 So let's talk about the arithmetic of fractions. The important thing to keep in mind is that arithmetic is bookkeeping. What we're trying to do is to keep track of how many of which units. If you keep this idea in mind, then the arithmetic of fractions is essentially identical to the arithmetic of the whole numbers. For example, suppose we want to add 1 and 2 thirds and 2 and 3 fourths. So we'll set down 1, and then we'll take 2 thirds of another unit. Next, we'll set down 2 and take 3 quarters of another unit. We'll combine them to get 3 and some additional pieces, which are going to be 2 thirds and 3 fourths. Now, we could write this as 3 plus 2 thirds plus 3 fourths, but we won't. Our fractional parts were thirds and fourths, so a common unit fraction is going to be 3 times 4 twelfths. So these 2 thirds are going to become 8 twelfths, and these 3 fourths are going to become 9 twelfths. Since they're 12ths, we can bundle 12 of them together to form 1, and there are now 5 left over. So our sum is going to be 4 and 5 twelfths. And we can set this up in abstract form as well. Remember, arithmetic is bookkeeping, and by now you have had enough experience in the bookkeeping that you can set up your own books. What about the subtraction? 2 and 1 half minus 1 and 2 thirds. So here we'll want to start with 2 and 1 half, from which we're going to want to remove 1 and 2 thirds. It goes without saying, so I won't say it, we'll remove 1. Now, we want to remove a couple of thirds, but we don't have any. But that's no problem. We'll just make some by cutting this unit into thirds. Now we could remove two of them. And OK, I have to say it, arithmetic is bookkeeping. We want to record how much we have left, which is 1 third, 1 tooth, well, actually we call this a half, and we could write our remainder as 1 third plus 1 half. But we won't, since our unit fractions are 2ths and thirds, a common unit fraction is going to be 2 times 3 sixths. 1 half is 3 sixths, 1 third is 2 sixths, and so what's left over is 5 sixths. For the abstract version, arithmetic is bookkeeping, and you can make your own books. For multiplication, 3 times 2 and 4 fifths, then we'll start by putting down 2 and 4 fifths. So we'll take a unit, cut it into five pieces, and take four of the pieces. And the definition of multiplication says that this product is formed by taking three sets of 2 and 4 fifths. So we'll make three sets. Since our fractions are fifths, we can bundle five of them together to form a single unit. And remember, arithmetic is bookkeeping. So this gives us our final answer, 8 and 2 fifths. We can perform the abstract multiplication, and you know what I'm about to say. So I won't say it. How about the division, 5 and 2 thirds divided by 4? We could read this partitively by taking 5 and 2 thirds, then distributing this amount among four recipients. Each person gets one. To distribute the rest, we'll break one into four equal parts and hand them out. We should remember that each of these equal parts is a fourth. We also have these thirds. So we'll break the thirds into four equal parts. We should remember that each of these equal parts is going to be a 3 times 4 twelfth. And it goes without saying, so we won't say it. And we can determine that each person received one. One of these parts, which was a fourth, and two of these parts, each of which was a twelfth. So our correct but not complete answer is that our quotient is 1, 1 quarter, and 2 twelfths. But we don't leave the answer this way. We do want to write this as a single mixed number. So remember that 1 fourth is 3 twelfths. So all together we have 5 twelfths. And so our quotient is going to be 1 and 5 twelfths. And you know what happens next.