 So, let's take a look at the arithmetic of fractions. And again, the important rule to keep in mind is arithmetic is bookkeeping. So, the basic rule of arithmetic, arithmetic is bookkeeping. So, what you're trying to do is to keep track of how many of which units. And if you keep this in mind, then all of the operations that we do with fractions are essentially identical to the arithmetic of the whole numbers, and you already know how to do arithmetic with the whole numbers. And I'll emphasize, this already know how to do arithmetic means that you understand the basis of the arithmetic with the whole numbers, and not that you are able to push things around on paper. Pushing digits around on paper is not understanding. Understanding is being able to recognize what you're actually doing and how it relates to this process of bookkeeping, how many of which units. So, for example, take this sum, fifteen and three-fifths plus seven and four-fifths. And so, arithmetic is bookkeeping. So, we first want to identify what units we have. And so, we'll set up a place value chart. Since we already know how to handle whole numbers, this mixed number is a whole number and a fraction. So, I'll deal with all of the whole numbers as a single thing. So, our place value chart, our units are going to be numbers. That would be our fifteen and our seven. And our fractions, these are fifths. And so, our units for the fractions are going to be fifths. And what do we have? Well, we're adding. So, we want to combine fifteen and three-fifths. There's our first term, together with seven and four-fifths. So, there's our seven and our four-fifths. And we're adding. So, we want to combine them. So, what do I have here? I have fifteen plus seven. I have twenty-two numbers. And I have three and four. I have seven-fifths. And so, my sum is twenty-two and seven-fifths. Well, I'm not quite done, because when we do arithmetic, we do have this final bundle in trade we often have to deal with. So, because my units are fifths, I know that five is going to get me one. So, I do want to bundle that seven. There's seven-fifths there. And I can take that apart. That's five and two. I can trade five-fifths. Get me one more. I'll go ahead and combine these. I have twenty-two and one. I have twenty-three numbers. And then my final answer, twenty-three and two-fifths. And I do want to write my final answer in the same form it was given. So, because my numbers were originally given as mixed numbers, I do want to write a final answer as a mixed number. Subtraction is no different. We are still doing the same thing. We want to subtract thirty-five minus twelve and two-ninths. So, here our units are going to be numbers and ninths. So, I'll go ahead and set down my place value chart. It's a subtraction problem. So, I can rephrase this as a from something, remove something problem. So, from thirty-five, I'm going to remove twelve and two-ninths. So, this is the amount I'm removing, twelve and two-ninths. I want to remove twelve from thirty-five. Not a problem, but I do want to remove two-ninths from. I have nothing here. So, I have to do something to address that problem. And so, I'll unbundle. So, I'm going to take that thirty-five. I'm going to break it up a little bit. So, there's still thirty-five, but now I'm going to trade that one away. One becomes nine. And now I have twelve. I can remove from thirty-four. Two, I can remove from nine. And what do I have left? Twenty-two, seven. And finally, I want to summarize this in the same form that the question was asked in. This is twenty-two and seven-ninths. So, I'm going to write that as twenty-two and seven-ninths. Multiplication is essentially the same process. And since this is a multiplication problem, we're going to do something completely and totally different from everything else we've ever done with. No, we're going to do pretty much the same thing. We'll set up our place value chart. And so, remember, this is saying that I'm going to have six copies of this two and three-quarters. So, what I'm going to do is I'm going to take my two and three-fourths. There's my place value chart. And I want to multiply that by six. So, I'm going to take six sets. So, that's six twos. Well, I know what that is. And six threes. Well, I know how much that is. And so, what's my answer? Twelve and eighteen-fourths. Well, I don't really want to say eighteen-fourths because I can always bundle and trade. And so, here I have a whole mess of fourths. And I'm going to look for sets of four. So, that eighteen, I'll split off a set of four, another set of four, another set of four, another set of four. And I can trade each of these for one more unit. So, there's a one, two, three, and four. And I'll do that consolidation. I have twelve and a whole bunch of ones. That's actually sixteen over here. And then, my final answer, sixteen and two-fourths. Again, I can write it in this form. Those of you who remember your rules of fractions know that this two-fourths can be reduced till further. Those of you who understand the rules of fractions know that it doesn't really make a difference if you reduce it. There is no particular importance to simplifying fractions other than it makes them easier to write.