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GMAT: Math 25

129-132, pgs. 169-170. Created by Sal Khan.

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  • leaf red style avatar for user Jason Louis McConnell
    For #130, I need someone to explain to me in a different way why choice E, |x - -1| is <= 4, is the correct answer. The video begins around . More specifically, why does Hal use a negative minus 1 which turns into a plus positive 1?
    (2 votes)
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    • aqualine sapling style avatar for user Glenn D'Abrera
      I don't have the printed material to refer to so just going by the video.
      This refers to the Modulus of a number. Look up Modulus on the internet. It refers to an unsigned number or a number separate from its (+) or (-) signature.
      Now (-) x (-) gives you a (+) so that takes care of the x+1
      The Modulus |x+1| is now less than or equal to 4 because at this point we are only talking about an absolute value of x and disregarding its positive or negative signature.
      (2 votes)
  • blobby green style avatar for user mailbox4emails
    Q 129: In the 11th edition book, the answer uses 284 and 11.4. I can't understand why. Does it have something to do with that fact you stated that it never approaches 11.5 and it's always 11.4999..?
    (1 vote)
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  • blobby green style avatar for user prateek prateek
    At In Q 130 , why do we take midpoint i.e -1 and secondly why do we subtract -1 from it?
    (1 vote)
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  • hopper happy style avatar for user 🐣🐥kacyrp🐈🐇
    Wait so it this kind of like percents?
    (1 vote)
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  • blobby green style avatar for user mikhasstedt81
    what is the square root of 20x to the 6 power over 25 to the simplest form
    (1 vote)
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    • purple pi purple style avatar for user doctorfoxphd
      hmm, so you mean √((20x⁶)/25) ? or do you mean (√(20x⁶)/25) ?
      for √((20x⁶)/25)
      Let's use this handy equivalent rule: √(A/B) = √A/ √B
      √((20x⁶)/25) = √(20x⁶)/ (√25) = √(20x⁶)/ 5
      Now we can use this other handy equivalent rule: √AC = √A ∙ √C
      so, √(20x⁶) =√20 ∙ √x⁶
      The square root of a power (base with an exponent) is going to be that exponent divided by 2 . Because x³ ∙ x³ = x⁶, then √x⁶ = x³

      Making progress!
      What we have left is (x³ ∙ √20)/5
      All we need is √20)
      For that we will use √DE = √D ∙ √E
      √20 = √(4 ∙ 5) = √4 ∙ √5 = 2√5
      At last, putting them all together:
      √((20x⁶)/25) = 2/5 x³√5
      Likewise, if you meant
      Then the answer, simplified, is
      2/25 x³√5
      Sorry, hope you weren't in a hurry for that!
      (1 vote)

