 Alright, let's take a look at another way of performing multiplication by what we might call decomposition. And a good example of this is what I sometimes call the Abbott and Costello problem, find 7 times 13. And you might wonder why this is called the Abbott and Costello problem. And you might want to do a Google search on videos because it gives you one of the very classic demonstrations of what happens when you don't understand what you're doing when you move digits around when you're multiplying. In any case, the importance of this is that I can, by definition, of what my multiplication is, 7 times 13 is the sum of 713, so I'm going to add 713's together. And something I can do to make my life much easier is to remember that 13 is 10 and 3. So each of these 13's is really a 10 and a 3, so when I add them together, what I'm going to get are 710's and 73's. And the advantage to this is that I might not know offhand what 7 times 13 is, but I know what 7 times 10 is, and they know what 7 times 3 is. And so this is 7 times 10, that's 70 and 21. These two together are 91, so that tells you what 7 times 13 is. Now this is actually an example of a very, very, very useful theorem in mathematics that's known as the distributive property, which tells us that if I have three numbers, a, b, and c, then the product a times the sum of b plus c is the sum of the product a times b plus the product a times c. In other words, I can convert a product that may be more difficult into a sum that may be easier, or I can also go the other direction. I may have a sum that I don't particularly feel like doing today, and I can go and convert that sum back into a product. Now for future reference, when you start doing algebra, there are these two common terms expand and simplify. Roughly speaking, expand goes from this side to this side, and simplify is going to go from this side to this side. And they're really two parts of the same thing, and they're both based on the distributive property. So in the case that we looked at, we did 7 times 13, well I can break that 13 up into a 10 and a 3, and I can distribute that 7 times among the two add ends, and again what makes this useful is that I can find 7 times 10 and 7 times 3 without too much thought, and I can add them together, and I've reduced a somewhat difficult multiplication into a bunch of easier multiplications and an addition, and I can find that product. And the key here is you can apply the distributive property in any number of ways. For example, if I wanted to find 7 times 27, well one possibility is I might break that 27 into 20 and 7, and so when I find that product, that 7 times 20 plus 7 times 7, and maybe I know that's 140 plus 49 is 189. And that works. That's a perfectly good use of the distributive property. On the other hand, many people think in quarters, and so the reason that that's useful is that I know many multiples of 25 very easily. And so this 27, well that's 25 and 2. So that 7 times 25 plus 7 times 2, that's 7 quarters, $1.75, and 14 cents, and once again I can add them together and I get my sum, 189. Another thing that's worth keeping in mind is we can rearrange where we write the partial products. So going back to this particular example, what I did is I had to find 7 times 20 and 7 times 7. And I wrote the partial products and sums horizontally, but I can also write them vertically. 7 times 20, 7 times 7, add them together as my product. And again, here we have the beginnings of something that should look as a very familiar way of multiplying two numbers.