 So what about the multiplication of mixed numbers? We can perform the arithmetic of mixed numbers using our two theorems. Any quotient gives us a mixed number, and any mixed number can be converted into an improper fraction. Actually, there's no good reason why we'd want to do this. So, for example, if we wanted to multiply 5 and 3 eighths by 12 and 2 fifths, we could convert our two mixed numbers into improper fractions. Our theorem says that 5 and 3 eighths will become the improper fraction, and 12 and 2 fifths will become... So our two mixed numbers become two improper fractions, which we could multiply. But since our first step is multiplying 43 by 62, we might not want to do that. And in fact, finding the product this way will be painful, and so let's see if we can find a better way. Instead, we'll use an area model for multiplication. Now, if you are very lucky, you were taught how to multiply two numbers this way. If you weren't so lucky, you were taught some other algorithm. But in either case, here's an overview of how an area model works. This is based on the idea that the product A times B is the area of a rectangle with length A and width B. So if I want to multiply 27 by 45, we'll draw a rectangle with the width of 27 and a length of 45. The area of this rectangle will be equal to the product 27 times 45. Now, it defeats the purpose of using the area model if we just multiply 27 by 45. The reason that the area model is useful is that we can break the sides up into smaller, more easily managed pieces. So this length 45, well, that's 40 and 5, while the width 27, well, that's really 20 and 7. And this means our original rectangle can be broken up into four smaller rectangles whose areas are much easier to calculate. So this rectangle has width 20 and length 40, so its area is going to be 800. This rectangle has width 20 and length 5, so its area will be 100. This rectangle has width 7 and length 40, so its area will be 280. And this rectangle will have width 7 and length 5 for an area of 35, and the area of the rectangle will be the sum of the areas of all the smaller rectangles. So the product will be 1215. So how does that work with mixed numbers? Well, let's go back to our problem. We want to multiply 5 and 3 eighths by 12 and 2 fifths, so we'll start with a rectangle that's 5 and 3 eighths wide by 12 and 2 fifths long. Remember that a mixed number is the sum of a whole number plus a fraction, so this 5 and 3 eighths really is 5 and 3 eighths. Likewise, the 12 and 2 fifths is 12 and 2 fifths. So let's pull those numbers apart. And now our original rectangle can be broken apart into four smaller rectangles. And that gives us the areas of four rectangles to compute. So this rectangle, which is 5 by 12, has area 60. The area of this rectangle will be the product of 2 fifths and 5, which is 2. This rectangle here is 12 by 3 eighths, so that area is going to be the product 12 times 3 eighths. This last rectangle has width 3 eighths by length 2 fifths, so its area is going to be the product 3 eighths times 2 fifths. And so the product will be the sum of all of these areas. Now, we can add the whole numbers together, 60 plus 2 plus 4 is. We'll need to add those fractions together, 1 half plus 3 twentieths. Well, we'll add those together and get 13 twentieths. And so our final answer, 66 and 13 twentieths.