Topic G: Repeated addition of fractions as multiplication
We've already seen that the fraction 2/5, or fractions like the fraction 2/5, can be literally represented as 2 times 1/5, which is the same thing, which is equal to literally having two 1/5s. So 1/5 plus 1/5. And if we wanted to visualize it, let me make a hole here and divide it into five equal sections. And so this represents two of those fifths. This is the first of the fifths, and then this is the second of the fifths, Literally 2/5, 2/5, 2/5. Now let's think about something a little bit more interesting. What would 3 times 2/5 represent? 3 times 2/5. And I encourage you to pause this video and, based on what we just did here, think about what you think this would be equivalent to. Well, we just saw that 2/5 would be the same thing as-- so let me just rewrite this as instead of 3 times 2/5 written like this, let me write 2/5 like that-- so this is the same thing as 3 times 2 times 1/5. And multiplication, we can multiply the 2 times the 1/5 first and then multiply by the 3, or we can multiply the 3 times the 2 first and then multiply by the 1/5. So you could view this literally as being equal to 3 times 2 is, of course, 6, so this is the same thing as 6 times 1/5. And if we were to try to visualize that again, so that's a whole. That's another whole. Each of those wholes have been divided into five equal sections. And so we're going to color in six of them. So that's the first 1/5, second 1/5, third 1/5, fourth 1/5, fifth 1/5-- and that gets us to a whole-- and then we have 6/5 just like that. So literally 3 times 2/5 can be viewed as 6/5. And of course, 6 times 1/5, or 6/5, can be written as-- so this is equal to, literally-- let me do the same color-- 6/5, 6 over 5. Now you might have said, well, what if we, instead of viewing 2/5 as this, as we just did in this example, we view 2/5 as 1/5 plus 1/5, what would happen then? Well, let's try it out. So 3 times 2/5-- I'll rewrite it-- 3 times 2/5, 2 over 5, is the same thing as 3 times 1/5 plus 1/5. 2/5 is the same thing as 1/5 plus 1/5. So 3 times 1/5 plus 1/5 which would be equal to-- well, I just have to have literally three of these added together. So it's going to be 1/5 plus 1/5 plus 1/5 plus 1/5 plus-- I think you get the idea here-- plus 1/5 plus 1/5. Well, what's this going to be? Well, we literally have 6/5 here. We can ignore the parentheses and just add all of these together. We, once again, have 1, 2, 3, 4, 5, 6/5. So once again, this is equal to 6/5. So hopefully this shows that when you multiply-- The 2/5 we saw already represents two 1/5s. We already saw that, or 2 times 1/5. And 3 times 2/5 is literally the same thing as 3 times 2 times 1/5. In this case, that would be 6/5.