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GMAT: Data sufficiency 21

91-94, pgs. 285-286. Created by Sal Khan.

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Video transcript

We're on problem 91. By what percent did the median household-- so median is middle-- by what percent did the median household income in country Y decrease from 1970 to 1980? So they want a percentage. All right. Statement number one tells us in 1970, the median household income in country Y was 2/3 of the median household income in country X. So let's write Y in 1970-- let's just write '70, because the 19 is redundant. The median income in country Y in 1970 is equal to 2/3 of the median income of country X in 1970. That's what statement one tells us. But one again, we know nothing still about what happened in 1980 in country Y. So we still can't answer the question. Statement number two. In 1980, the median household income in country Y was 1/2 the median household income in country X in 1980. Now, you might be tempted to say, oh, two linear equations and two unknowns, maybe I can solve for it, et cetera. But no. There's actually two linear equations in four unknowns, right? Y in 1970 is different than Y in 1980. And X in 1970 is different than X in 1980. So there's actually four variables. But you say, oh no, no. But we don't need to figure out all the variables, we just need to figure out the percent decline. Right? They just say by what percent did the median household income in country Y decrease from 1970 to 1980? So essentially if we could figure out this, Y80 over Y70, we'll know what the percent decline was. Right? If this number is 0.8, then it would be a 20% decline. If this number was 0.5, then it would be a 50% decline. So whatever this number is, that's essentially 1 minus that is the percent decline. So maybe we just have to figure out this. But look. I can prove to you mathematically that we still can't figure it out without this. Because Y80 divided by Y70, that's equal to 2/3X-- oh no, sorry, Y80. So Y80, that's 1/2X in 1980 divided by 2/3X in 1970. Right? And then let's see. If we could take this 3/2 up, we get, let's see-- divide by 1/3. 3 becomes 3/2. So it becomes 3/4X80 over X70. They never told us what the median household income in country X was in any year. So we still can't solve this problem. So there's not enough information given. Next problem, 92. A certain group of car dealerships agreed to donate x dollars to a Red Cross chapter for each car sold during a 30-day period. What was the total amount that was expected to be donated? So the total amount-- let's see. Donation is going to be equal to x dollars times the number of cars. And this is what we need to figure out. We need to figure out what the donations are equal to, or the expected donations. It says a total of 500 cars were expected to be sold. So it's essentially telling us that C is equal to 500. We still can't figure out what D is because we don't know how many dollars are we getting for each car. So that is not enough information by itself. Statement two. 60 more cars were sold than expected. So the total amount actually donated was $28,000. Interesting. So essentially it's saying that C plus-- so if you had an extra 60 cars than expected, right? 500 were expected to be sold. Actually we could just say 500 were expected to be sold. 60 more cars were sold than expected. So this is the actual number that were sold. And then that times the amount donated per car is equal to $28,000. Well, now we do have enough information to figure out-- well, let's think about it. We have enough information from this to figure out x, right? Well, we definitely have enough information now to figure out x if we use statement number one. I just want to be careful to make sure that we can't solve this just with statement number two. If I just said 560x is equal to $28,000, then you get x is equal to 28,000 over 560. So then the amount that was expected to be donated would be 28,000 over 560-- that's what x is, whatever that number is-- times the number of cars that were expected to be sold. Times 500. And actually, yeah, you have to have statement one there, because statement two-- 60 more cars were sold. Actually, let me think about that. So both combined, when I use both of the information, it definitely works. So let me see if I can figure it out just using statement two alone. I don't think I can, but I have a nagging feeling that they may be giving more information than I'm-- so let me say that the cars expected is C. So this is what statement two actually is telling us. I assumed the 500, which I shouldn't have done. So statement two is actually telling, when I have 60 more than the cars expected to be sold times x, I raised $28,000. Right. This alone is not-- you don't know what x or C is. Because at the end of the day you need to know what x times C is. x times C is our goal. So if you distributed this out, you get xC plus 60x is equal to $28,000. And you get xC is equal to 28,000 minus 60x. So this is as much information as you can glean just from statement number two. So that alone is not enough. If you could figure out what x is, you're done. You actually don't need to know the expected number of cars sold. You would just know. Well actually, that's true with statement one. So you need both of these statements to solve the problem. Anyway, as you can see, I haven't done these problems before. So sometimes I'm not sure. And sometimes I might even get them wrong. Problem 93. While driving on the expressway, did Robin ever exceed the 55 mile an hour speed limit? Well, who knows? Statement one. Robin drove 100 miles. Well, that doesn't tell me whether she ever went more than-- I'm assuming it's a she. I guess it's an androgynous sounding name. That still doesn't tell me whether she went over 100 miles. Two. Robin drove for 2 hours on the expressway. So time is equal to 2 hours. So each of these independently give me no information about how fast she went. But if I use both of them, I can figure out her average speed. I can say, well, she went 100 miles in 2 hours. Average speed is equal to 50 miles per hour. Now, the question is, did Robin ever exceed the speed limit? Well, I don't know. It's completely possible that she just went up to 50 miles an hour. Well, we don't know how fast her car accelerates. She might just gone-- she accelerated really fast, got to 55, stayed there or went down a little bit and ended up averaging at 55. She's never exceeded it. But she could have easily gone 80 miles an hour at some point, and then slowed down and taken a break, and had a picnic. We don't know. So this by itself is not enough information to figure out if she ever exceeded the speed limit. If her average speed was 56 miles per hour, then we'd know that she had to exceed the speed limit. Because, well, we can assume that at some some point she was going at least 56 miles per hour. And especially if we can assume that she started at a standstill, because then you'd have to go even faster than the average to make up for the time that you're going slower. But anyway, there's not enough information here to figure it out, just knowing that her average speed was 50. Next problem. 94. In Jefferson School, 300 students study French or Spanish or both. OK. This sounds like a Venn Diagram. French or Spanish or both. So this is French, this is Spanish. And this right here is both, in the intersection. If 100 of these students do not study French-- OK, they study French or Spanish or both. There's not an option to do neither. If 100 of these students do not study French, how many of these students study both French and Spanish? So when they tell us that 100 of these students do not study French, that tells us that this area-- let me color it in a suitably garish color. Oh, no. That's not what I want to do. That tells us that this area right here is 100. So essentially, the people who are studying Spanish but not French is 100. Right? And they're asking, how many of these students study both French and Spanish? So they essentially want to know the intersection of French and Spanish. That purple area is what the question asks. So let's explore the statements. Statement number one tells us, of the 300 students, 60 do not study Spanish. 60 no Spanish. So that's essentially telling us this area. Not this, just this. So that tells us how many study French only. That's 60. It still doesn't help us to know the intersection. Statement number two says, a total of 240 of the students study Spanish. Well, I think this alone is enough information, right? Because they're telling us that this purple area plus this yellow area is equal to 240. We want to figure out the purple area, right? So the total number of people who study Spanish-- it's the people who study French and Spanish, which is this purple area-- plus the people who only study Spanish, which they gave us in the problem was 100. And then statement number two says that that is equal to 240. So the people who study French and Spanish has to be the purple area, which is 240 minus 100, which is 140. Which we don't have to figure out the number, we just have to know that two gives us enough information, by itself, to solve the problem. And we don't even need statement number one. and I'm out of time. See you in the next video.