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GMAT: Data sufficiency 22

95-98, pg. 286. Created by Sal Khan.

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Video transcript

We're on question 95. A certain sales person's weekly salary is equal to a fixed base salary plus a commission that is directly proportional to the number of items sold during the week. So let me write this. 95. So the salary is equal to some base plus a commission that is directly proportional to the number of items sold during the week. So let's say the number of items sold is x. x items times the commission per item. So, I'll call that lowercase c. And this is the total commission. Fair enough. If 50 items are sold this week, what will be the sales person's salary for this week? OK, so they actually told us that there's 50 items. Fair enough. What will be the sales person's salary this week? So statement number 1 tells us, last week 45 items were sold. So 45 in last week. That alone doesn't tell me much. Because I don't know how much he made last week. Statement 2. Last week's salary was $405. Well, does this help me still? Last week. Because in order to figure out-- we need to figure out-- right now we need to know how much of his salary in any given week is base, and how much commission does he get per unit. And the only way we can figure that out is if you tell us that last week we know that he sold 45 items. So, for example, last week he made $405. If I use both statements, let me see if I can get anywhere. He made $405 last week, which is equal to his base plus 45 units sold times his commission per unit. So that's all this information gives me. Actually if I use both statements, that's all it gives me. That 405 is equal to b plus 45 times c. And we need to figure out s is equal to b plus 50c. What is s equal when you sell 50 units. And so we have two linear equations. But we have how many unknowns? We have one unknown, two unknown, three unknowns. Sorry. One unknown, two unknown, three unknowns. b and c was already there. So we have 3 unknowns, but only two linear equations. So we don't have enough information to solve it. So the answer is E. All of these statements still do not give us enough information. 96. If Juan had a doctor's appointment on a certain day, was the appointment on a Wednesday? So we want to know, was it on a Wednesday? 1, exactly 60 hours before the appointment it was Monday. 60 hours before the appointment it was Monday. This is interesting. OK, so how many days? 60 hours is 2 days. 2 days is 48 hours. It's exactly 2 and 1/2 days. 48 and then another 12. So this is equal to 2 and 1/2 days. So this is interesting. If we said that 2 days before, it was a Monday, then his appointment had to have been on a Wednesday, right? Because if you pick any hour in Wednesday and you go exactly 48 hours ago, it would be that same exact hour on Monday. So if you said 48 hours ago it was Monday then this would be enough information. This is 2 and 1/2 days ago. So for example, if his appointment-- and you know it might sound weird-- but if his appointment was at Thursday 1 a.m. and you go 2 and 1/2 days before-- let's see if you go 2 days before you to Wednesday 1 a.m., Tuesday 1 a.m., and then you go 12 hours before. You would end up at Monday at 1 p.m. So, I've given a case that meets condition 1, where his appointment wasn't on Wednesday. It was on Thursday. But of course, I can give a condition where his appointment was on Wednesday. His appointment was on Wednesday at 10 p.m. Then if you go 1 day back you're at Tuesday 10 p.m. I'm sure you can't read that. Then you go another day. You're at Monday 10 p.m. And then you go a half a day. You're at Monday 10 a.m. So, statement 1 does not give us enough information. That's because of this pesky half day. Now let's see what statement 2 gets us. Statement 2. The appointment was between 1 p.m. and 9 p.m. Now, this by itself, obviously, is useless. I mean, you could have an appointment any day between 1 p.m. and 9 p.m. But used in conjunction with each other, it seems like I have enough information. Because let's say the appointment was at 9 p.m. and 2 and 1/2 days ago it was a Wednesday. So let me see, sorry, 2 days ago it was a Monday. So if I have it at Wednesday 9 p.m., if I go back 1 day, I'm at Tuesday 9 p.m. Sorry. 1 day back I'm at Tuesday 9 p.m. 2 days back I would be at Monday 9 p.m. And then if I go 1/2 a day more, I'm at Monday 9 a.m. And if I take the lower bound. Wednesday at 1 p.m. Do the same logic and you're going to end up at Monday 1 a.m. So at either end of this range, if I know that if you go 2 days ago, you end up at Monday, the only day that works is Wednesday. You could try this with Thursday or Tuesday. You won't end up on Monday if you go 2 and 1/2 days back. So for this problem, both statements are necessary in order to know whether his appointment was on Wednesday. Next problem. 97. What is the value of 5x squared plus 4x minus 1. First they tell us, x times x plus 2 is equal to 0. Well let's see, can we use this alone? Let's just multiply it out. We get x squared plus 4 x is equal to 0. And that tells us that x squared is equal to minus 4x. Well maybe we could substitute this in for 4x, or for x squared. So we get 5 times x squared. Well we know that x squared is minus 4x. Minus 4x plus 4x minus 1. That's minus 20x plus 4x minus 1. No it doesn't get us anywhere, even playing around with the algebra. So we get minus 16x minus 1. So statement 1 by itself, at least as far as I can figure out, does not help us solve this problem. Statement number 2. x is equal to 0. Well that's all we need. x is equal to 0, then this is 0. This is 0. And we're just left with negative 1. So this is the only piece of information we need to solve this problem. And statement 1 doesn't really help us much. So only statement 2 is necessary. What is that, B? Next problem. 98. At Larry's Auto Supply Store, brand x antifreeze is sold by the gallon. So x by gallon. And brand y is sold by the quart. Excluding sales tax, what is the total cost for 1 gallon of brand x antifreeze-- so 1 of x-- and 1 quart of brand y? OK, so we want to know the cost. Equals how much dollars? So let x be the number of gallons of x and y be the number of quarts of y. So we'd have to know their prices in order to figure out how much the combination costs. So let's see, at least I think. Statement 1. Excluding sales tax, the cost for 6 gallons of brand x antifreeze and 10 quarts of y-- so plus 10 of y-- is equal to $58. Well this, once again, by itself does not help me much. I have two unknowns with one linear equation. So I can't solve for what x plus y is equal to. If I could have factored out-- if this was 10x plus 10y I could have factored out a 10, and I would have had x plus y sitting there. So actually I probably could have solved, if it was that way. But this isn't quite as easy. So statement number 2. Excluding sales tax, the total cost for 4 gallons of brand x plus 12 quarts of brand y-- so it's 12 quarts of brand y-- is equal to $44. And actually, just to be exact, I realize I misspoke something. Let x be the cost of x antifreeze per gallon. And y is equal to the cost of y antifreeze per gallon. And so we are still trying to figure out x plus y, but I just wanted to be exact. Because we've got 6 gallons. So the cost would be 6 times x plus 10 quarts of y. So this would be 10 times y to get 58. Et cetera, et cetera. But now we have two equations with two unknowns. So it's trivial now to solve for x and y. This is your algebra 1 problem. So, both equations combined are enough to solve for it. But each independently are not enough. Because you can't just factor out a number and just be left with x plus y here. That could have been a trick, if this was a tricky problem. But it's not. Anyway, see you in the next video.