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GMAT: Data sufficiency 20

87-90, pg. 285. Created by Sal Khan.

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Video transcript

We're on problem 87. It says an employee is paid 1.5 times the regular hourly rate for each hour worked in excess of 40 hours per week. OK. So after 40, they get overtime. 1.5 times. Excluding sunday. And 2 times the regular hourly rate for each hour worked on sunday. So each hour worked on Sunday, regardless of whether or not they're above 40 hours. How much was the employee paid last week? OK. So let's see. Statement one tells us the employee's regular hourly rate is $10. $10 per hour is normal. Now based on that, we know what their overtime is. Overtime is 1.5 times, so it's $15 an hour. And it tells us that sunday is double that. $20 an hour. All of that was said in the problem description, I guess you could call it. But we still don't know how much they were paid because we don't know how many hours they worked and when those hours happened to be. So it's still not enough information. Statement number two. It says last week the employee worked a total of 54 hours but did not work more than 8 hours on any day. OK. So let's think about that. If you worked no more than 8 hours in a day, how many days would you have to work? Well you would have to work-- I guess at minimum, you'd have to work 6 days for 8 hours. And 1 day you would work for 6 hours. Right? Or there's other ways you could say it. You could have all 7 days you could be working-- for how many times does 7 go into-- you could be working 7 6/7 hours a day. Now neither of these help us because we don't know where the hours were allocated. We don't know how many of these hours ended up on sunday. Because that's a critical question. And we know that there was some overtime in some form, right? But we don't know if that was sunday overtime or if that was regular overtime. Imagine working 7 6/7 hours every day. Then you would have 7 6/7 hours for monday through friday, then you get some overtime for saturday as soon as you got above 40 hours. And then on sunday you would get paid double. But then you have this situation. Maybe sunday is a day you worked 6 hours. Or maybe sunday is one of the days you worked 8 hours. So we don't know. So, this problem, there's not enough information to solve this problem. 88. What was the revenue that a theater received from the sale of 400 tickets, some of which were sold at the full price, and the remainder which were sold at reduced price? OK, fair enough. The number of tickets sold at full price-- so this is statement number one-- so immediately we know that there were 400 tickets sold. And that is the number of full-priced tickets plus reduced priced tickets. It's equal to 400, right? Wait. Some of which were sold at full price and the remainder of which-- oh, they want to know the revenue. OK. They want to know the revenue. So we have to know how much we got for each of these tickets in order be able to figure it out. So number one. The number of tickets sold at full price were 1/4 of the total number of tickets sold. So 1/4 times-- what was the total number of tickets sold? Well they already told us that. 400. So it equals 100. And then we could just look at that and that tells us that the reduced price were equal to 300 tickets were sold at the reduced price. But that still doesn't tell us the total revenue because we don't know how much full price was or how much reduced price was. So that's not enough information just yet. Problem two says the full price of a ticket was $25. But we still don't know what the reduced price is. We know the 25 100, or $2,500 were generated from the sale of the full-price ticket. But the reduced price, they don't tell us that. How much did they reduce the price, right? Some of which were sold at full price and the remainder of which were sold at a reduced price. We don't know what that is. It was 25% off? 50% off? We don't know. So unless we know the price of the reduced price ticket, we can't figure this out. So once again, not enough information to solve the problem. Problem 89. What is that, a circle? The circle represents one of the operations plus, minus and times. OK. This is interesting. So they say a circle represents one of the operations plus minus and times is k circle l plus m equal to k circle l plus k circle m for all numbers k, l, and m? Well essentially, what are they doing? They're doing the distributive property, right? They're saying that k-- whatever this operation times l plus n-- is the same thing as k times whatever this operation is with l, plus k whatever this operation is with m. And the only places where the distributive property works is either with multiplication or division. And division isn't one of these properties. So essentially this doesn't work with addition or subtraction. So if essentially we're able to prove or disprove that multiplication is this operator, then we have enough information. This question could be rephrased as: is circle equal to multiplication? So if we can answer this with the statements, then we can answer this top one. Because only multiplication works with this. Or division, but division isn't one of the options. Statement one says k circle 1 is not equal to 1 circle k for some numbers k. Well this immediately tells me that this is not multiplication, right? In fact, this tells me that this is subtraction. Right? Because the only time where k minus 1 is different than 1 minus k, with addition, they'd always be equal to each other. With multiplication they'd always equal each other. So this has to be if you believe statement one, then circle is subtraction. Which tells you that this statement up here is not true. So statement one alone is sufficient to determine whether this statement is true. Or another way to phrase problem, whether circle is equal to multiplication. So statement one is sufficient. Statement one actually tells us that the circle is subtraction. Statement number two. Circle represents subtraction. OK well they just told it outright there. Well so this and this are equivalent information and so they're enough to determine that this up here is not true. Remember, they're not saying, is the statement true? They're just saying, do you have enough information to figure it out? And we know that this isn't true because circle is subtraction. And this statement only holds true if the circle is equal to multiplication. So each statement alone is sufficient to solve that problem. Problem 90. How many of the 60 cars sold last month by a certain dealer had neither power windows nor a stereo? OK. So it tells us 60 sold. And we want to know how many had neither power windows nor a stereo. Statement one tells of the 60 cars sold, 20 had a stereo but not power windows. Fair enough. But that still doesn't tell us how many had neither. So let me draw a little circle Venn Diagram here. OK. So that's all 60 cars that were sold. That pool right there, that set. 20 had stereos with no power windows. So let me draw some Venn Diagrams. [SNEEZES] Excuse me. All right. That sneeze that I was talking about a couple of videos ago finally happened and I feel-- [SNEEZES] excuse me. All right. Back to the problem. 20 had stereos with no power windows. So let's say that this is the pool that had stereos. This is the pool that had power windows. And what we care, actually, is what had neither stereo nor power windows. So we care about this outside. Oh, that looks tacky. That's a little garish. But anyway. So if this is stereos, this is power windows, this would be stereos and power windows. You're saying 20 had stereos but no power windows. So that's this right here. So I'll do another fill. Yeah. That's not pleasant to look at. But this is 20 stereo, no power window. But that alone doesn't tell us what this green area is. So what's the second statement? That's too dark. Statement two tells us, of the 60 cars sold, 30 had both power windows and a stereo. So that tells us that this range right here-- let me do it in another tacky color. That tells us that that's how many had power windows and stereos. Let me make sure I have a good color here. So that's 30. I know that you can't see that. OK. So we could answer a couple of questions. We can answer, how many cars sold had stereos in general? Well 50, right? 20 had a stereo, no power windows. 30 had a stereo with power windows. So a total of 50 stereos were sold. We know that. 50 had stereos. But that still doesn't answer our question. Of the 10 that remained-- 60 were total, and there's 10 left within this space and this space. We don't know how those 10 fall out. Maybe 5 had neither and 5 had only power windows with no stereo. Or maybe there were no cars with only power windows and no stereos, and all 10 had neither. So you don't know, even using both statements. So, at least for this one, they haven't given us enough information to figure out how many had neither. They did give us enough information figure out how many stereos got sold, or how many cars with stereos got sold. Anyway. See you in the next video.