Adding and subtracting fractions with unlike denominators
- [Voiceover] So we want to figure out what 1/4 plus 3/5 minus 3/10 is. And I encourage you to pause the video and see if you could figure this out on your own. All right, I'm assuming you've had an attempt, let's work through this together now. So the first thing that might jump out at you is look, I have these fractions that I'm adding and subtracting, but they all have different denominators. So in order to add and subtract them in a reasonable way, we'd want to rewrite them so they all have the same denominator. So what we really want to do is find a common multiple of four, five and 10. And like I always say, we can use any common multiple, but we simplify things a little bit if we use the smallest common multiple, or the least common multiple. And one way to find the least common multiple is take the largest of these numbers and look at their multiples and keep increasing the multiples until you find one that is divisible by the other two. So for example, I could start at 10. 10 is divisible by five but is not divisible by four. So then we can go 20. 20 is divisible by both five and four, so the least common multiple of four, five and 10 is 20. So let's rewrite all of these fractions as something over 20. So let's start with 1/4. 1/4 is what over 20? Well, to go from four to 20, you have to multiply by five, so you have to do the same thing to the numerator. You have to multiply it by five. One times five is five. One over four is the same thing as 5/20. Four is four times one, 20 is four times five. Then we want to think about, well, what happens to 3/5 If I write it as something over 20? Well to go in the denominator from five to 20, you have to multiply it by four, so you have to multiply the numerator by four as well. So three times four is going to be equal to 12. And then we're going to subtract 3/10, but how do we write that as something over 20? Well, to go from 10 to 20, you multiply the denominator by two, so if we want to have the same fraction, we need to multiply the numerator by two as well. So three times two is six. So what is this going to be equal to? Well now I have 5/20 plus 12/20 minus 6/20. So what is this going to be? Well, there's a bunch of ways you could think about it. You could say, well, this is just going to be, this is going to be 5/20 plus 12/20 plus 12/20 minus 6/20, minus 6/20. Or another way of thinking about it, the way I just wrote it here, five plus 12 minus six, all of that, over 20. This is how many 20ths are going to result. So what is that equal to? Well that's going to be equal to all of this, we are counting 20ths, is one way to think about it. So five plus 12 is 17 minus six is 11. So we get 11/20. I have 5/20, I add 12/20 and I take away 6/20, I'm going to be left with 11/20. Let's do one more example of this, just for good practice. So here I have 4/9 minus 1/6 plus 1/3. And like before, I encourage you to puase the video and see if you can work this out. Well, what I want to do is I want to rewrite all of these fractions so that they have a common denominator. And to find a common denominator, you need to find a common multiple of nine, six and three. And like I did before, what I could do is I could start with a nine and say okay, nine, it's not divisible by six. It is divisible by three, but that's not good enough. I want to find the common multiple of nine, six and three, and as small as one as possible. So let's look at the next multiple of nine, we get to 18. And 18 is divisible by nine of course, six and three. So let's go with 18. Let's write everything as something over 18. So 4/9 is what over 18? Well, nine times two is 18, that's what I did to the denominator, so I have to multiply the numerator by two as well. So this is going to be 8/18. Well, what's 1/6 over 18? Well, to go from six to 18 in the denominator, I have to multiply it by three, so I have to multiply the numerator by three as well. So one times three is three. And then last but not least, what is 1/3 as something over 18? Well, three times six is 18, so one times six is going to be six. Or you could do it another way, you could say, well, what's 1/3 of 18? It's going to be six. What's 1/6 of 18? It's going to be three. 4/9 of 18? That's a little bit harder to think in your brain but that's going to be 8/18. Either way, I've just rewritten these fractions, and I've rewritten them in the corresponding color. So this up here is the exact same thing as 8/18 minus 3/18 plus 6/18. So what is this going to be equal to? It's going to be a certain number of 18ths, and so if I have 8/18 minus 3/18, that's going to be 5/18, plus 6/18 is going to be 11/18. And we are done.