they keep going. Three circles, and then one more. It's going to be right there. Four circles. I think that's about right. And they say essentially that the circles just keep going on and on. That's what I think the drawing implies. They say the inside of a rectangular carton is 48 centimeters long, 32 centimeters wide and 15 centimeters high. OK. So that's the inside. Kind of from the other wall. This is the floor of the inside of the container and then there would be another up here. But I think you get the idea. The carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns as indicated in the figure above. OK. So this is like a top view of it. So we could say that this side up here, this is 48 and then this is 32. So we don't care so much about the height, I think. So, let's see, the carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns as indicated in figure above. If the cans are 15 centimeters high-- OK, so each of the cans are exactly 15 centimeters high. So they literally are exactly as high as the carton. If the cans are 15 centimeters high, what is the value of k? So we have to figure out how many of these cans will fit in this area essentially? And then they tell us point number one: each of the cans have a radius of 4 centimeters. Radius is 4 centimeters. Well, if we know that each of them have a radius of 4 centimeters then we know exactly how much square area each of these circles will take up, assuming that they're packed exactly like this, right? Because if you think about it, what is this area right here? Well, there's a bunch of were different ways you could think about it. The easiest is that if the radius is 4 centimeters, that the diameter right here is 8 centimeters. So if this is 8 and you have a 48 length, you can only do 6 of these, right? 6 times 8 is 48. So you can only do 6 that way. And then if the diameter is 8 this way and this length is 32, you can only do 4 this way. So statement number one, alone, is enough to figure out how many you can put. It would actually be 4 times 6. You could put 24, k would be 24 cans that you could fit in. So statement one, alone, is enough. Now what does statement two tell us? 6 of the cans sit exactly along the length of the carton. So they're telling us that these cans, there's 6 cans that fit right along the length, which we figured out from statement number one. But that also gives you the same information as statement number one. Because you know that these are cylindrical cans. I think it's a safe assumption to say that these are circles. Yeah. A cylinder, you're assuming that the tops are circles. So if you say that 6 can fit that way, then you know that the diameter of each of them is 8 centimeters and then you can make the same argument that the diameter's 8 centimeters to say that 4 can fit the other way down. So two, alone, is also sufficient by itself. So the answer is, what is that, C? Both statements-- no D, each statement, alone, is sufficient. See you in the next video.