If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Dividing a whole number by a fraction with reciprocal

Dividing by a fraction involves finding the reciprocal and multiplying. For example, when dividing 8 by 7/5, first find the reciprocal of 7/5, which is 5/7. Then, multiply 8 by 5/7, resulting in 40/7 or 5 5/7. This method simplifies fraction division and enhances understanding. Created by Sal Khan.

Want to join the conversation?

Video transcript

- [Instructor] In this video, we're gonna do an example that gives us a little bit of practice to think about what does it mean to divide by a fraction? So if we wanna figure out what eight divided by 7/5 is, but we're gonna break it down into two steps. First of all, we're gonna use these visuals here to think about how many groups of 7/5 are in one. Or another way of thinking about it is how these 7/5 are in a whole? So pause this video, and just think about this first part. All right, so let's look at 7/5. 7/5 is everything from here, all the way to there. And then one is this. So how many 7/5 are in one? Well, you can see that one, which is the same thing as 5/5, is less than 7/5. So it's actually going to be a fraction of a 7/5 that is one, or that is in one. And you can see what that fraction is. One is what fraction of 7/5. Well, if you look at the fifths, 7/5 is of course seven of them, and a whole is 5/5, so five of the 7/5 make a whole. So the answer right over here is 5/7. 5/7 of a 7/5 is equal to one. You can also see this right over here. If you take each of these to be a fifth, each of these to be a fifth, then this whole bar is equal to 7/5. And the blue part is equal to one. So how many 7/5 are in the blue part? Well, we can see it's 5/7 of the whole bar. Once again, 5/7 of the whole bar. So we can also think about this as one divided by 7/5. This is another way of saying how many 7/5 are in one, or how many groups of 7/5 are in one? And this is equal to 5/7, which we've learned in other videos is the reciprocal of 7/5. The numerator and the denominator is swapped. So now what is eight divided by 7/5 going to be? Well, if one divided by 7/5 is 5/7, or if you have 5/7 of a 7/5, of a 7/5 in one, I know the oral language gets a little bit confusing. Well, you're going to have eight times that many in eight. So this is going to be the same thing as eight times, we could do it this way, eight times one, divided by 7/5. Or you could just view this as eight times the reciprocal of 7/5, which is five over seven. And we've learned how to multiply this before. Eight times 5/7 is going to be equal to 40/7, and we're done. You could obviously also write that as a mixed number if you like, this would be the same thing as 5 5/7. So the big picture is when we think about how many of a fraction are in one, that's the same thing as saying, what's one divided by that fraction? As you see visually here, you essentially get the reciprocal of that fraction. And so if you take any other number other than one divided by that fraction, you're essentially just gonna multiply it by that reciprocal, because it's that number times one. So when you divide by that fraction, it's that number times the reciprocal.