Dividing fractions and whole numbers
- [Instructor] Let's see if we can figure out what 2/3 divided by five is equal to. Pause this video and see if you can figure this out. Well, there is a couple of ways that we can approach it. We can first do it in a conceptual way, think about it visually. To do that, let me represent 2/3, so let's say that what I'm drawing right over here is a whole. This is a square, and it represents a whole. Now, I can divide into three equal sections. I'm gonna try to hand-draw that. So, this is, that looks pretty good, three equal sections here. Each is 1/3, and we have 2/3, so we are really representing all of this stuff right over here. That is two of my thirds. Now, I wanna divide those 2/3 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. So, one, two, three, and then four, and five equal sections. So, what is one of those five equal sections of my original 2/3? Well, this right over here is one of those five equal sections of my original 2/3. 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 2/3. But, what does just one of them represent? And if we figure out what this represents of the whole, then we know what 2/3 divided by five is. Well, when I took my thirds and I divided them into five equal sections, I essentially constructed 15ths. 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 15th. You have three times five, and you could count 'em if you like. And, what we have circled off in red is two of these 15ths. We have 1/15 right over here, and then 2/15 right over there. So, this is going to be equal to 2/15. Now, another way that you could think about, and over time this is the way you will approach it, but it's nice to think about it conceptually, when you divide by any number, it's the same thing as multiplying by the reciprocal. So, five is the same thing as five wholes, or 5/1. And so, 2/3 divided by five is the same thing as 2/3 times the reciprocal of five, or the reciprocal of 5/1, which is you just swap the numerator and the denominator, which is 1/5. And so, another way of thinking about this is this is 1/5 of 2/3, which it once again will be this section right over here. The way you could compute this, conceptually, you see that this is 2/15, but you could also say well, I could just multiply it. When I multiply fractions I can just multiply the numerators. Two times one is two. We do that same red color. Two times one is two, and then I could multiply the denominators. Three times five is 15. And, 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.