Adding mixed numbers with regrouping

- So we have here 2 2/3 plus 8 3/4 and I encourage you to pause the video and see if you can compute what this is. So now let's work through this together and there's a bunch of ways that you can tackle this. I'll do it in a couple of different ways. So one way that you might see is to just rewrite this expression, so just rewrite 2 2/3 plus, plus 8 3/4. Plus 8 3/4 and you might be saying why whould I rewrite it this way? Well, when you rewrite it this way, it's very clear that these are the fraction parts of the mixed numbers and these are the whole number parts of the mixed numbers, and maybe we could add them separately, but before we even start adding the fraction parts of the mixed numbers, we notice that we have different denominators here. We have a three and we have a four, so it would be nice to find a common denominator and we've seen multiple times in the past, when you're adding fractions with unlike denominators, a good common denominator would be their least common multiple, the least common multiple of three and four. Well, what's that going to be? Well, let's look at the multiples of four. Four is not perfectly divisible by three. Eight is not perfectly divisible by three. 12 is, in fact, 12 is four times three, so I can rewrite both of these fraction parts with 12 as their denominator. So this is going to be equal to, let me just write it this way. So I can rewrite it as two and something over 12. Now what would be that something over 12? To go from three to 12 in the denominator, we multiply it by four, so we have to multiply the numerator by four as well. So two times four is eight. 2/3 is the same thing as 8/12, so 2 2/3 is the same thing as 2 8/12, and then we can do the same thing down here. 8 3/4 is gonna be the same thing as eight and something over 12. Well, to go from four to 12 in the denominator, you multiply it by three, so we have to multiply the numerator by three as well. Three times three is nine, and now, we can add. We can add, and what do we get? Well, we have 8/12 plus 9/12. That's going to be 17, 17/12. Well, you might recognize, 17/12, this is greater than or equal to one, so we can actually, this is an improper fraction right over here, so we can actually rewrite this, we can rewrite this as a mixed number. Let me make it clear. 17/12, so, do it right over here. 17/12 is the same thing as 1 5/12. How did I figure that out? 12 goes into 17 one time, and then I still 5 left over. 17/12 is the same thing as 1 5/12. So what I could do is, I could write the 5/12 part in the fraction part of our, or I guess, say, the fraction place on our numbers, and then I could regroup the one, put it in the whole number column, and now I can just add the one plus two is three, plus eight is 11. So we get 11 5/12, 11 5/12. Now there is other ways that we could have tackled this. What we could've done is gone from this place right over here, the 2 8/12, and the 8 9/12, and we could've converted these into improper fractions then added those improper fractions, so we could've said, and this is the same thing, let me rewrite it as, let's see, two is the same as 24/12 plus 8, is going to be 32/12, so I could rewrite this top mixed number as 32/12, clearly an improper fraction, and I could rewrite this bottom mixed number as, let's see, eight is how many 12s? That's going to be 96/12 plus another 9/12, is 105/12. 105/12. If what I just did looks a little bit confusing, you should review the Khan Academy videos on converting mixed numbers to improper fractions or vice versa, but now I could think about adding these two, and what, what would I get out? 32/12 plus 105/12, which would be, what, 137, 137/12, and then if we wanted to rewrite this as a mixed number, we could say, OK let's see, 12 will go to 137, 11 times, so it goes 11 times. 11 times 12 is 132, so you would have five left over, so it's 11 5/12. So you could've done it either way. I think this one, the way we did it the first time would've been a little bit easier because you didn't have to deal with these really big numbers like 137 and 105 and 32, and you just dealt with more straightforward numbers. Say, hey look, 8/12 plus 9/12 is 17/12, which is the the same thing as 1 5/12, and then you had 1 5/12 plus two plus eight, is gonna give you 11 5/12.