We're on problem 47 on page 281. If the two floors in a certain building are 9 feet apart, how many steps are there in a set of stairs that extends from the first floor to the second floor of the building? So I think I can draw that. So this is the second floor, this is the first floor. They want to know how many there's-- they're 9 feet apart, that's what they tell us already-- 9 feet, we need to figure out how many steps there are to go from the first to the second floor. So piece of information one, they give us, each step is 3/4 foot high. Well, this by itself should give us enough information, right? Because each of these distances-- I'll do it in green-- each of these are 3/4 foot high. So to figure out how many total steps to get to the top you just say 3/4 times the number of steps is going to be equal to 9. And then you could just solve this by multiplying both sides by 4/3. But we don't have to solve it. We just have to know that we can solve for it. So one, alone, is enough, right? And you could, what? If you multiply both sides of this you get 12 steps. But anyway, we want to avoid having to actually do math. We just want to figure out whether we can do the math. Piece of information two, each step is 1 foot wide. Well, that's clearly useless because it doesn't tell us how much altitude we're making on each step and that's what you have to figure out to say how many steps you need to go up 9 feet in the air. So this is useless. So the answer is A: statement one alone is sufficient. Next problem. 48. If xy is equal to 0, oh, does not equal 0. So that tells us that neither x nor y is 0, right? Is x divided by y going to be less than 0? In order for this to be true one of these numbers-- so I mean if x divided by y is negative, that means that either x or y, but not both of them, are negative, right? That's the only way you can get a negative number when you're dividing. If both of them are negative this would be a positive number. So let's see what we can do with their information. Number one. x is equal to, and this is the question, is xy less than 0? And one says x is equal to minus y. Well, immediately let's just substitute that back in. If x is equal to minus y then what's x/y? x is equal to minus y, so you have minus y/y and that will equal-- and we know y doesn't equal 0. So for any other value other than 0-- if this was 0 this would be undefined-- but then this is equal to negative 1. So in this case x divided by y is equal to negative 1, which is definitely less than 0. So that proves our statement. So statement one alone is all we need. Now let's see what statement two gives us. I'll do it here. Statement two. Minus x is equal to minus y. So that tells us that minus x is equal to y and then we can just substitute the same thing in again. Well, now we could substitute for x, or we could just multiply both sides by negative, you get x is equal to minus y, which is the same thing as this here. So statement two, alone, is also sufficient to solve this problem. And so the answer is D: each statement, alone, is sufficient. Switch colors, problem 49. How many people are directors of both company k and company r? OK. Directors of k and r. Statement number one. There were 17 directors present at a joint meeting of the directors of companies k and r and no directors were absent. At a joint meeting of the directors, so that's all the directors of k and r. So me draw some Venn Diagrams. So if that is k and then that is r. So they're saying that when you add both of these together, because this is a joint meeting of all the directors of both, we got 17. So there's 17 in this entire universe of directors. That's what statement one tells us. If you take this whole circle and then you add up the extra-- don't double count the intersection-- you add up the extra, you get 17. But that alone doesn't tell us how many joint directors there are. Joint directors are these people. People who are on the board of both k and r. So this is k and this is r. So statement one alone doesn't help us. Although, I'm suspecting, maybe in conjunction with something else, it could. Statement number two is company k has 12 directors and company r has 8 directors. OK. k is equal to 12 directors and r is equal to 8 directors. OK, so everything in the k circle combined is 8, right? Sorry, everything in the k circle is 12, right? k directors. Everything in the r circle is 8. Now, if we wanted to get the total of the k and the r-- so by itself that doesn't tell me how many overlap. So when you take statement one or statement two independently that doesn't help us. But what if we were to figure out if we would use them together? So how many total directors are there going to be? There's going to be the total directors in k-- so k's directors-- plus the director's in r. But if you were to just add those two up, you would double count the people who are in both k and r, right? You would count them twice. You would count them when they're in k and you would count them when they're in r. So if you wanted to figure out how many total directors there are you would then subtract out the people who are in k and r, right? You don't want to subtract them twice, you just don't want to double count them, right? Because when you we do k plus r you're counting them for k and then you're counting them for r. So let's subtract them out once so that you only count them for k or that you only count them for r. You only count them once. So minus kr-- so this notation that's people who are in k and r. And what does that equal? Well, statement one told us that. It told us that they're total of 17 directors. And so statement two told us there are 12 directors in k plus 8 directors in r minus the joint directors is equal to 17. And, by the way, we don't even have to do this. We could've just recognized that if we know the total number of directors and we know how many are in each of the groups that we can figure it out. And we would just answer the question that both statements together are sufficient. But I'll just show you that we can figure it out just so that you're happy with it. Let's see, we get 20 minus kr is equal to 17. And so you get kr is equal to 3. So there are 3 directors that overlap with both and we were only able to answer that question by using statement one and statement two. Next problem, 50. If x and y are positive is xy greater than 1? Statement number one tells us xy is greater then 1. So let's see, does that help us at all? And that's not obvious. That just tells us that x is greater than 1/y or y is greater than 1/x. So let's just think about it, this statement is equivalent, if we multiply both sides by y. And we can do that without changing the inequality because we know y is greater than 0. So if we do that, we get x is greater than y. If we can show this, we can show that. And remember, the only reason why I didn't have to change the inequality is because I knew that when I'm multiplying both sides by y, that y is greater than 0. If y was less than 0 I'd have to switch the sign right here when I multiply both sides by it. But anyway, if I can prove x is greater than y, we're all set. This doesn't help me. Let's see, statement number two is x minus y is greater than 0. Well, if I add y to both sides of this equation what do I get? I get x is greater than y. So this proves exactly what I need to prove, and if you want to go all the way to what originally they asked, divide both sides by y-- and we don't have to change the sign because y is positive-- you get x divided by y is greater than 1. Which is exactly what we needed to prove. So statement two, alone, is sufficient. See you in the next video.
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