 So one of the unfortunate things about having to deal with fractions is having to deal with something called a mixed number. And on the one hand, this isn't too bad. On the other hand, the mixed numbers can lead to a lot of confusion later on. So what's a mixed number? Well, it's an expression that contains both a whole number and a fraction. So for example, something like this, we have our whole number, portion three, we have a fraction, portion five eights, and it's called the mixed number because you kind of mix them together in this sort of hodgepodge number. Now, what does it mean? In general, you should always read and rewrite a mixed number as a whole number plus a fraction. So this three five eights, you should immediately rewrite this as three plus five eights. And the reason is that now that it's an addition, all the properties we have for the addition of numbers come into play. So this means commutativity and associativity and everything else. And that's extremely useful because it means that we can handle mixed numbers like we handle whole numbers and like we handle fractions as opposed to having to do something entirely new with them. So for example, let's consider a typical problem. We're going to add two mixed numbers, 17 and five eights to 11 and three fifths. So again, it's very useful to rewrite these mixed numbers as whole numbers plus fractions. So this 17 and five eights, I'm going to split that 17 five eights, the 11 and three fifths, 11 and three fifths. So I split them up into a whole number plus a fraction. And the thing that I can rely on is I know how to add whole numbers already. So associativity and commutativity, say I can rewrite this, I can get the whole numbers together and I can add 17 and 11 immediately. Again, strictly speaking, we can actually go from this expression over here at the 17 and 11 gives us 28, five eights and three fifths are left over. And well, that's a fraction. So five eights plus three fifths, I can add those. I could find the common denominator and then combine them. And one last note is that this is actually an improper fraction. The numerator is greater than the denominator. This is 49 40th. So I'm going to split that up. I'm going to split that into 40 40th. And I know this is equal to one and then I have my leftovers. So that's going to give me one and whole number addition. I can do that. Not a problem. That gives me 29 and 9 40s. And then finally, here's an important idea. The final answer that you write should be expressed in the same dialect as the problem. What is that? Take a look at how the problem is expressed. So we have this 17, five eights. And this is an expression as a mixed number. So what that means is that the question is asked, first of all, it is a mathematical question. That's the language. But the actual dialect is the dialect that includes mixed numbers. And what we want to do is we want to write our final answer in that same dialect. So we start with mixed numbers. We want to end with mixed numbers, which means our final step is we take this perfectly usable expression, 29 plus 9 40th. And we mash it up as 29 and 9 40th as a mixed number. Now the importance of this particular example is that mixed number addition is really addition of whole numbers, addition of fractions, and we can treat the two parts separately. Also, anything that we did to add or subtract two whole numbers or two fractions, we can also apply to the addition or subtraction of mixed numbers. There is no difference in the arithmetical operations of mixed numbers and the arithmetical operations of whole numbers and the arithmetical operations of fractions. So, for example, let's consider the subtraction, mixed number minus another mixed number. And one of the things we did with subtraction is to break the subterhent apart into its two components. So here, this is 15 and 2 7. So I'm going to rewrite that as the mixed number 15 plus 2 7s. But I'm subtracting 8 and I'm also subtracting 3 4ths. So I'll rewrite this problem 15 and 2 7. So there's my mixed number rewritten as a whole number plus a fraction and then minus 8 minus 3 quarters. Now, if you remember our trades before method of doing subtraction, one of the things that we did that made our lives a lot easier is we set things up so that we could just do the subtraction immediately. And so 15 minus 8, not a problem. But at some point I'm going to have to subtract this 3 quarters from something and I really don't feel like doing a subtraction of two fractions. And that's a lot of work. On the other hand, thinking trades before, if I set this up so I can subtract 3 quarters from something, my life will be much easier. What's that something going to be? Well, what I can do is I can split off a 1 from my 15 to help me do the subtraction. Well, so now I have a bunch of whole numbers and I have some fractions that I'll have to deal with. So I'll rearrange things a little bit. I'll subtract the whole numbers 14 minus 8. I can do that, that's 6. And now I want to consider the subtraction 1 minus 3 quarters. So I'll rearrange that and again, 1 minus 3 quarters. Well, that's something I can do without really having to think too hard about that. That's just going to be 1 quarter. And now my horrible subtraction problems become this nice little addition problem. I have a whole number and then I have to add these two fractions 1 quarter plus 3 7s. So I'll apply my common denominator, get my fraction. It's a nice proper fraction, numerator less than the denominator. And again, the dialect the problem was asked is the dialect that includes mixed numbers. And so the final answer that I want to give is also going to be expressed in the same dialect, 6 and 15 28s. Now it's worth keeping in mind that this is just a operations on whole numbers, operations on fractions. And what that means is anything we did to subtract using whole numbers has an analog when dealing with mixed numbers. So let's consider this subtraction problem again. So one of the problems that we had with subtraction, one of the ways we had of solving subtractions was to use this notion of equal add-ins. And the idea is that I don't really want to subtract 8 and 3 quarters. I'm going to increase this by something to make it easy to subtract. And I have to do the same thing to the other number. So let's see, that's 8 and 3 quarters. If I increase this by 1 quarter, that takes me up to 9. So my equal add-ins, I'm going to increase both of my terms by a quarter. So there's my 15 and 2 7s increased by a quarter. 8 and 3 quarters increased by a quarter is 9. Mixed numbers rewritten as a fraction plus a whole number. And I can handle 15 minus 9, my whole number subtraction. And then finally, I have this fraction to add, 15, 28. And again, I arrive at the same answer. Or there's another way I could do that subtraction. Again, same subtraction problem. But here's one where counting up might actually be easiest to use. Because here I have 8 and 3 quarters. And so I'm going to go and count up from 8 and 3 quarters up to 15 and 2 7s. So let's see, easy step. If I increase by a quarter, that takes me to 9 plus 6 takes me to 15 plus 2 7s, takes me to my final answer 15 and 2 7s. And how far did I count up? Well, it's 6 plus a quarter plus 2 7s. And once again, I have this expression here that I can add as a set of fractions and get my final answer, 6 and 15, 28s.