Topic G: Foundations
- [Voiceover] Let's see if we can calculate what 5/6 plus 1/4 is, and to help us, I have a visual representation of 5/6, and a visual representation of 1/4. Notice I have this whole whole, I guess you could say, broken up into one, two, three, four, five, six sections, and we've shaded in five of them, so this is 5/6, and then down here, we have another whole, and we have one out of the four equal sections shaded in so this is 1/4, and I want to add them, and I encourage you at any point, pause the video, and see if you could figure it out on your own. Well, whenever we're adding fractions, we like to think in terms of fractions that have the same denominator, and these clearly don't have the same denominator, but in order to rewrite them, with a common denominator, we just have to think of a common multiple of six and four, and ideally, the smallest common multiple of six and four, and the way that I like to do that is I like to take the larger of the two, which is six, and then think about its multiples. So I could first think about six itself. Six is clearly divisible by six, but it's not perfectly divisible by four, so now, let's multiply by two, so then we get to 12. 12 is divisible by both six and four. So 12 is a good common denominator here. It's the least common multiple of six and four. So we can rewrite both of these fractions as something over 12. So, something over 12 plus something, plus something over 12 is equal to. Now, there's a bunch of ways to tackle it, but what I want to do is I just want to visualize it here on this drawing. So, if I go, if I were to go from, if I were to go from six equal sections to 12 equal sections, which is what I'm doing if I'm going from six in the denominator to 12 in the denominator. I'm essentially multiplying each of these sections by, or, I'm essentially multiplying the number of sections I have by two, or I'm taking each of these existing sections and I'm turning them into two sections, so let's do that. Let's do that. Let me see if I can do it pretty neatly, so, I can do it a little bit neater than that. So, it'll look like that. And, whoops. Let me do this one. I want to divide them fairly close to evenly. I'm doing it by eye so it's not going to be perfect. So, and you have that one. And then last not, last but not least, you have that one there, and then notice, I had six sections, but now I've doubled the number of sections. I've turned the six sections into 12 sections by turning each of the original six into two, so now I have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12 sections. So if I have 12 sections now, how many of those 12 are now shaded in? Instead of having five of the six, I now have 10 of the 12 that are shaded in. So I now have 10/12. 5/6 is the same thing as 10/12. Another way you could have thought about that, to go from six to 12, I had to multiply by two, so then I have to do the same thing in the numerator. Five times two is 10. But hopefully you see that those two fractions are equivalent, that I didn't change how much is shaded in, I just took each of the original six and I turned it into two, or I multiplied the total number of sections by two to get 12, and then instead of having 5/6, I now have 10/12 shaded in. Now let's do the same thing with the four, with the 1/4. Right here, I've depicted 1/4, but I want to turn this into something over 12. So to turn it into something over 12, each section has to be turned into three sections. So let's do that. Let's turn each section into three sections. So, that's one, two, and three. So then I have one, two, and three. I have, I think you can see where this is going. One, two and three. I have one, two, and three. And so notice, all I did is I multiplied, before I had four equal sections. Now I turned each of those four sections into three sections, so now I have 12 equal sections. And I did that, essentially, by multiplying the number of sections I had by three. So now what fraction is shaded in? Well, now, this original that was one out of the four, we can now see is three out of the 12 equal sections. It's now three out of the 12 equal sections, and so what is this going to be? Well, if I have 10/12, and I'm adding it to 3/12, well how many twelfths do I have? I'm going to have 13/12. And you could see it visually over here as well. Up here in green, I have 10/12 shaded in. Each of these boxes are a twelfth. Let me write that down. Each of these boxes are 1/12. That's 1/12. This is 1/12. So how many twelfths do I have shaded in? I have the 10 that are shaded in in green, and then I have an 11/12, a 12/12 and then finally, the 13/12 is one way to think about it.