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# Comparing proportionality constants

## Video transcript

- We're told that cars A, B, and C are traveling at constant speeds and they say select the car that travels the fastest and we have these three scenarios here. So, I encourage you to pause this video and try to figure out which of these three cars is traveling the fastest, car A, car B, or car C. Alright, now, let's work through this together. So, car A, they clearly just give its speed, it's 50 kilometers per hour. Now, let's see, car B travels the distance of D kilometers in H hours based on the equation 55h is equal to D. Alright, now, let's see if we can translate this somehow into kilometers per hour. So, 55h is equal to D or we could say D is equal to 55H and here I'm doing, this is this scenario right over here, not scenario A. And so, another way to think about it is distance divided by time, so if we divide both sides by hours, we would have distance divided by time, and so if we have D over H, then we would just be left with 55 on the right hand side. All I did is I divided both sides by H. Now, this is distance divided by time, so the units here are going to be, we're assuming, and it tells us D is in kilometers, H is in hours, so the units here are going to be kilometers per hour. So, car B is going 55 kilometers per hour while car A is only going 50 kilometers per hour. So, so far, car B is the fastest. Now, car C travels 135 kilometers in three hours. Well, let's just get the hourly rate or I guess you could say the unit rate. So, 135 kilometers in three hours, and so we can get the rate per hour, so 135 divided by three is what? That is going to be, let's do it in our head, I think it's 45 but let me just verify that, three goes into 135, three goes into 13 four times, four times three is 12. You subtract, you get, yep, three goes into 15 five times, five times three is 15. Subtract zero. So, this is equal to 45 kilometers per hour. So, car A is 50 kilometers per hour, car B is 55 kilometers per hour, car C is 45 kilometers per hour, so car B is the fastest. Let's do another example. So, here, we're asked which relationships have the same constant of proportionality between Y and X as the equation 3y is equal to 27x. And we have three choices here and actually there's two more below the fold, but just for the three that you see, pause the video and see which of these have the same constant of proportionality as 3y is equal to 27x. Alright, now let's work through this together. So, first, let's figure out this constant of proportionality with 3y is equal to 27x. One way to figure out the constant of proportionality is you can get it in the form Y is equal to R times X where R would be your constant of proportionality. So, let's do that. Well, to get it in that form, you just divide both sides by three, divide both sides by three, and you are left with Y is equal to 27 divided by three is 9x. So, our constant of proportionality that we're looking for is nine. Well, that's exactly what choice A right over here is, Y is equal to 9x, constant of proportionality is nine. So, I like that answer. Now, 2y is equal to 18x, if we were to solve for Y on the left hand side, divide both sides by two, you're gonna get Y, Y is equal to 9x again. Once again, same constant of proportionality. So, B is a good choice. Now, let's see over here. When X is equal to one, Y is equal to nine. Or another way to think about it is Y, in this case, is nine times X. So, this is the point one, comma, nine, and so you see that Y is nine times X, Y is nine times X. So, our constant of proportionality again is equal to nine. So, actually, the three that we can see on screen, and they say choose three, so these are going to be our three answers. Well, let's just verify that the other two choices that you don't see don't have a constant of proportionality of nine. Constant of propor, propor. (laughs) Trouble saying that, alright. Okay. So, here, we're given the relationship between X and Y at a table. And so, to go from three to one third, let's see, to go from three to one third, what do you multiply by? Well, to go from three to one third, you multiply by one ninth. To go from six to two thirds, you multiply by one ninth, or another way to think about it, you divide by nine. So, here, our constant of proportionality is one ninth, not nine. Y is equal to one ninth times X. Now, over here, let's see, to go from two to 18, you do multiply by nine, so that's interesting. And then, here, but to go from four to 27, you don't multiply by nine, you're multiplying by something other than nine. And then to go from six to 36, you're multiplying by six. So, this last table, it's not even a proportional relationship, much less have the same constant of proportionality.