Video transcript

We're on problem 129. On a recent trip, Cindy drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons of gasoline, rounded to the nearest gallons. The actual number of miles per gallon that Cindy's car got on this trip must have been between? So they want us to kind of say these are the rounded numbers, now what were the actual numbers between, and then from that what's the range of miles per gallon? So if this is rounded to the nearest 10, then her miles had to have been greater than or equal to 285, because anything 285 or more would have been rounded to 290. But then less than 295 because anything less than 295 would be would be rounded down, but 295 itself would be rounded up to 300. And her gallons had to have been greater than or equal to 11.5 gallons, by the same logic, and less than 12.5 gallons. So what was the lowest possible miles per gallon? So the lowest possible miles per gallon would have the lowest miles and the highest gallons. You want to minimize the numerator and maximize the denominator. So lowest possible miles would be 285. And the highest possible gallons, at least it'll approach 12.5. So 12.499999 is the possible number of gallons, so it'll approach this. But it can never be exactly 12.5 gallons, so her miles per gallon are going to be greater than this. It would've been greater than equal to this with like 12.499999, but since 12.5 can never get there, we're going to be greater than this in terms of miles per gallon. And then the highest possible miles per gallon is the highest possible miles, which is 295. But we're never going to be able to go twice quite there. We're always going to be less than 295. And then the lowest number of gallons-- going the highest mileage with the lowest possible gallons, you want to minimize the demonimator-- divided by 11.5. Let's see if that is a choice. I don't see that exactly, but choice D looks interesting because choice D says-- so they say the actual number of miles per gallon that Cindy's car got on this trip must have been between. Choice D looks close to what we have because we have this. Choice D says between 284 divided by 12.5. That's going to be even less than that, because you have the 12.5 but the numerator's less. So that's going to be even lower than the range that we have. And then on choice D, 295 divided 11.4. This number is larger than this number, because you have the same numerator and you have a smaller denominator. So you're dividing by a smaller number. So this number's larger. So if her miles per gallon is between these two numbers, and these two numbers are between these two numbers, then D is right. Her miles per gallon need to be between 284 divided by 12.5 and 295 divided by 11.4. So it's choice D. Problem 130. Which of the following inequalities is an algebraic expression for the shaded part of the number line above? So this number line, let's try to draw it. That's the line. And I'm not going to draw all of the dots on it, but it essentially starts at positive 3 and it goes backwards. Let's switch colors. It starts essentially from minus 5 to positive 3. That's all of what this includes. This includes minus 5 to positive 3, then you have 2, 1, 0, and then minus 5. So you could say that x is less than or equal to 3 because it's filled in at 3, and greater than or equal to minus 5. And that's not one of the choices. They want it in terms of absolute value. So how we can view this, this is a range around some number. Whenever you're dealing without absolute value, you're essentially saying the distance from some number. So this is a range of minus 5 and 3. This has a length of 8, so they're both 4 away from minus 1. Think about it. You could go 4 backwards or 4 forwards. So one way to think about it is the difference between x and minus 1, and that might be a negative or positive difference depending on whether we're positive or negative, depending on whether we're above negative 1 or below negative 1. But we just want the absolute difference. We just want the distance away from negative 1. And that's given by this, the difference between any x and negative 1. And we took the absolute value in case x is less than negative 1. The distance is going to be less than or equal to 4, because if we go above negative 1 we're at most at 3, which is 4 more than negative 1. If we go below negative 1, we're at minus 5 which is 4 less than negative 1. So it's less than or equal to 4. So if you simplify this, you get x plus 1, the absolute value is less than or equal to 4. And that is choice E. And this is a useful skill. You have probably taken the SAT already, but in general, a lot of standardized tests think of absolute value as a distance in 1 dimension. So the distance along the number line. So they're saying all of these points, they're within some range from 1 point. And you can say if you take the midpoint of this, you're at negative 1. And every point here's within 4 of negative 1. So the distance between every point on this line, every x that satisfies this and negative 1, the distance is less than or equal to 4. Anyway, I kind of just did the problem twice for you. Problem 131. A factory has 500 workers, 15% of whom are women. So 15% times 500 is equal to women. If 50 additional workers are to be hired and all of the present workers remain, how many of the additional workers must be women in order to raise the percent of women employees to 20%? So how many women workers do we have right now? 5 times 15, we have 75 workers. 15 times 500. We have 75 women right now. If we wanted to find the proportion of women, it's 75 over 500, which is once again 15. It's 15%. We're going to add a total of 50 workers to this population. And they want to know how many of the additional workers must be women in order to raise the percent of women population to 20%. So we're adding 50 total workers. So how many of those have to be women in order for the new proportion-- this is going to be the new number of women, the number of women that are added to the 75 that were already there-- divided by the 500 workers who were there, plus the 50, some of whom might be women. So this now has to equal 20%. We get 75 plus w over 550 is equal to 0.2. So we get 75 plus the number of women that need to be added is equal to 0.02 times 550. We multiply both sides times 550. That equals 110. And then subtract 75 from both sides. Women has to be equal to 110 minus 75. And that's equal to-- 25 plus 10-- that's equal to 35. 35 of the 50 employees have to be women. And that's what they asked. How many of the additional workers must be women? So that's choice E. Next question, 132. At a small snack shop, the average arithmetic mean revenue was $400 per day over a 10 day period. So the average over 10 days was $400. During this period, if the average daily revenue was $360 for the first 6 days-- so the average of 6 days was $360-- what was the average daily revenue for the last 4 days? So we've done a lot of problems like this. Hopefully this is a bit of second nature if you've watched the solution to the other ones. But what's the average of all of them? It's the average of the first 6 days. We can almost assume that the first 6 days had exactly $360 in revenue. So their average times 6. So this would be the total revenue of the first 6 days plus what's the total revenue of the next 4 days? It's going to be the average of the next 4 days times 4. That would be the total of those 4. We're not saying that they all have to be exactly the average, but when you sum them all up they're equal to the average times 4. Now if you divide all of those, this is a total of the revenue all 10 days. If you take that sum and divide it by 10, you get the average for the 10 days. And that is equal to $400. Let's simplify this a little bit. Let's see what we could do. So we get 360 times 6 plus the average for 4 days times 4 is equal to 10 times 400, which is 4,000. Before I get too involved in multiplication, let's divide both sides of this equation by 4. So 360 divided by 4, that's 90, times 6 plus this divided by 4 plus A4 is equal to 1,000. 90 times 6 is 540 plus the average of the 4 days is equal to 1,000. So the average of the 4 days is equal to 1,000 minus 540. Which is what? That's $460 per day. And that's choice D. And I'm out of time. See you in the next video.