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Adding mixed numbers with unlike denominators

Addition and subtraction of fractions

Video transcript

Add. Simplify the answer and write as a mixed number. And we have three mixed numbers here: 3 and 1/2 plus 11 and 2/5 plus 4 and 3/15. So we've already seen that we could view this as 3 plus 1/12 plus 11 plus 2/5-- let me write that down. This is the same thing as 3 plus 1/12 plus 11 plus 2/5 plus 4 plus 3/15. The mixed number 3 and 1/12 just literally means 3 and 1/12 or 3 plus 1/12. And since we're just adding a bunch of numbers, order doesn't matter, so we could add all the whole numbers at once. So we have 3 plus 11 plus 4, and then we can add the fractions: the 1/12 plus 2/5 plus 3/15. Now, the blue part's pretty straightforward. We're just adding numbers. 3 plus 11 is 14 plus 4 is 18, so that part right there is just 18. This will be a little bit trickier, because we know that when we add fractions, we have to have the same denominator. And now we have to make all three of these characters have the same denominator and that denominator has to be the least common multiple of 12 and 5 and 15. Now, we could just do it kind of the brute force way. We could just look at the multiples. We could pick one of these guys and keep taking their multiples, and then figuring out whether those multiples are both divisible by 5 and 15. Or the other way we can do it is take the prime factorization of each of these numbers, and just say that the least common multiple has to contain the prime factorization each of these guys, which means it contains each of those numbers. So let me show you what I'm talking about. If we take the prime factorization of 12, 12 is 2 times 6, 6 is 2 times 3, so 12 is equal to 2 times 2 times 3. That's the prime factorization of 12. Now, if we do 5, prime factorization of 5, well, 5 is just 1 and 5, so 5 is a prime number. It is the prime factorization of 5. There's just a 5 there. This 1 is kind of useless. So 5 is just 5. And then 15, let's do 15. Actually, when I did the prime factorization of 5, I should have said, look, 5 is prime. There's no number larger than 1 that divides into it, so it's actually silly to even make a tree there. And now let's do 15, 15's prime factorization. 15 is 3 times 5, and now both of these are prime. So we need something that has two 2's and a 3, so let's look at the 12 right there. So our denominator has to have at least two 2's and a 3, so let's write that down. So it has to be 2 times 2 times 3. It has to have at least that. Now, it also has to have a 5 there, right? Because it has to be a common multiple of 5. 5's another one of those prime factors, so it's got to have a 5 in there. It didn't already have a 5. And then it also has to have a 3 and a 5. Well, we already have a 5. We already have a 3 from the 12, and we already have a 5 from the 5, so this number will be divisible by all of them, and you can see that because you can see it has a 12 in it, it has a 5 in it, and it has a 15 in it. So what is this number? 2 times 2 is 4. 4 times 3 is 12. 12 times 5 is 60. So the least common multiple of 12, 5 and 15 is 60. So this is going to be plus. We're going to be over 60. So all of these are going to be over 60. All of these three fractions are over 60. Now, to go from 12 to 60, we have to multiply the denominator by 5, so we also have to multiply the numerator by 5, so 1 times 5 is 5. 5/60 is the same thing as 1/12. To go from 5 to 60 in the denominator, we have to multiply by 12, so we have to do the same thing for the numerator. 12 times 2 is 24. The last one, 15 to 60, you have to multiply by 4, so you have to do the same thing in the numerator. 4 times 3 is 12. And now we have the same denominator. We are ready to add. So let's do that. So this is going to be 18 plus, and then over 60, we have 5 plus 24, which is 29. 29 plus 12, let's see, 29 plus 10 would be 39 plus 2 would be 41. It would be 41. And as far as I can tell, 41 and 60 do not have any common factors. 41 actually looks prime to me. So the final answer is 18 and 41/60.