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GMAT: Math 10

55-60, pg. 159. Created by Sal Khan.

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Video transcript

We're on problem 55. If y is equal to 4 plus x minus 3 squared, then y is lowest when x is equal to-- so think about when will this expression be the lowest possible number? Well, if you think about this whole term, right here, this x minus 3 squared. Any number squared, if we're dealing with real numbers, is going to be a positive number, right? So you could say that x minus 3 squared is going to be greater than or equal to 0, right? 0 is kind of its minimum point. So this whole value is going to be as small as possible when x minus 3 squared is as small as possible, right? Because that's the variable part of it. And this is going to be as small as possible when x minus 3 is equal to 0. And when is x minus 3 going to be equal to 0? We could write it out. When is x minus 3 equal to 0? Well, add 3 to both sides. It's when x is equal to 3. So y is lowest when x minus 3 is equal to 0 or x is equal to 3. And that's choice D. And all you have to recognize is the lowest value that this function can attain is when this term goes to 0, because otherwise it's a positive term. It will only add to the 4. Next problem. 56. Which of the following is not equal to the square of an integer? So they say the square root of the square root of 1. The square root of 1 is 1. And the square root of that is going to be equal to 1. You could have said negative 1, but then that would have been undefined. Fair enough. One of the roots of the roots is 1. And this is the square of an integer, right? This is equal to 1 squared. 2. Square root of 4. Square root of 4, that equals 2. Now is that the square of an integer? No. That's equal to square root of 2 squared, but this clearly is not an integer. Right? So the choice is 2. 2 is not equal to the square of an integer. We could look at the other ones. 18 divided by 2. Yes. 9 is the square of 3. 41 minus 25. That's 16. 36. So all of the other ones are squares of integers. That's a fairly straightforward problem, I think. We're on problem 57. Fermat primes are prime numbers that can be written in the form 2 to the K plus 1, where K is an integer and a power of 2. Which of the following is not a Fermat prime? So A, 3. Well 3 is equal to what? It's equal to 2 to the 1 plus 1. Right? And in this case K is 1, which is a power of 2. That's 2 to the 0. B is 5. Well, 5 is what? That equals 2 squared plus 1. 4 plus 1. And this is definitely a power of 2. 2 is a power of 2. 2 to the 1st. C. 17. That's 2 to the 4th plus 1. Right? 2 to the 4th is 16 plus 1 and 4 is 2 squared. So that's a power of 2. Choice D. 31. The closest power, if we subtract 1 from it. 2 to something does not equal 30, right? That would have to be true if 2 to the K plus 1 is equal to 31. Then 2 to the K would have to equal 30. And I can't think of an integer where I take 2 to some power and get 30. So the choice is D. D is not a Fermat prime. And if you wanted to make sure that it's not E. E is 257. Well that's just 2 to the 8th plus 1. 2 to the 8th is 256. And that's 257. So the answer is D. 58. If x squared is equal to 2y to the third, and 2y is equal to 4, they want to know what x squared plus y is equal to. OK, x squared plus y. Well, they tell us that x squared is equal to y to the third. So let's just substitute that. Right? So then we get 2y to the third plus y, using this information and that information. And they tell us 2y is equal to 4. Well, just divide both sides by 2, you get y is equal to 2. So let's substitute that into this. What's 2 to the third power? Well, that's 8 plus 2. So I don't want to divide. So that equals 16 plus 2, which is equal to 18. Which is choice E. Next problem. 59. A glucose solution contains 15 grams of glucose per 100 cubic centimeters of solution. If 45 cubic centimeters of the solution were poured into an empty container, how many grams of glucose would be in the container? So we can just set up a ratio. We have 15 grams per 100, so how many grams are we going to have per 45 centimeters cubed? How many grams, right? So they still want to know grams. All the units are the same. And so let's see if we can cross multiply. We could say 15 times 45 is equal to 100 times x. So 15 times 45 is equal to 100x. And just to make the math easy, let's divide by some numbers. Let's divide both sides by 5. So then this becomes 20 and this becomes 9. We could divide both sides by 5 again and then this becomes 4 and this becomes 3. And now the numbers are easy to deal with. 3 times 9 is 27 is equal to 4x. Divide both sides by 4. You get x is equal to 27/4. 27/4 and 4 goes into 27. what is that? That is equal to and 6 and 3/4, and all their numbers are in decimals, so that's 6.75. Which is E. Right? 4 times 6 is 24, and you have 3 left over. Right. Next problem. Problem 60. In the figure of P, Q, R, S is a parallelogram. Let me draw this. This looks interesting. So I have to draw a couple of lines. That's one line. And they have another line that looks something like that. It's a parallelogram. Let me connect the lines. It looks like that. Let's see, this line actually keeps going. Let me draw that in. They tell us, this is P. This is Q. This is R. And this is S. And they tell us that this is 140 degrees. They tell us that this is 2y. And they tell us that this is x. And then they say, in the figure above, what is y minus x? So since we know this is a parallelogram, we know that this line is parallel to this line, and we know that this line is parallel to that line. That actually tells us a lot of information, especially if we continue the lines over. Then everything we learned in geometry about parallel lines starts to hold. Let me draw it in a different color. So if this line, in green, is parallel to this line, then you can view this line right here as a bit of a transversal. And 140 degrees and 2y are corresponding angles. So they're actually going to equal each other. And you can even eyeball it and think about, well, that make sense. Those lines are parallel. So 140 degrees is equal to 2y. y is equal to 70. Fair enough. And now let's consider-- let me switch colors-- that this line right here, this magenta line and this magenta line, those two are parallel. And we can now view this as a transversal. Right? If this is a transversal, then the 70 degrees and this angle are corresponding angles, so they're going to be equal. So that's 70. And then this angle up here, that 70 degrees that I just drew, and the x angle, they're supplementary, right? They add up to 180. So you have x plus 70 is equal to 180. So you get x is equal to 110 degrees. Oh, no, no, sorry. I made a mistake. I made a mistake. So this angle isn't 70. y is equal to 70. But this angle is 2y, so this is 140 degrees. Right? We figured out that this is 140 degrees, so this has to be 140 degrees. And what my brain caught, I was like, boy, that doesn't look like 70 degrees. 70 degrees would be a lot narrower. That looks more like 140. But anyway, this is 140. So if this is 140, the corresponding angle up here is also going to be 140 degrees. And so, I did that wrong. x plus 140, right? x plus this 140 degrees. The logic is still the same. They're supplementary. Equals 180. So x is equal to 40 degrees. And they wanted to know what? They wanted to know what y minus x is. So that's 30. 70 minus 40 is 30. And that's choice A. See you in the next video.