GMAT: Math 33

We're on problem 167. If 1 plus 1 over x is equal to 2 minus 2 over x, then what does x equal? The first thing I would do is-- I don't like these x's in the denominator. So let's just multiply both sides of this equation by x. So I'll get x plus-- x over x is just 1-- equals 2x minus 2. Now I can subtract 1 from both sides. Actually, let's add 2 to both sides, so you get x plus 3 is equal to 2x. Let's subtract x from both sides. You get 3 is equal to x. And that's choice E. I think I have the hiccups. I hope you don't notice, or if you do, I hope it doesn't bother you. Problem 168. Last year, for every 100 million vehicles that traveled on a certain highway, 96 vehicles were involved in accidents. So 96 vehicles for every 100 million. So 100 times a million is 10 to the sixth vehicles. If 3 billion vehicles traveled on the highway last year, how many of those vehicles were involved in accidents? You could view this as the fraction that will be involved in accidents, so we just multiply that times the total number of vehicles. That's 3 times a billion. And they even write a billion as 1 with 9 0's, so it's 3 times 10 to the ninth. So if we simplify this a little bit, let's see what we can do. So 100 times 10 to the sixth, this is the same thing as 1 times 10 to the eighth. You could view this as 10 to the second times 10 to the sixth is 10 to the eighth. If you have a 10 to the ninth in the denominator divided by 10 to the eighth, this cancels out and you're just left with a 10. So the denominator becomes 1, and the numerator becomes 96 times 30. And 96 times 30 is-- let's put a 0 here and then 3 times 6 is 18, 3 times 9 is 27 plus 1 is 28. How many vehicles are involved in accidents? 2,880, C. I think this number is low compared to the reality. I think actually it's closer to the 40,000 range. That's what want. How many of those vehicles are involved in accidents? We multiply the fraction that have an accident times the total number of vehicles, and then you get your answer. 2,880. Next question, 169. 30% of the members of a swim team have passed the lifesaving test. So 30% passed. Among the members will have passed the test, 12 have taken the preparatory course. So this the 70% not passed. And they say that 12 have taken a course, so 12 course. And 30 have not taken the course. How many members are in the swim club? So many people have not passed the test? They said that 12 have taken a course, 30 haven't. I'm assuming that's all of them. So the 70% that have not passed, that is 12 plus 30 is equal to 42. And this is 70% of the total people in the swim club. So 70% of the total is equal to 42. So the total is equal to 42 divided by 0.7. 42 divided by 7 would be 6, but since we're dividing it by 1/10 of that, it's going to be equal to 60. And that make sense. 70% of 60 is 42. So there's 60 people in the swim club, and that's A. Problem 170. In a certain company, the ratio of the number of managers to the number of production line workers is 5:72. If 8 additional production line workers were to be hired, the ratio of the number of managers to the number of production line workers would be 5:74. So if we hire 8 more workers, then the ratio is equal to 5:74. How many managers does the company have? So this is interesting. Let's think of it this way. There's a quick way to think about it, and then there's an algebraic way, and we'll do both of them. But when we add 8 to the bottom of this-- when we add 8 to the number of workers-- the ratio actually only increased by 2. So what's happening is we probably have 4 times as many workers as this ratio predicts. So the ratio is probably 20:288. And let's think about that logic. Because 20/288-- this is 5/72 if you were to divide the top and the bottom by 4-- and if you were add 8 to the bottom, you would get 20/296. And then if you divide this by 4, if you just take this to the lowest reduced form, you get 5. Divide that by 4. This divided by 4 is what? It's 74. So if you see that intuition that I'm talking about, that when you add 8 workers it only increased this reduced form by 2, that means that this reduced form is probably 1/4 of the real number. Because 2 is 1/4 of 8. If that doesn't make sense to you, don't worry about it. But if you did it this way, you could say there must be 20 managers. And that would be a very fast way of solving this problem. But we can also do it alegebraically. So you could use this information to say 5 times the workers is equal to 72 times the managers. And then you can use this information to say 5 times the workers plus 40-- I'm just cross-multiplying, so 5 times w8-- is equal to 72 times the managers. And now we can do some algebra. So let's subtract this from. Let me say minus 5w is equal to minus 72 times the managers. Subtract this from this. These cancel out. You get 40 is equal to 74 minus 72 is 2 times the managers. So then you get the managers are equal to 20, which is what we figured out the first time. Maybe this is simpler for you. And I would probably do that on the exam to make sure I didn't make a careless mistake. Next question, 171. If x minus 1 squared is equal to 400, which of the following could be the value of x minus 1? x minus 1 squared is equal to 400, so that means x minus 1 is equal to positive 20, the positive square root of 400. And don't forget this. This also means that x minus 1 could equal minus 20, because minus 20 squared is also 400, the negative square root. In this case, if you add 1 to both sides, you get x is equal to 21. In this case, if you add 1 to both sides, you get x is equal to minus 19. So that's what the values of x are. So what can be a value of x minus 5? That's what they ask for, they're not asking for the value of x. x minus 5 in this case would be-- what's 21 minus 5? That's 16, minus 19 minus 5. That's minus 24. So x minus 5 could be 16, or it could be minus 24. And they only have this one there, so it's choice C. So they were just testing to see if you remembered that x minus 1 could be the minus square root of 400, the negative square root. Problem 172. Which of the following describes all values of x for which 1 minus x squared is greater than or equal to 0? So let's add x squared to both sides of this equation. So then you get 1-- x squared plus minus x squared, that zero-- is greater than or equal to x squared. I'm just going to switch it because that's just how I think. x squared is less than or equal to 1. We're dealing with real numbers, so x squared is going to be a positive number. So the only way that I can square a positive number and have it be less than or equal 1, is if x itself is less than or equal to 1. Or it could be greater than or equal to negative 1, because those numbers will square to being positive. So x would be less than or equal to 1. But it can also be greater than or equal to negative 1. And you could try it. Try negative 1/2. Negative 1/2 squared is positive 1/4. No matter what, whether it's positive or negative, this is going to be a positive number. And the magnitude can be at most 1. So that takes us from minus 1 to 1. And even 0 works. So let's see if that's one of the choices. That's choice E. And I'll see you in the next video.