Adding and subtracting fractions with unlike denominators
- [Instructor] In this video we're gonna try to figure out what 1/2 plus 1/3 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 1/2 and 1/3. So this is a visualization of 1/2 if you viewed this entire bar as whole, then we have shaded in half of it. And if you wanted to visualize 1/3 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. And 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 and the least common multiple of two and three is six. So can we express 1/2 in terms of sixths and can we express 1/3 in terms of sixths? So we can just start with one over two and I made this little fraction bar a little bit longer 'cause you'll see why in a second. Well if I wanna express it in terms of sixths, to go from halves to sixths, I would have to multiply the denominator by three. But if I want to multiply the denominator by three and not change the value of the fraction, I have to multiply the numerator by three 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 if I multiply it 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. So I now have 3/6 instead of 1/2. 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 sixths? Well the way that I could do that, it's one over three, I would want to take each of these thirds and make them into two sections. So to go from thirds to sixths 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. And so this is the same thing as 3/6 plus this is going to be 2/6. And you can see it here. This is 1/6, 2/6, and now that everything is in terms of sixths, 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 5/6. 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 sixths, one, two, gets us to 5/6. But you could also view it as this green part was the original half and this gray part was the original 1/3, but to be able to compute it, we expressed both of them in terms of sixths.