Mixed number addition with regrouping

Let's see if we can add five and two-fifths to three and four-fifths. Pause this video and see if you can figure out what this is.

All right, now let's do this together. We've had a little bit of practice adding mixed numbers in the past, and so one way to think about it is you could view five and two-fifths as five plus two-fifths. To that, we're going to be adding three and four-fifths, which you could view as three plus four-fifths. Then you could just change the order with which you are adding and say, "All right, well I could say five plus three." So that's five plus three, and then to that, I could add two-fifths and four-fifths. So plus two-fifths plus four-fifths.

What is that going to get me? Well, five plus three is going to be equal to eight. Eight plus, and then if I have two-fifths and I add four more fifths to that, well now I'm going to have six-fifths. Two of something plus four of that something is going to be six of that something, and the something in this case are fifths. So now I'm going to have six-fifths.

Some of you might be tempted to say, "Hey, isn't this just going to be equal to eight and six-fifths?" and you wouldn't be completely wrong if you said that. But pause this video and think about why this feels a little bit off. Well, the reason why this isn't standard is that the fractional part of this mixed number, six-fifths, is greater than one. So there's a whole inside of this six-fifths.

The standard way to do this is to see if we can break out that whole. What do I mean by that? Well, I could rewrite eight plus six-fifths as eight plus six-fifths. This is the same thing as a whole or five-fifths plus one-fifth. And why is this useful? Well, five-fifths is the same thing as one. So now I can say this is going to be equal to eight plus one whole is nine. That's eight plus the fifths, and then what I have left over is one-fifth. So nine and one-fifth, and this is the direction that people will traditionally go in.

Now, there's another way that you could approach it, which is really the same idea. We're just writing things a little bit differently. We could write this as five and two-fifths plus three and four-fifths. Notice the way that I wrote it; I put all the fractions in the fraction column, I guess you could call it that way, and I put all of our whole numbers underneath each other. If I had multiple digits here, I would align them according to place value.

Then what we could do is say, "Okay, two-fifths plus four-fifths is going to be six-fifths." We could write six-fifths there. We say, "Hey, there's something a little bit fishy about six-fifths." That's really the same thing as five-fifths plus one-fifth, or you could say that's the same thing as one and one-fifth. Six-fifths is equal to one and one-fifth.

So what you could do is you could write the one-fifth part in the fractions column, and then the one, well now you're going to be regrouping that into our whole numbers. So you put a one right over there. Notice, two-fifths plus four-fifths is one and one-fifth, which is the same thing as six-fifths.

Then you add the whole number parts: one plus five plus three is nine. So you get nine and one-fifth, but hopefully, you realize that these are really the same idea, just different ways of writing things.