 One of the important reasons for introducing fractions is it allows us to talk about mixed numbers. And in particular, what we're going to be able to do is that we're able to express quotients as a fraction. And this is easiest to see if we use our concrete representations of number and our concept of division as a quotative division. So remember that when we're dealing with a quotative division, the divisor tells us the size of each share, how big the piece of cake is, and the quotient is going to be the number of recipients. So for example, let's consider the division 11 divided by 4. And so for that, I'm going to take 11 objects, so for example, 11 flashlights. And I'm going to try and arrange these into sets of 4. So I'll form a set of 4, another set of 4, and at this point I don't have enough to form another set of 4. So I can say that I have two complete sets and I have three objects left over. And previously we'd have written this quotient as 11 divided by 4 is equal to 2 with remainder 3. However, let's take a close look at what our remainder is. Now a full set would be four flashlights, four objects. And so in some sense, this last group of flashlights here, it's missing a flashlight to form a full set. Well, how would we describe this amount? Well, our full set is a unit. Each individual flashlight is one of one, two, three, four objects that should be in that unit. And so each flashlight is one fourth of the set. And so this remainder 3 is going to be represented by the fraction 3 fourths. So rather than saying our quotient is 2 with remainder 3, we can say instead that a quotient is 2 and 3 fourths. Then this combination of a whole number and a fraction is referred to as a mixed number. And when we write it, we do the same thing we do with whole numbers and we omit the word and. And so we write this 2, 3 fourths. Well, let's take a look at another example. Let's say I want to find 5 divided by 2 thirds. And so if we view this as a partative division, well, this is kind of complicated because we have to figure out what we mean to give out 2 thirds sets. On the other hand, we can view this as a quotitive division. And so what this says is that we have 5 objects. Well, how about 5 pieces of cake? And we want to distribute portions that are the size 2 thirds of each of these objects. And so what that means is that each of these cakes, each of these pieces of cake is going to be divided into three parts and one serving is going to be two of those parts. So let's cut some cake. So we've cut the cake into three parts. One serving is two of those three parts, is two thirds of a piece. And so here the first person gets their serving of cake. The next person, well, there's only one third of a piece there. So we need to cut the next piece into three parts so that the next person could also get two thirds. So they get their serving. We can give one more serving out. And we say, well, that's it, no more cake. Well, actually, there's still cake left over. So we'll go ahead and serve the next person. The next person. Another person. Another person. And then one last person comes in and they don't get a full serving of cake. They were a little bit late to the table and they don't have enough cake to get a full serving. So, well, how much cake do they get? Well, again, it's worth keeping in mind that a full serving of cake consists of two thirds of a slice. And so that's going to look something like any of these. And so what this last person has is they have one of the two pieces of cake that they should be having. They have one half of a serving. And so that gives us our quotient. One, two, three, four, five, six, seven, and a half. And so our quotient, five divided by two thirds, is equal to seven and a half.