Adding fractions with unlike denominators introduction

In this video, we're gonna try to figure out what one-half plus one-third is equal to. And like always, I encourage you to pause this video and try to figure it out on your own.

All right, now let's work through this together, and it might be helpful to visualize one-half and one-third. So this is a visualization of one-half. If you view this entire bar as a whole, then we have shaded in half of it. And if you wanted to visualize one-third, it looks like that.

So you could view this as this half plus this gray third here. What is that going to be equal to? Now, one of the difficult things is we know how to add if we have the same denominator. So if we had a certain number of halves here and a certain number of halves here, well then we would know how many halves we have here. But here, we're trying to add halves to thirds. So how do we do that?

Well, we try to set up a common denominator. Now, what do we mean by a common denominator? Well, what if we could express this quantity and this quantity in terms of some other denominator? A good way to think about it is: Is there a multiple of two and three? And it's simplest when you use the least common multiple.

The least common multiple of two and three is six. So can we express one-half in terms of sixths, and can we express one-third in terms of sixths? So let me just start with 1 over 2. I made this little fraction bar a little bit longer because you'll see why in a second.

Well, if I want to express it in terms of 6, to go from halves to 6, I would have to multiply the denominator by 3. But if I want to multiply the denominator by 3 and not change the value of the fraction, I have to multiply the numerator by 3 as well. And to see why that makes sense, think about this: So this what we have in green is exactly what we had before.

But now, by multiplying the numerator and the denominator by three, I've expressed it into sixths. So notice I have six times as many divisions of the whole bar, and the green part, which you could view as the numerator, I now have three times as many. So these are now sixths. I now have three sixths instead of one half.

So this is the same thing as three over six, and I want to add that. Or if I want to add this to what? Well, how do I express 1/3 in terms of 6? Well, the way that I could do that is 1 over 3. I would want to take each of these thirds and make them into two sections.

So to go from thirds to six, I'd multiply the denominator by two, but I'd also be multiplying the numerator by two. And to see why that makes sense, notice this shaded in gray part is exactly what we have here. But now, we took each of these sections and we made them into two sections. So, you multiply the numerator and the denominator by two.

Instead of thirds, instead of three equal sections, we now have six equal sections. That's what the denominator times two did. Instead of shading in just one of them, I now have shaded in two of them because that one thing that I shaded has now turned into two sections.

And that's what multiplying the numerator by two does. So this is the same thing as three-sixths plus this is going to be two-sixths. Then you could see it here: this is one-sixth, two-sixths. And now that everything is in terms of six, what is it going to be?

Well, it's going to be a certain number of sixths. If I have three of something plus two of that something, well, it's going to be five of that something. In this case, the something is sixths, so it's going to be five-sixths. I have trouble saying that.

And you can visualize it right over here. This is three of the sixths: one, two, three, plus two of the six: one, two, gets us to five-sixths. But you could also view it as this green part was the original half, and this gray part was the original one-third. But to be able to compute it, we expressed both of them in terms of sixths.