Visually dividing a fraction by a whole number

Let's see if we can figure out what 2 over 3 divided by 5 is equal to. Pause this video and see if you can figure this out.

Well, there's a couple of ways that we can approach it. We can first do it in a conceptual way, think about it visually, and to do that, let me represent two-thirds. So, let's say that this, that what I'm drawing right over here is a whole. This is a square and it represents a whole.

Now, I could divide it into three equal sections, and I'm going to try to hand draw that. So, this is that looks pretty good, three equal sections here. Each is a third, and we have two thirds. So, we are really representing all of this stuff right over here, that is two of my thirds.

Now, I want to divide those two thirds by five. Well, the way I could do this is I could divide it into five equal sections, but if I'm doing it, I might as well just divide everything, all the thirds, into five equal sections. So, let me do that: one, two, three, and then four and five equal sections.

So, what is one of those five equal sections of my original two-thirds? Well, this right over here is one of those five equal sections of my original two-thirds. Notice I could draw that, I could draw another one here, another one here, another one there, and another one there, and I would have five equal sections that make up those two-thirds.

But what does just one of them represent? If we figure out what this represents of the whole, then we know what two-thirds divided by five is. Well, when I took my thirds and I divided them into five equal sections, I essentially constructed fifteenths. How do I know that? Well, I had one, two, three thirds, and then I divided it into one, two, three, four, five sections.

So, each of these squares right over here is a fifteenth. You have three times five, and you could count them if you like. What we have circled off in red is two of these fifteenths. We have one fifteenth right over here and then two fifteenth right over there. So, this is going to be equal to 2 over 15.

Now, another way that you could think about it, and over time this is the way you will approach it, but it's nice to think about it conceptually, is when you divide by any number, it's the same thing as multiplying by the reciprocal. So, five over one is the same thing as five wholes, or five.

And so two-thirds divided by five is the same thing as two-thirds times the reciprocal of five, or the reciprocal of five over one, which is you just swap the numerator and the denominator, which is one over five.

And so, another way of thinking about this is this is one-fifth of two-thirds, which is once again will be this section right over here. The way you could compute this conceptually, you see that this is 2 over 15, but you could also say, well, I can just multiply.

When I multiply fractions, I can just multiply the numerators: 2 times 1 is 2. Let me do that same red color: 2 times 1 is 2. And then I can multiply the denominators: 3 times 5 is 15.

Hopefully, what we just drew out may help make sense of why dividing by something is the same thing as multiplying by the reciprocal. Then, when you multiply fractions, it's the same thing as multiplying the numerators to get our new numerator and then multiplying the denominators to get our new denominator.