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GMAT: Data sufficiency 19

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We're on problem 84. On a company-sponsored cruise, 2/3 of the passengers were company employees, and the remaining passengers were their guests. So we could say, employees is equal to 2/3 of passengers. And I guess we could also say that guests are equal to 1/3 of passengers. The remainder are guests. If 3/4 of the company employee passengers were managers, what was the number of company employee passengers who were not managers? OK. So the second statement, in at least the problem statement, it says if 3/4 of the company employee passengers were managers-- so 3/4 of E were managers, if I read that statement-- what was the number of company employee passengers who were not managers? So essentially, they want to know what 1/4 of E is, right? This is what they're asking. Because they want to know, what were the number of company employee passengers that were not managers? And 3/4 were managers, 1/4 were not managers. So if we could figure out P, we would be able to figure out E, and we'd be able to figure out 1/4 E. If we were able to figure out guests, we could then use that to figure out P, which we could use to figure out E, which we could then figure out 1/4 E. So that's how we should be thinking about it, not that we should be spending that much time on it. Problem number one. There were 690 passengers. 690 is equal to P. Well, we'd be done, right? Because E is equal to 2/3 P, and then we want to know what 1/4 E is. So the non-manager employees would be 1/4 times E, which is just 2/3 times P. Times 690. And that would be our answer. So one alone is sufficient. And then statement number two. There were 230 passengers who were guests of the company employees. So they're essentially saying that guests are equal to 230. And then we could use this information, that guests are just 1/3 of P. And then, of course, from there you could solve. You could multiply 3 times both sides of the equation to figure out that P is equal to 690. Which is the same information that they gave in one, which was enough to solve the problem. So two alone is also sufficient. So they are each independently sufficient to solve the problem. That problem is confusing, just with the language. Company, employee, manager, passenger, guest. It was very confusing. 85. In the x-y plane, does the point 4 comma 12 lie on line k? OK, this is interesting. So let me draw a line k, and I'll worry about the point 4 comma 12 later. So let's see. Let me look at the points that they do say lie on k. So if this is the x-axis, that's the y-axis. Statement one tells us that the point 1, 7 lies on k. So the point 1 comma 7 lies on k. And then they also tell us that the point--- that alone isn't going let me know if the point 4 comma 12 lies on it. So 1, 2, 3, 4. 4 comma 12. It might be up here. This is the point that we care about. If you can see that. Let me scroll down. 4 comma 12. So if the line looks like that, maybe it goes through, but the line could be like that. So statement number one alone isn't enough. Statement number two says the point minus 2 comma 2. x is minus 2, y is positive 2. So they're saying that lies on it. Eyeballing this line, k, it seems like 4 comma 12 could very well lie on it. But we have to do a little bit of math to figure it out. So first of all, just statement two by itself doesn't help us, because this line could go anywhere. You need two points to just know what the line is. And so how do we figure out if the point 4 comma 12 is on this line? Well, the easiest way to do it is to figure out the slope-- well, I don't know if it's the easiest way, but it's the way I would think about doing it-- is to figure out the slope of the line, and then extend that slope here and see if the point 4 comma 12 lies on it. So let's see, what's the slope of this line? So when we went up 7 minus 2. So the rise is 5. When you go up 5, we went over how much? 1 minus minus 2. We went over 3, right? So for every 3 that you move over, you go up 5. That's the easiest way to think about it. So we're going from x is equal to 1 to x is equal to 4, so we're going to the right by 3. And so we should be going up 5. So 5 plus 7, sure enough, is equal to 12. So if you just extend the slope, it does hit the point 4 comma 12. Well, actually you know what? I just wasted a lot of your time, because we just have to know whether the data is sufficient to answer the question. We don't have to prove that the question is true. I've just proven to you that 4 common 12 does lie on the line. But you could have just said, oh, well, you know what? Statement number one gives me a point on the line. That alone isn't enough to solve the problem. Statement number two gives me another point. And then at this point, if you're really taking the GMAT and time matters, you just say, hey, I got two points for this line. Two points is everything I need to know about a line. Once I know two points on a line, I can figure out if any other point is on that line. I'm done. Both statements combined are sufficient. You wouldn't have to do all of this stuff that I did, which would waste your time. But just to prove to you that you could figure it, I did that. But anyway. I have to admit I did waste your time a little bit. You should just immediately say, oh, I got two points on a line. Once I have two points, I can completely figure out if any other point lies on that line. 86. The length of the edging that surrounds circular garden K-- OK, so we have two gardens, it looks like. So we have circular garden K, and then we have circular garden-- let's see, what does it say? It is 1/2 the length of the edge that surrounds circular garden G. So this is circular garden G. This is K. This is G. So they're saying the length of the edging that surrounds circular garden K is 1/2 the length of the edging that surrounds circular garden G. So we could say circumference of K-- right, that's the edging that surrounds it-- is equal to 1/2 the circumference of G. Fair enough. What is the area of garden K? Assume that the edging has negligible width. Well, a couple of things. If we know the circumference, we know the area, right? Because the circumference is 2 pi r, and once we know r, area is pi r squared. So if we can figure out the circumference of K, then we know its area. And likewise, if we know anything about the radius, or the diameter, or the area of G, we can figure out its circumference. We know its area. We can use the formula, area equals pi r squared to figure out what r is. And then once you figure out what r is, you could just figure out that circumference is equal to 2 pi r. So if you have any of this information, you're able to figure out any of the rest of it. So statement number one says, the area of G is equal to 25 pi square meters. So let me write it. So immediately you should say, hey, if I know the area of G, I can figure out the radius of G. And if I know the radius of G, I can figure out the circumference of G. If I know the circumference of G, I can figure out the circumference of K. If I know the circumference of K I can figure out the radius of K, and if I know the radius of K, I know the area of K. You shouldn't have to calculate any of that. So this is enough. This is sufficient. Statement two. And maybe I'll do that for you, just to show you that it can be done. But hopefully it's making some sense. The edging around G is 10 pi meters long. So they're actually telling us that the circumference of G is equal to 10 pi. So this is even easier. They're giving us this. That means the circumference of K is 5 pi, which means that we can figure out the radius of K and we could figure out the area. So this is also sufficient to solve the problem. And just so you don't take my word for it, let me just show you how quickly you could figure it out. If the area of G is 25 pi, that means that pi times the radius squared of G is equal to 25 pi. That means that the radius is equal to 5, right? Cancel out the pi on both sides. r squared is equal to 5. That means the radius is 5. That means the circumference is 2 pi times this. So that means that the circumference of G is equal to 2 pi times 5, which is 10 pi, which is exactly what statement two told us. And then from there, we could figure out the circumference of K, which is 1/2 of that. So circumference of K is equal to 5 pi. And then we know that 2 pi times the radius of K is equal to 5 pi. Divide both sides by pi. We know that the radius is equal to 5/2. And then the area of K would be pi times this squared. So it would be pi times 25/4, and we'd be done. But all of this was a waste of time. I just wanted to show you that once you have the radius or the circumference, or the area of either of these, you're able to figure out everything else. And I'm out of time. See you in the next video.