Modeling with rational functions
Let's say that someone's running a fairly unusual pet store that only has three types of animals in it. It has cats, and we represent the number of cats with the letter c. It has dogs, and we represent the number of dogs by the letter d. And it has bears. That's what makes it unusual. And we're going to represent the number of bears with the letter b. Now, this person also tells us that at this unusual pet store, the number of cats is greater than the number of dogs, which is greater than the number of bears. Now based just on this information, they then ask us which expression is greater. b divided by c plus d plus b or 1/3. And I encourage you to pause the video now and come up with your own rationale behind which of these is greater. Or maybe neither is greater. Maybe they're equal. Or maybe you just don't have enough information from what this pet store owner told us to actually figure out. So pause the video now. So let's reason through what this expression is actually representing. This is b, the number of bears, over the number of cats, plus the number of dogs, plus the number of bears. So there's the number of bears over the sum of all the animals. So this really is the fraction that are bears. Now so this question really reduces to is the fraction that are bears greater than a third, less than a third, equal to a third, or can we not tell? And there's a bunch of different ways to do this problem. And I'll try to expose you to many ways in this video and the next. So let's do a visual one doing different cases. Let's do the scenario where the fraction that are bears are greater than one third. So let's visualize this. So this little diagram right over here. Let's say this is all of the pets in the pet store. And I've divided it into thirds right over here. If the fraction that are bears is greater than one third, then it might look something like this. So this is a third under the dotted lines. If I want to be greater than 1/3, I'll go a little bit more than greater than 1/3. So we go greater than 1/3 right over there. But if the fraction of bears is greater than 1/3, well the fraction of dogs has to be even greater than that. So it has to be at least as much as this blue area. So it's going to be even greater than that. And the fraction that are cats has to be even greater than that. And I didn't even make this one that great. It has to be more than this blue area. And the fraction that are cats has to be even greater than that. And you see, you can't have three things that are all greater than 1/3 adding up to a whole. You can't have all three parts of it being greater than 1/3. So this scenario breaks down. Another way you could have thought about it is if this expression right over here is greater than a third, then c plus d over this has to be less than 2/3. But that means one of them, c or d, would have to be definitely less than a third in order for that to work out. So at least thinking about this way, you know that this thing cannot be greater than a third. Let's think about whether it can be equal a third. Let's imagine if this expression, the fraction that are bears, were equal to a third. So let's make our diagram again. So in this situation, the bears are exactly 1/3. So I'll just color in this third right over here. So exactly 1/3 are bears. But we know that the number of dogs is greater than the number of bears. So the fraction that are dogs has to be greater than the fraction that are bears. So the fraction that are dogs would have to be greater than 1/3. But if the fraction that are dogs are greater than 1/3, then all we have left is something less than 1/3 for the cats. That's actually the largest fraction of the animals are cats. So we know that this also is not a possibility. So what we're really left with is that this has to be less than 1/3. And let's make sure that's reasonable. So, let's paste another example right over here. So if the bears are less than 1/3-- let me make it a little bit more dramatic, so that you can make it clear. So this is less than a third. let's say that the purple, the dogs, let's say they're roughly exactly 1/3. Then you could make a scenario where you have the largest proportion of cats. So this would be the cats right over here. That's not the same color as the cats. Let me do the same color as that pink color. So this right over here is our cats. So this is complete believable. c over c plus d plus b does indeed look bigger than d over c plus d plus b, which does indeed look bigger than b over c plus d plus b. This is completely believable to be less than 1/3. So based on just this visual argument that we just made, you could say hey, look. This right over here is the larger quantity. In the next video, I'll give what I would call more of an analytical argument, where I won't draw diagrams to show that this has to be less than 1/3.