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GMAT: Math 30

154-160, pg. 173. Created by Sal Khan.

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Video transcript

We're on problem 154. An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity. So you could say you can fill 3/5 of a pool every 8 hours, which is just another way of saying that you can fill, let's see, 3/5 divided by 8 is 3 over-- what's 5 times 8? 3/40 pools per hour. I have a feeling that they're just going to ask us something along just the rate of the fill. So then they say, how much more time will it take to finish filling the pool? Oh, well, this actually is a little bit easier. So if it takes 8 hours to fill the pool to 3/5 of capacity, how many hours will it take to finish filling the pool? So how many hours will it take to fill it essentially what? What's left? 2/5, right? To fill it 2/5 more. So this is going to be the same thing as 8/3 is equal to x/2, And you could do that by dividing both sides of this equation by 5, right? And you get a 5 in the denominator in both places, which cancels out. And this is easy, 8/3 as x/2 so you have 16 is equal to 3x. And then you get x is equal to 5 1/3 hours. That's 5 hours and 20 minutes, because a third of an hour is 20 minutes. And that is choice B. Problem 155. A positive number x is multiplied by 2 and this product is then divided by 3. And they told us x was positive. If the positive square root of the result of these two operations equals x, what is the value of x? Let's solve this. Let's square both sides of this, so we get 2x/3 is equal to x squared. And then multiply both sides by 3, you get 2x is equal to 3x squared. And they tell us that x is positive. Well, first of all, that tells us that x does not equal 0. They tell us that x is greater than 0. So let's see, let's divide both sides of this equation by x. Well, there's a couple of ways we could think about it. Maybe the easiest way-- yeah, let's just divide both sides of this equation by x, and we can do that because we know that x cannot equal 0 and you get 2 is equal to 3x. And then divide both sides by 3, you get x is equal to 2/3, and that's choice D. I was thinking about saying, oh, well, let's subtract 2x from both sides. You get 3x squared minus 2x is equal to 0. You could factor out an x, and then you get 3x minus 2 times x is equal to 0. And that would tell you that either 3x minus 2 is equal to 0, which would give you that answer, or x is equal to 0. but? X can't equal 0, because they told us that x is positive. But anyway, it gets you the same answer, 2/3. Next question. Problem 156. A tank contains 10,000 gallons of a solution that is 5% sodium chloride by volume. What is that? NaCl? This isn't chemistry. Let's see, 2,500 gallons of water evaporate from the tank. The remaining solution will be approximately what percent sodium chloride? So they're essentially saying, well, if water evaporates, the salt in the solution doesn't go away, right? The salt doesn't evaporate with the water. So if we're 5% sodium chloride, what's 5% of 10,000? 5% of 100 is 5, 5% of 1,000 is 50, 5% of 10,000 is 500. I guess you could say there's 500 gallons of sodium chloride. If we lose 2,500 gallons of the water, now the percentage of sodium chloride is still 500, but it's going to be 500 over what? We don't have 10,00 gallons of water anymore. We have 10,000 gallons of water minus 2,500 gallons of water, so we have 7,500 gallons of water. Let's see, that's equal to 5/75, which is equal to 1/15, and what is that as a percentage? 15 goes into 100 6 times? It's not 7. Yeah, 6 times. 6 times 5 is 30. 6 times 1 is 6 plus 90. And then I have a 10, so it keeps going on, 0.0666 repeating. So it's 6 and 2/3 percentages, or 6 point-- well, let's see, they have choice-- yeah, choice D looks right. 6.67% if you round this. It just keeps repeating with 6. So that is choice D. Problem 157. If x plus 5y is equal to 16 and x is equal to minus 3y, then what does y equal? So let's just substitute this over here. So we get minus 3y plus 5y is equal to 16. Add those together, you get 2y is equal to 16, or y is equal to 8. And that is choice E. Problem 158. A committee is composed of w women and m men, so w and m. If 3 women and 2 men are added to the committee and if one person is selected at random from the enlarged committee, then the probability that a woman is selected can be represented as? OK, the probability of a woman being selected is essentially the fraction of the population that are women. So how many women are there? There's going to be w plus 3 women. And what's the total population? There's w plus 3 women, and then there's m plus 2 men. So this is the probability we select a woman, and then we can just simplify this a bit. That equals w plus 3 over w plus m plus 5. And that is choice E. Problem 159. If the product of the integers w, x, y, and z is 70. So let me just write that, w, x, y, z is equal to 770. And if 1 is less than w, which is less than x, which is less than y, which is less than z, what is the value of w plus z? Ok, this is interesting. So they're telling us that these are integers. We have four integers, and they're all greater than 1. Let's see if we can do a prime factorization of 770, because I have a feeling that that is what it's going to come down to, if we just look at the factors of 770. So let's see, that's equal to 7. 7 times 11 is 77. 7 times 11 times 10, right? 7 times 11 times 10 is 770. 110. And then 10 is-- so that's equal to 7 times 11 times-- let's see, what's 10? 10 we can rewrite it as 2 times 5. So that's its prime factorization. And lo and behold, we have exactly four numbers, and let's put them in order. So that equals 2 times 5 times 7 times 11. And so this is what w, x, y, and z have to be because they're different numbers. They're all greater than 1, and they're all integers. So let's see, they want to know w plus z, so w plus z is 2 plus 11 is 13, and that's choice B. And looks like we have time even for another one. Let's do this one. The figure above shows a circular flowerbed. Let me see if I can draw it. So that's the outside and then they drew the inside. So if I do it like-- oh, I didn't do a good job. Let me see if I can draw the inside, maybe if I put it there. It's not showing up the way I want to. Let me move it around a little bit. Let me just erase it and try to draw it one more time. Well, I think you get the idea. And they tell us that the distance from the center to here is 8 feet. The figure above shows a circular flowerbed with it's center at 0 and surrounded by a circular path that is 5 feet wide. So this is 5. That is 3 feet wide. I'm having trouble reading. 3 feet wide. What is the area of the path in square feet? So it's essentially the area of this outside circle minus the area of this inside circle. That would be the area of this path. So what's the area of the outside circle? Well, the area of the outside circle, you go 8 to get to this edge, then you have 3 more. So its radius is 8 plus 3 is 11, so the area of the outside is equal to pi times 11 squared. And what's the area of the inside? The area of the inside is equal to pi times the inside radius times 64 squared. So the area of our path is going to be area of the outside minus the area of the inside, which is equal to 121 pi. That's 11 squared times pi minus-- oh, sorry, it's 8 squared, which is 64. I should have written 8 squared there, so that's minus 64 pi. And that is equal to-- what's 121 minus 64? 121 minus 64 is-- let's see, we have 21 plus 36, that's equal to 57 pi, and that is choice D. See you in the next video.