Dividing a whole number by a fraction with reciprocal

In this video, we're going to 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 want to figure out what eight divided by seven-fifths is, we're going to break it down into two steps.

First of all, we're going to use these visuals here to think about how many groups of seven-fifths are in one. Another way of thinking about it is how many seven-fifths are in a whole. So pause this video and just think about this first part.

All right, so let's look at seven-fifths. Seven-fifths is everything from here all the way to there, and then one is this. So how many seven-fifths are in one? Well, you can see that one, which is the same thing as five-fifths, is less than seven-fifths. So it's actually going to be a fraction of a seven-fifth that is one or that is in one.

You can see what that fraction is: One is what fraction of seven-fifths? Well, if you look at the fifths, seven-fifths is, of course, seven of them, and a whole is five-fifths. So, five of the seven-fifths make a whole. The answer right over here is five-sevenths. Five-sevenths of a seven-fifths 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 seven-fifths, and the blue part is equal to one. So how many seven-fifths are in the blue part? Well, we can see it's five-sevenths of the whole bar, once again five-sevenths of the whole bar.

So we can also think about this as one divided by seven-fifths. This is another way of saying how many seven-fifths are in one or how many groups of seven-fifths are on one. This is equal to five-sevenths, which we've learned in other videos is the reciprocal of seven-fifths. The numerator and the denominator are swapped.

So now, what is eight divided by seven-fifths going to be? Well, if one divided by seven-fifths is five-sevenths—or if you have five-sevenths of a seven-fifths of a seven-fifths in one—I know the word 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 seven-fifths, or you could just view this as eight times the reciprocal of seven-fifths, which is five over seven. We've learned how to multiply this before. Eight times five-sevenths is going to be equal to forty-sevenths.

And we're done! You could obviously also write that as a mixed number if you like. This would be the same thing as five and five-sevenths.

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 if you take any other number other than one divided by that fraction, you're essentially just going to 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.