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How to add fractions that have different denominators

Addition and subtraction of fractions

Video transcript

We're asked to add 4/9 and 11/12 and to write our answer as a mixed number, and then simplify and write our answer as a mixed number. So here we have two fractions we're adding together, but we have different denominators. So whenever you add fractions, the first thing you have to do is check the denominators. If they're the same, you can add, but if they're different like this, you have to make them have the same denominator. So what we have to do is find a number that both 9 and 12 will divide into, and that will be our common denominator, and you'll see why both 9 and 12 have to divide into it. So let's think about what that number is, and there's two ways of coming up with that what we could call a least common multiple, the smallest multiple of both 9 and 12 that is common. One way is just to kind of look at the multiples of 9 and see if any of them are divisible by 12. So if you start with 9-- we can do it over here. So you have 9, that's not divisible by 12. 18 isn't divisible by 12. 27 isn't divisible by 12. 36, well, that is divisible by 12. That is 12 times 3. So 9 goes into 36 and 12 goes into 36. So what we want to do is write a common denominator. So we're going to write 4/9 as something over 36, and we're going to write 11/12 as something over 36. Now, to turn your 9 into a 36, you have to multiply it by 4, right? 9 times 4 is equal to 36. Now, you can't just multiply the denominator by 4. You also have to multiply the numerator by the same thing. So if you multiply the numerator by 4, you get 4 times 4 is 16. So 4/9 is the exact same thing as 16/36. If you wanted to simplify this one to 4/9, you divide the numerator and the denominator by 4. Now, we do the same thing over here. 36, 12 times 3, so we're multiplying 12 by 3 to get 36. Well, if we did that to the denominator, we also have to do that to the numerator, so 11 times 3 is 33. And just like that, we've now rewritten each of the fractions so that they have the same denominator. Both of their denominators is 36. So now we're ready to add. If you add these two things, we'll have 36, because we're considering kind of parts of 36 or fractions of 36, and then we have 16 plus 33 in the numerator. Let me write that down. 16 plus 33 in the numerator. And 16 plus 33 is what? 6 plus 33 would be 39 and then you have another 10, so it's 49. So it's equal to 49/36. Now, can we simplify this? 49, it's 7 squared, so it has 1, 7 and 49 as factors. This has 1-- it has a bunch of numbers, but it's not divisible by 7, so this is actually in simplest form, but this is an improper fraction. The numerator is larger than the denominator. So let's write it as a proper fraction. To do that, we divide 36 into 49. 36 goes into 49 how many times? Well, it only goes one time, so it equals 1. And how much will be left over? If I divide 36 into 49 one time, or 1 times 36 is 36, then I have 13 left over to get to 49. So it's 1 and 13/36. And you can do that manually, if you like you. You'd say 36 into 49. 36 goes into 49 one time. 1 times 36 is 36, and then you subtract. 9 minus 6 is 3. 4 minus 3 is 1. You have a remainder of 13. So that's our answer: 1 and 13/36.