Multiplication and division relationship for fractions

You are likely already familiar with the relationship between multiplication and division. For example, we know that three times six is equal to eighteen.

But another way to express that same relationship is to say, "All right, if 3 times 6 is 18, then if I were to start with 18 and divide it by 3, that would be equal to 6." Or you could say something like this: "That 18 divided by 6 is equal to 3."

Now, we're just going to extend this same relationship between multiplication and division to expressions that deal with fractions. So, for example, if I were to tell you that 1/4 divided by—I'm going to color code it—divided by 2 is equal to 1/8, is equal to 1/8. How could we express this relationship but using multiplication?

Well, if 1/4 divided by 2 is equal to 1/8, that means that 1/8 times 2 is equal to 1/4. Let me write this down. Or I could write it like this: I could write that 1/4 is going to be equal to—it's going to be equal to 1/8 times 2 times 2.

And we could do another example. Let's say that I were to walk up to you on the street and I were to tell you that, "Hey, 42 is equal to 7 divided by 1/6." In the future, we will learn to compute things like this, but just based on what you see here, how could we express this same relationship between 42, 7, and 1/6 but express it with multiplication?

Pause this video and think about that. If 42 is equal to 7 divided by 1/6, that means that 42 times 1/6 is equal to 7. So, let me write that down. This is the same relationship as saying that 42 times 1/6 is equal to 7.

Now, let's say I walk up to you on the street and I were to say, "All right, I'm telling you that one-fourth divided by six is equal to some number that we will express as t." So can we rewrite this relationship between 1/4, 6, and t, but instead of using division, use multiplication?

Pause this video and try to think about it. So, if 1/4 divided by 6 is equal to t, based on all of the examples we've just seen, that means that if we were to take t times 6, we would get 1/4. So we could write it this way: t times 6 is going to be equal to 1/4.

If this isn't making sense, I really want you to think about how this relationship is really just the same relationship we saw up here. The only new thing here is instead of always having whole numbers, we're having fractions and representing some of the numbers with letters.