 So we've looked at ratios, and there's an important qualification we have to make here. Strictly speaking, when you find a ratio, you can never have a ratio between different types of quantities. So, for example, you cannot have a ratio between birds and cats because a bird is not a cat. However, because we can assign numbers to many things, we can have a ratio between the number of birds and the number of cats because both of them are numbers. And this allows us to expand the concept of ratio to consider what might be deemed proportionality problems. They're really ratio problems. There is really no difference. The only change that we've made is while we speak about the ratio between two different things, we're really referring to the ratio between the numbers of those two different things. And so, in practice, we don't really care whether we're trying to compare two different things because what we're actually going to be looking at is the number, and we don't have to worry about that. The difference is worth remembering because it does become important later on. Well, let's take a look at the problem. So here's a fairly typical situation. We have a 100-foot length of wire with a given weight of 2 ounces, and I know that the wire that I have in front of me weighs 8 ounces. And I want to find out what its length is. Well, I could stretch it out and measure it, but let's see if there's an easier way of doing that. Now, the starting point here is that the problem actually gives us a relationship between two things. On the one hand, we have a length of wire, 100 feet. On the other hand, we have the weight of the same length of wire, and that's the important thing. We can't compare this length of wire to this weight here because it's not the same wire. But we do know that this length here, 100 feet, has a weight of 2 ounces, and they are the same object, so it is meaningful to consider this ratio between the 100 feet of length and the weight of 2 ounces. And so because I have a relationship between two quantities, there is a ratio involved here, and because I want to focus on the numbers, I'm going to worry about the numbers 102 and not necessarily about the length and weight. Well, there's a little bit of a problem if I focus just on the numbers. I could draw a picture very easily representing this ratio, numbers 100 to 2. So that's my ratio. So I'm going to first draw 100 boxes. Well, I really don't want to do that. 100 boxes, each of these boxes we are going to represent one of our feet rather than drawing 100 boxes each representing one foot. Maybe I'll just draw one box, and that's going to represent 100 feet. And likewise, my 2 ounces, rather than drawing 2 boxes each representing 1 ounce, again, 2 isn't too bad, but we might as well make our lives easier. Instead of 2 boxes each representing 1 ounce, I'll draw 1 box that represents 2 ounces. So this is what my ratio is. 100 feet, 2 ounces, and what's worth noting here is that I've converted in some sense my ratio. Again, the ratio should really be 100 each representing a foot to 2 each representing 1 ounce. But instead I've simplified it and now I have a ratio one of these to one of these. And that's a much easier ratio. It's a 1 to 1 ratio. And here's the important thing. As long as I maintain this ratio, I can compare any length of wire to any weight of wire. So for example, what if I take 2 of each of these boxes? So now I have 2 of these boxes, to 2 of these boxes, and it's worth noting this is still a 1 to 1 ratio, except in this case our 1 is a bigger thing. And now I still have the 1 to 1 ratio, but this time it represents 200 feet and 4 ounces. So now I have a 200 foot length of wire with a weight of 4 ounces, and I can keep adding boxes until I get my 8 ounce weight of wire. So I'll add another box. That's 2, 4, 6 ounces. I'll add one more box. There's 2, 4, 6, 8 ounces of wire here. 1, 2, 3, 400 feet of wire over here. And again, I still have this ratio 1 thing to 1 thing. And so my ratio is still there, but this time I have 400 feet of wire representing 8 ounces of weight. And so I can now answer the question, if I have a coil of wire that weighs 8 ounces, well, here it is. Then what's the length? Well, here it is, 400 feet. Well, it could take a few of this problem from another angle. So again, same problem as before, but this time, more efficiently, we can just start by setting down our ratio boxes. And the real question was, I want to get 8 ounces of weight over here. So how many of these 2 ounce boxes is going to be enough to make 8? So how many 2s make 8? And the answer to that, 4 2s make 8, and I can just fill those out right away. And once again, I'm back at my same solution. 8 ounces of weight, 400 feet of wire. They are the same thing. Now, here's a slightly different way, an even more efficient way of looking at this problem. I don't actually need to write the boxes. Those boxes are there, they're nice, they're useful, they're visual, but they're not really necessary. What's important is I have a whole bunch of ounces over here. I have a whole bunch of lengths over here. And it's worth noting that what I actually have on the right is I have 4 2s. So I can write that as 4 times 2. And over on the left, I have 400s. So I can write that as 4 times 100. And the thing to notice is that I have as many things here as I have over there, which means that the multiplier here is going to be the same as the multiplier over there. And that suggests a very general approach to solving any problem involving ratios. So again, we'll take a look at our problem one more time. And again, we still want to set down our ratios. 100 feet is the same as 2 ounces. Now, the question is, I would like to get 8 ounces of weight. And that's going to be over here in this column. So I'll set that down. And the question at hand is, 2 times what is going to get me 8? How many boxes of 2 do I need to make 8? Well, that's the equivalent to a multiplication problem. 2 times what gives you 8. And while that's actually the same thing as a division problem, what is 8 divided by 2? And I know what that is. That's 4. So if I take these 2 ounce boxes, I take 4 of them. I get an 8 ounce box. Well, I'm going to take the same number of boxes of the 100 foot length. So I'm going to take 4 of those 100 foot boxes. And that gets me 400 feet of length. And I arrive again at my correct answer. That 8 ounce coil of wire is a coil of wire with a 400 foot length. All right, well, let's do a more difficult problem. Under pressure, as you descend underwater, pressure increases by 14 psi that's pounds per square inch for every 10 feet. So if you get a camera that claims it can be taken underwater, there's a little footnote that says it's waterproof to 30 psi. And so the question is, well, how far underwater does that work out to be? Well, again, the statement gives us a relationship, so let's set those down. And this time I know the pressure. So that's going to be over here in this column. So it's 30 psi of pressure. And the question is, well, how many feet of depth is that going to correspond to? So we'll ask the same question. How many of these do I need to make up this? 14 times what gives me 30? So that's the same as asking what is 30 divided by 14 and that is a number that is not going to give me a whole number quotient. Fortunately, we know how to express it as a rational number. 30 divided by 14 is the rational number 30 divided by 30 over 14. I might want to reduce that. That's actually 15 over 7. And I know that 14 times 15 over 7 should give me 30. So I have 15 over 7 boxes on the left-hand side, which means I want 15 over 7 boxes over on the right-hand side. So I'm going to take 15 over 7, multiply that by the 10, and get 150 over 7. And I do want to reduce that to a mixed number that works out to be 21 and 3 7s. So this 30 psi and this 21 3 over 7s feet, they represent the same amount. So 30 psi means you can take that camera down to a depth of 21 and 3 7s feet.