 So rather surprisingly, multiplication of fractions turns out to be a little bit more difficult conceptually than the division of fractions. And part of that is we actually have two ways of looking at division. We can either view it quotatively or partatively. With multiplication, we only have one good way of looking at it, which is as we're heated addition, which is fine if we're doing a whole number multiple, but if we're doing a fractional multiple, it's a little bit more complicated. So let's start off with the easy problems first. Let's find the product 5 times 2 thirds. And so by our definition of multiplication, what that means is we're going to take five things each of which are 2 thirds. So we're going to produce a piece of size 2 thirds. So we can take a bar divided into three equal parts, and I'm going to take two of those, and so this represents 2 thirds, and I want five of those all together. So there's one, two, three, four, five, and this forms my product. Now, how big is that product? The thing to keep in mind is that three of these rectangles made up our one unit. So what I can do is I can count holes by finding sets of three rectangles. So here we go, one, two, three, and there's one third left over. And so I can write my result as a mixed number. Five times 2 thirds, one, two, three, and one third left over, and there's my product 3 and a third. Now, unfortunately, this doesn't work if we multiply a fraction by a fraction, because there's something like 2 fifths times 3 quarters, and we don't have any good way of taking 2 fifths, 3 quarters, and adding them together. It's kind of hard to wrap our minds around what that is. So let's take the problem apart. So first off, this 2 fifths, if we think about what that looks like, this is the same as 2 times a fifth. It's a fifth and a fifth that's a repeated addition, so it translates into a multiplication. Again, if you think about where our fractions and rational numbers came from, this 1 fifth itself comes from a quotient 1 divided by 5. So this product, 2 fifths times 3 quarters, I can view this as 2 times 1 divided by 5 times 3 quarters. And because I can do my division and multiplication in any order that I want to, I'll rearrange things a little bit. I'll do the 2 times 1 because I can, and then the divide by 5 I'll relegate to after the 3 quarters. So this product, 2 fifths times 3 quarters, is the same as 2 times 3 quarters divided by 5. And what that suggests is I might take the 3 quarters, divide it into 5 pieces, and then take 2 of the pieces that I produce. So let's take a look at that. So we'll get our 3 quarters first, so we'll take a bar. We're going to divide it into 4 parts. I'm going to take 3 of the 4 parts, and there's my area representing 3 quarters. Now to do the multiplication by 2 fifths, I want to divide this into 5 equal pieces. There we go, and I'm going to take, so here's my 1, 2, 3, 4, 5 equal pieces, and I want to take 2 of those equal pieces. So there's my 2 pieces that I'm going to take. Again, here's 1 piece is this entire thing, there's a second piece, and so I'm going to take those 2 pieces, and I'll just count. We have 1, 2, 3, 4, 5, 6 pieces out of a total of 20, and so my product 2 fifths times 3 quarters is going to be 6 out of 20, 6 over 20, and I could reduce this to the proper reduced value, 3 out of 10. However, we're not going to do that right at this point.