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GMAT: Math 32

163-166, pg. 174. Created by Sal Khan.

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  • aqualine ultimate style avatar for user Prashantraj Singh
    if the length of the longest rod that can be placed in a cubical room is 4*2^0.5 m
    ,then the area of the four walls of the room is
    (1 vote)
    Default Khan Academy avatar avatar for user
    • purple pi purple style avatar for user doctorfoxphd
      Well, the dimensions of a rectangular room are length, width and height (l, w, h) so
      area will be the sum of the areas of each wall.
      For a cubical room, if all we want is the area of the 4 walls, this simplifies to the sum of 4 squares of side s
      which is 4s²
      So A₄ = 4s²
      For the rectangular room, the distance of a diagonal is √(l² + w² + h²)

      For a cubical room, we can use s = w = l = h
      so the length of the diagonal is √(s² + s² + s²) = √(3 s²) which simplifies to s√3
      Now you were given a length of 4√2, so we set that equal to s√3 to solve for s
      s√3 = 4√2
      We COULD divide both sides by √3 and then rationalize the denominator, but,
      since we need s², in order to solve the final part, I am going to square both sides
      (s√3)² = (4√2)²
      s²∙3 = 16 ∙ 2
      s² = 16 ∙ 2/3
      4s²=4 ∙ 32/3
      =128/3 m² or 42 2/3 m²
      (1 vote)

Video transcript

I was interrupted in the last video by a phone call. I was actually expecting a phone call from someone whose call I wanted to take, but that ended up being one of those robocallers. So, anyway, I was on problem 163. Let me just start it over so that I don't interrupt your train of thought. A fruit salad mixture consists of apples, peaches, and grapes in the ratio of 6:5:2. So the ratio of apples to peaches to grapes is equal to 6:5:2 by weight. If 39 pounds of the mixture is prepared, the mixture include how many more pounds of apples than grapes? So they essentially want us to figure out what the apples minus the grapes are equal to. So we have 39 pounds, so the pounds of apples plus the peaches plus the grapes, that equals 39 pounds. And to do a ratio problem like this, the easiest way is to say let's just define the number of pounds of apples being 6 times some number x. If the apples are 6 times some number x, what are the peaches? We know that the ratio of the apples to the peaches are 6:5, so the peaches are going to have to be 5 times some number x. By the same logic, the grapes are going to have to be 2 times some number x. And if you try out the ratio the apples to the grapes is 6x over 2x, which is 6:2. So it all works out. So let's use this information to substitute back there and see what x is equal to. So we get 6x plus 5x plus 2x is equal to 39. 6 plus 5 is 11 plus 2 is 13. 13x is equal to 39. x is equal to 3. And so how many apples are there? 6x, there are 18 apples. We don't have to figure out the peaches, but we can do it fast. There are 15 peaches, and then there are 6 grapes. And if you want to know how many more apples there are than grapes, 18 minus 6 is equal to 12-- actually these are all in pounds. There are 12 pounds more of apples than grapes. And that's choice B. Problem 164. Louise has x more dollars than Jim has. So L is equal to Jim plus x, and together they have a total of y dollars. So L plus J is equal to y. Which of the following represents the number of dollars that Jim has? So they want Jim equals, and they want it in terms of x and y. So let's take this L and substitute it here so that we have an equation that this has J's, x's, and y's in it. So if L is equal to J plus x, we can substitute. So we get J plus x, instead of an L, plus J is equal to y. That is that, just that. And you get 2J plus x is equal to y. Subtract x from both sides. 2J is equal to y minus x. And we get J is equal to y minus x over 2. And that is choice A. Problem 165. Let's switch colors. During a certain season, a team won 80% of its first 100 games, and 50% of the remaining. If the team won 70% of its games for the entire season, so they won 70% of the total, what was the total number of games that the team played? So how do we figure out the total number of games that they played? It'll be 100 plus remainder. So let's think of it this way. If we wanted to figure out the total percentage that they won, we would want the number that they won. So how many games did they win? They won 80% of their first 100 games, so that's 80 games, plus 50% of the remainder games. So plus 0.5 times the remainder. That's how many games they've won. Now, how many games did they play? The total number of games is going to be 100 plus whatever this remainder number of games are. That's their total number of games. And the problem tells us that this ratio, the total percentage-- this is of all the games they won divided by all the games they played-- that that is equal to 0.7 or 70%. If we multiply both sides of this equation by 100 plus r, we get 80 plus 0.5r is equal to-- 0.7 times 100 is equal to 70 plus 0.7r. Now let's subtract 70 from both sides. You get 10 plus 0.5r is equal to 0.7r. And then if we subtract 0.5r from both sides, we get 10 is equal to 0.2r, or r is equal to 10 over 0.2. And 10 over 0.2, 0.2 is the same thing as 1/5. So this is 10 times 5, so that's equal to 50. So the remainder games were 50, so they want to know the total number of games the team plays. So it's going to be the first 100 games plus the remainder games. So 100 plus 50. They played 150 games, which is choice D. We're on problem 166. Of 30 applicants for a job, 14 had at least 4 years of experience. So experience greater than 4, we have 14. I just make up notation as we go. Of 30 applicants for a job, 14 had at least 4 years of experience, 18 had degrees-- so we say degrees, we have 18-- and 3 had less than 4 years of experience and did not have a degree. So experience was less than 4 years' experience and no degree, and that is 3 people. How many of the applicants had at least 4 years of experience and a degree? So let's see how we can think about it. There's 30 applicants for a job. So let me draw our universe. If I did all of the applicants, let me draw it like this. So that's the whole applicant universe, and that's 30. And we know that 14 had more than 4 years of experience. Let me do that in this color. So this attribute, I'll make it look like that, so that's the 14 that have experience greater than 4 years. And that's 14 of them. And then 18 have a degree. And I'm assuming that there's some overlap between the degrees in the 14. We don't know that for sure, but let's say that this point right here I'll draw another circle. So let's say that circle looks something like that. So let's call this the degreed people. And there's 18 of them in this whole circle. And then they say that there were 3 people with experience less than 4 and no degree. So they're outside of both of these circles. So this right here is 3. Now, what are they asking? I've already forgotten that. How many of the applicants had at least 4 years experience and a degree? So they're asking for this intersection right here. So there's one question we can answer. What's the sum of this population, the population that has more than 4 years of experience or a degree? So let me circle. What's this sum right here? This and this. It's going to be whatever's left after you take out the 3 people who have nothing. So that's going to be 27 people. So 27 people have at least 4 years of experience or a degree. Now, how can relate that to the 14 and 18? So what happens if we just add the 14 and the 18? When we have the 14, we're going to count these people once. And if we were add that to 18, we would count these people twice. So if we wanted this number-- if we wanted the number of people who are in this circle or this circle-- what we would want to do is add the 14 to the 18, but then subtract out once the people who are this intersection. Let's call them B. That's a B. I know you can't read it. Let's say there B people in that intersection. You'll subtract that out once. I want that to really make sense for you. If we want to know the total number of people in this circle and this circle combined, when we say 18 people, that's including some people here. And if we were add that to 14 people, where once again some of those 14 are here again, are these people. So when we just add 18 to 14, we're counting this twice. So if we want to know the actual number in the circle or the circle, since we add 18 to 14 we count it twice, we wants to subtract these people out once. Hopefully that makes sense. But anyway, now we're ready to solve. We get 27 is equal to-- 14 plus 18 is equal to 32 minus B. 27 from both sides, you get 0 is equal 5 minus B. Add B the both sides. You get B is equal to 5. And that is choice E. See you in the next video.