 Let's talk about ratios and proportions, and so what happens sometimes is we want to consider two objects, and we want to consider a relationship between the two of them. So we have this thing here, and we want to compare it somehow to this thing over here. And we can make a couple of fairly obvious conclusions. We could say that this one is bigger than the other one, and this is a qualitative comparison. It refers to a quality that the one bar has relative to the other, the quality of being bigger or the quality of being smaller. However, a quantitative comparison requires that we assign a specific value to the relationship, mostly so that we might be able to compare the relationship that these two have to each other to some other relationship that two other things might have to each other. So the question is, how can we produce a quantifiable relationship between the two objects? Now it might not be possible to do that, but in the cases where we can do that, if we can identify a quantifiable relationship between the two objects, we call the relationship the ratio between two or potentially more quantities. Now one possibility is suppose each bar actually represents a whole number. So here the top bar actually represents the whole number four, and the bottom bar actually represents the whole number three. For what? Three what? It doesn't really matter, though the important thing here is that whatever each of these boxes here represents, it's the same in some quantifiable sense as what the boxes on the bottom represent. So here I have the two bars representing whole numbers, and in this case I could say that the relationship between these two is the ratio one, two, three, four to one, two, three. So we have the ratio four to three somehow representing the relationship between top bar and bottom bar. Now we express that as the ratio of four to three, we write that out that way, or we can use four colon three to represent that ratio. Now what makes ratios useful is that there's a lot of things that may have the same relationship as top bar to bottom bar that don't necessarily look like this picture. So we might begin as follows. Suppose I actually do take two bars representing this ratio of four to three. Now what if I divide each bar in the same way? So again, the pieces that I have are now of the same size and shape, but I have more of them. And if we count, we know that the top bar has 12 pieces, the bottom bar has 9 pieces, and so my ratio is now 12 to 9. And that expresses the relationship between these two objects. However, it's still the same relationship that we started with. In other words, this ratio that I'm looking at right now, this 12 to 9 ratio, is still four to three. Nothing has really changed, and that suggests that the two ratios are the same ratio. This four to three ratio is the same as the 12 to 9 ratio. And in general, this suggests the following. For any ratio whatsoever a to b, and any number n, then the ratio of a to b is going to be the same as the ratio n times a to n times b. And that emerges because if I partition both of these by the same number, then I haven't changed what the relationship is. I've only changed how it's expressed. Rations are among the most powerful of tools that we have for solving algebra problems except we don't really call them algebra problems, they're really arithmetic problems. So consider this in a classroom, the ratio of boys to girls is four to three, if there are 12 girls, then how many boys are there? And so one thing we might do is we can draw a picture representing the ratio of four to three, and so here's my ratio of four to three, and my ratio represents the ratio of boys to girls, so I'll label the two bars just to keep everything straight. And again, it's important to keep in mind that the bars have to, in some sense, be equivalent. They have to be the same thing. And what that is, we'll talk about in a moment. So now what do we know? Well, we know they're 12 girls. So these three boxes down here must represent the 12 girls. And since each of the boxes is the same size and these 12 girls altogether, then that means that each of those boxes must represent four girls. So now we have each box here representing four persons. So each box up here must also represent four persons. So there's four, four, four, and four, and at this point we can just count. There's going to be this many boys altogether 16. Well, we could have ratios involving more terms. The ratio of wolves to rabbits to bobcats and the wildlife preserve is two to five to three. And suppose I know there's a hundred animals altogether, how many are there of each? Let's draw a picture. It's going to be convenient to show all the animals on the same bar. So here I might have my wolves, there's my two, here's my rabbits, there's my five, and my bobcats, there's my three. So now I have my ratio of wolves to rabbits to bobcats, two to five to three. Now what I notice is that there's ten bars altogether, and I know that there's a hundred animals altogether. So these ten bars also have to represent those hundred animals. And so what does that tell me? Well, that tells me that each of those bars must represent ten animals. So I'll go ahead and fill that in. And at this point I can just pick off what the numbers are. So I know that there are ten, ten, there's twenty wolves, there's ten, ten, ten, there's fifty rabbits, and then there's thirty bobcats. And I can just read off the numbers from my drawing representing the ratio. And I've actually solved a problem that we could express algebraically, but I didn't need to use algebra. All I needed to do was know how to count. Once you know arithmetic, you can do algebra.