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GMAT: Math 42

205-206, pg. 180. Created by Sal Khan.

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Video transcript

We're on problem 205. In the table above, what is the number of green marbles in jar R? OK. Let's see, they give us a table, they have jars P, Q, and R. They say number of red, is x, y, and x. And they tell us the number of green is y, z, and z. And then they tell us the total is 80, 120, and 160. And they say, in the table above, what is the number of green marbles in jars? So we need to solve for z. So what we have here is three variables and we have three unknowns. We actually have three equations because we know that the red plus the green is equal to the total for any one of the jars. y plus z is equal to 120 and x plus z is equal to 160. So let's just try to solve this equation. So let's try to get everything-- I don't know, what do we want to cancel out first? What happens if we subtract that equation from this one? Let me write that one over. So that's x plus y is equal to 80. I just rewrote that column down here. Now, if we were to subtract this from that-- let's just well multiply it by negative 1. So, minus x, minus y, minus 80. And now let's just add them. We get 0 plus z minus y, so we get z minus y is equal to-- what's 160 minus 80? Well that's just 80. And now, this is interesting, we have x minus y equals 80, and z plus y is equal to 120. So I think we're pretty close. Let's write that. So we could write this equation-- I'll write it in yellow. z plus y-- I just switched the order-- is equal to 120. But, in the back of my mind, I'm like, well, all I have to do is figure out these. If I can figure out these as fast as possible, all the better. Add these two equations. 2z plus 0 is equal to 240. z is equal to 120. And that's the number of green marbles in jar R. 120. And that is not one of the choices. Where did I make a mistake? I often make a mistake right near the end. Let's see. x plus z. Let me see. X plus y is equal to 80. y plus z is equal to 120. x plus z is equal to 160. Now, I took this equation-- no, I took this one and I multiplied it by negative 1 and put it here. Minus x, minus y, is equal to minus 80. Then I added these two. So the x's cancel out and you're left with z minus y is equal to 80. And then, this equation says y plus z is equal to 120-- oh, I see my mistake. You see, I made it right at the end. z minus y is equal to 80. z plus y is equal to 120. 80 plus 120 is not 240. 80 plus 120 is 200. So 2z is equal to 200. z is equal to 100. Boneheaded mistake. So the answer is 100. D. Next question. Question 206. In the circle above, PQ-- let me draw the circle above. This is the circle above. Let's see, in the circle above, PQ-- OK they then draw diameter. Let me draw diameter as well. Because I want to be accurate. So that's diameter and they labelled this as O and R. And they say PQ-- so they have this other line up here that this is parallel to the diameter. So let me draw it up here. Maybe it looks something like this. That's parallel. And they call this point P and this point Q. And then they have another line they say is a 35 degree angle. Let me see if I can draw that. You have a line that goes from here to here. And they tell us that this angle right here is 35 degrees. In the circle above, PQ is parallel to diameter OR. And OR has length 18. What is the length of minor arc PQ? Fascinating. So, they want to know what this is. So, we can figure out the circumference of this circle. The circumference, we know its diameter. It's just going to be the diameter times pi. So, we already know that the circumference of the circle is 18 pi or 2 times the radius times pi. But same thing. Now, in order to figure out this arc length, what would be useful-- and I don't know yet if I can figure it out, is if we could figure out this angle right here. If we could figure out this angle right here, we'd be done. We could easily figure it out because that angle as a proportion of 360 would be the same relative proportion of the total radius. So let's see if we can figure out what that angle is. Just given the information they've given us. This angle here's is going to be 35 degrees. And how did I know that? Well, those are those are opposite inside angles. Let me see. What matters is-- let me think about it a little bit. And this is on the circle. Oh, well this is interesting. We should use the information. This is a circle. So, what is the length of that right there? Well, if the diameter is 18 then the radius is 9, right? If that length is 9, than this length is also going to be 9. And if that length is 9-- oh this is interesting. This length is 9, this length is also 9. That length also has to be 9. They're both radiuses of the circle which tells us that this, right here, is kind of an isosceles triangle on its side. It looks like that. Where these sides are equal. They're both equal to 9. So, if this is a 35 degree angle, then this, right here, has to be a 35 degree angle as well. I know you can't read it. Because the base angles of an isosceles triangle are equal. I could redraw it like this. This is an interesting problem. So, if we know that length is equal to that length, we know that angle is equal to that angle. So, then we know that this angle right here has also got to be equal to 35. This is useful because if this angle is 35 and this angle is 35, what is this entire angle? What is that entire angle? Well that entire angle-- let me draw it up here. So that's the PQ side. So, this is also 35. So this entire angle right here is 70. 35 plus 35. I should have drawn this a lot bigger. So, this right here is 70 degrees. And how do I figure it out? I figure out this 35 because this is an isosceles triangle. And I figured out that this is 35. I figured out that this is right here is 35 because it's an opposite angle from a transversal between two parallel lines. Because that line is parallel to that line. So, that was all pretty useful. Now, what if there was only a way to figure out what this angle was? Because if we know this angle and this angle, then we know for sure this angle. Well, I'm going to use the same argument. This line right here, this is a radius of the triangle. So this side right here is also 9. That side. so, if we know that this side is equal to this side-- let me draw the triangle that I'm talking about now, because I think this is an important problem. So, I'm trying to figure out the value of this upside down triangle. And I'll label it so you know what I'm talking about. So, this is P. This is Q. And this is the center of the circle. And we know that the side is 9, this side is 9. And we just figured out that this angle right here is 70. This was 35 plus another 35. Once again, another isosceles triangle. If this side is equal to that side, that this angle is equal to this angle. So, this angle also has to be equal to 70. And now, if we know two angles of a triangle, we know the third. This has to be-- [PHONE RINGING] I'm on a roll. I'm not going to answer the phone. Oh I think it's my wife. I'm going to call her back right after this. So what was I doing? I lost my train of thought. I was on a roll. If this angle is 70 degrees, this angle is 70 degrees, then this angle has to be what? 70 plus 70 plus x is equal to 180 degrees. 140 plus x is equal to 40 degrees. Sorry, is equal to 180. My brain was skipping ahead. x is equal to 40 degrees. OK. So that angle is 40 degrees which is the same thing as that angle. So the length of-- so let's go back to the problem. We have to figure out what this arc length is. So, this angle, which is 40 over all the degrees in the circle, over 360, is going to be equal to this arc length-- let's just call it x-- over the circumference of the whole circle. And we know that the circumference of the full circle is 18 pi. Let's see if we can simplify this. 40 goes into 360-- that's 1/9. So to this becomes 1 over 9. Then we could cross multiply and get 9x is equal to 18 pi. x is equal to, 18 pi divided by 9, is equal to 2 pi. And, luckily, that's one of the choices. It's choice A. See you in the next video. I now have to call my wife.