Dividing mixed numbers example

Let's see if we can figure out what four and four-fifths divided by one and one-half is, and I encourage you to pause the video and see if you can figure it out on your own. And I'll give you a hint: see if you can rewrite these mixed numbers as what is sometimes called improper fractions.

All right, now let's do this together. So how can we rewrite four and four-fifths? Well, four and four-fifths is the same thing as if we take the four. That's the same thing as four plus four-fifths. Four plus four-fifths, and four is the same thing as twenty-fifths. So four is the same thing as twenty-fifths, and then plus four-fifths. Well, what are you going to get? You're going to get you add the numerators, you get twenty-four-fifths.

Twenty-four-fifths! Another way to think about it is take this denominator; it takes the fifths, multiply by four, you get twenty-fifths plus the four-fifths that you already have is twenty-four-fifths. And so this is the same thing as twenty-four-fifths divided by the same idea: one and a half is the same thing as one plus one-half. One is the same thing as two halves plus one-half, and so that's going to be add the numerators that's going to be three-halves.

So just like this, we're able to rewrite our expression as twenty-four-fifths divided by three-halves. And now the key realization is that that is the same thing as twenty-four-fifths times the reciprocal of three-halves. So times—pause the video—what's the reciprocal of three-halves? Well, the reciprocal of three-halves, you just swap the numerator and the denominator, is going to be two over three.

Now, what is this going to be? Well, there's a couple of ways to do it. You could just straight up multiply the numerators, and you would get forty-eight, and then multiply the denominators, and you would get fifteen. So you get forty-eight over fifteen. But you might be able to rewrite that in a slower or in a more, sometimes what's called a more simplified way.

But another way of thinking about this is you could just say, well, this is the same thing as twenty-four times two times two over five times three and simplify things before you even multiply them out: five times five times three. And you realize that, look, twenty-four and three are both divisible by three. So let me divide them both by three. So twenty-four divided by three is eight, and three divided by three is equal to one.

And then you could multiply the numerators and the denominators, and so you get in the numerator eight times two is sixteen, in the denominator you get a five. So you get sixteen-fifths, and then if you want to express that as a mixed number, sixteen over five, well, five goes into sixteen three times with one left over. So this is three and one-fifth.

And one thing to appreciate right over here: I simplified the twenty-four and the three at this step. Sometimes you'll see people simplifying at this step, so they'll say, "Hey look, eventually I'm going to have a twenty-four in the numerator and a three in the denominator, so let me divide both of those by three." So they'll say, "Twenty-four divided by three is eight, and then three divided by three is one." And this is sometimes called a cross reduction, but this is all that's going on right over here.