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GMAT: Data sufficiency 2

6-9, pg. 278. Created by Sal Khan.

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Video transcript

Let's keep going. We're on problem number six on page 278 of the data sufficiency sample questions. So let's see. They've drawn a little figure. And they say, in the figure above, lines k and m are parallel. I guess I better draw lines k and m. That's line k. That's line m. And then they have a transversal line. And we've gone over all of this in the geometry playlist. You might want to review it if this looks completely foreign to you. And so this is line k. This is line m. And they're calling this angle right here x degrees. This line here is z degrees. And then this angle here-- did I say line? That's an angle. This angle here is y degrees. And the question is-- question number six-- in the figure above, if lines k and m are parallel -- so they're both parallel -- what is the value of x? OK. In statement number one they tell us that y is equal to 120 degrees. So if y is equal to 120-- let me do this one in magenta-- so this is based on this is equal to 120. What can we figure out? Well, this y is supplementary to z, right? So when you add these two angles together, it's 180 degrees, right? Because you're completing kind of a whole 1/2 arc, or 1/2 circle. So z would be 60 degrees. And then you could say that -- there's all these words that people use in geometry class. But if you have two parallel lines and a transversal, then these opposite inside angles are going to be the same. So you know that x is equal to 60 degrees. Another way you could have done it, you could have said, OK, if y is equal to 120 degrees, then y and this angle right here are also supplementary, right? Because they complete this whole arc. So y plus this angle have to be equal to 180. So this angle would also be 60. And then you use what you learn in geometry class. That corresponding angles on a transversal intersecting two parallel lines, that they're also equal. So you'd also get to the same conclusion. That x is equal to 60. So either way, statement number one alone is enough to figure out x. Now, what did they give us for statement number two? And I'll do that in a different color. Statement number two. z is equal to 60. Well, this actually gives us the same information as that. Because if we know that z is equal to 60, then we know that y is going to be equal to 120, even if the book never told us this the first time. So z is equal to 60 is the same information as y is equal to 120. And so you can make the exact same argument as you did for the first one. So actually point number two alone is also enough to figure out that x is equal to 60. You actually didn't even have to figure it out. An important skill, eventually, when you're when you're taking the GMAT, is to be able just look at it and say, oh, I can figure that out, and then move on. Instead of actually having to figure out that x is equal to 60. But anyway. soon. So this one is, either of them alone are sufficient. So that's d. OK. Problem number seven. What percentage of a group of people are women with red hair? So women with red hair, percentage. So statement number one tells us, of the women in the group, 5% have red hair. 5% of women have red hair. That alone doesn't tell me what percentage of the entire group are women with red hair, because I don't know how large the whole group is. There could be 20 women, and there could be 10 million men. Or there could be 20 women and no men. So that still doesn't help me with what percentage of the group are women with red hair. Statement number two tells us, of the men in the group, 10% have red hair. So 10% of men have red hair. That's really useless. Once again, I don't know how big the group is. Think about it. If I have 20 women, then that tells me that there's one woman with red hair. And I don't know I have 20 women, right? But I still don't know how many men there are. If there are 20 women and one has red hair, there could be a million men. There could be no man. In which case this answer would turn out very different. What percentage of the group are women with red hair? So both of these combined are fairly useless questions. And actually, let me draw a Venn diagram, because I think it's useful. So the entire group is both women and men. I'll draw a Venn rectangle instead. So that's women and men. So some percentage of-- we don't know how many women there are, and how many men. So this area is women. That's the number of women. And this is the number of men. And this first point tells us that 5% of the women have red hair. So it just tells us that 5% of this area is red, right? Which is maybe, I don't know, I'll eyeball it. It's like that. And then this says 10% of the men have red hair. So maybe that area looks something like that. So we know the ratio of this to this box. And we know the ratio of this to this box is 10%. But we don't know the ratio of this to the entire universe, because we don't know how many-- we don't know what the total population sizes, so we'll never be able to figure it out. Anyway, so that is e. All right. Problem number eight. Maybe I missed something. Problem eight. If r and s are positive integers, you r is what percent of s? So r, s, positive integers. And we want to know r is what percent of s? So essentially, we just want to figure out what r over s is equal to, right? This'll give us some decimal. And then you multiply by 100, and you know the percentage. So if you can figure out this, you can figure out the percentage of r is what percentage of s. So statement number one. They tell us that r is equal to 3/4s. Well, let's just do a little algebraic manipulation. We're trying to get r over s, so let's divide both sides by s. So you get r over s is equal to 3 over 4. So there we got it. We got the answer. That was a helpful data point. All we needed was that data point, actually. Let's see what the second data point gives us. Data point two. r divided by s-- well, they wrote it like this; they wrote it the way you did in second grade-- r divided by s is equal to 75 over 100. Well, that's just another way of just writing r over s is equal to 75 over 100, which is exactly the same thing as this. So these are actually equivalent statements, almost. So each of them independently are enough to figure out r over s, or what percentage r is of s. All right. Problem number nine. I'll draw a line here, I don't want to get too messy. Is it true that a is greater than b? I sometimes find these statements slightly humorous. Is it true that a is greater than b? All right. The first statement is, 2a is greater than 2b. So I don't know if you remember from algebra, but you can operate on inequalities the exact same way you can operate on equalities, or I guess you call them equations. And you just have to remember that if you multiply or divide by a negative number, that you have to swap the inequality sign. Well, luckily in this case, we could divide both sides by a positive number. So if you multiplying or dividing by a positive on both sides, you don't have to change the inequality. So just divide both sides by 2. And you can test that with numbers, just to see why that make sense. So you divide both sides by 2. And you get a is greater than b. So that's all we needed. We just needed statement one. Now let's see what statement two does for us. Statement two tells us that a plus c is greater than b plus c. Well, once again, we can subtract c from both sides of this equation, or from both sides of this inequality, without changing the inequality sign. So you subtract c from both sides. And once again, you get a is greater than b. So each of these statements independently are enough for us to figure out that it is true, that a is greater than b. Let's do one more. Actually, I've run out of chalkboard space, so I might as well just wait until the next video. See