Subtracting Fractions: Do You Really Need the LCD?

Added: 2026-05-31

[00:00:00]All right. So, here's the problem. 916 - 3 over 20 3 20ths. So, when we're talking about fractions, let's talk about the different operations that you can face with fractions. Okay. So, we can multiply fractions. We can divide fractions. These uh operations when we have two fractions, we're multiplying or dividing. This has nothing to do with the LCD. Okay? Has nothing to do with LCD. Matter of fact, this is quite easy uh to do these operations. But when we're adding and subtracting fractions, this is when we have to be thinking about the lowest common denominator.

[00:00:38]Now, sometimes we don't need the LCD, but typically students when they struggle with fractions, it's because they are confused with the lowest common denominator and they just get really scared of fractions. And typically fractions get a bad reputation. You know, people don't like fractions. It's like, you know, uh I hate fractions. I don't want to do fractions. Listen, I get it. But once you understand this and you uh really have a mastery over it, you're like, "Oh, no. I totally understand it." Then you know, it's not going to be so bad. So, let's go ahead and figure this out. All right. So, when it comes to fractions, again, adding and subtracting fractions, let's just look at two fractions here. So, when we have a fraction like 9 over 16 or 3 over 20, we have two numbers, right? That basically make up a fraction. We have a top number and a bottom number. That top number is called the numerator. The bottom number is called the denominator.

[00:01:33]So here uh in this case the numerator is 9, the denominator is 16. Here the numerator the numerator is 9 and the denominator is 16. So here we have two fractions a numerator and a denominator.

[00:01:45]We want to add another fraction has a numerator and denominator. So the deal is when it comes to adding and subtracting fractions. Okay, it's basically the same uh procedure. The bottom number, i.e. the denominators must be the same number. Okay, so you could see in this situation, they are not the same. So, we're going to have to kind of fix this up. But let's take a look at a situation uh where the um numbers are the same, when we have the same denominator. Now, again, when the uh bottom numbers or the denominators are not the same, that's when you need to be thinking LCD. And a lot of you um you know if I gave you nice easy numbers like 1/3 + 15 uh you could tell me the LCD. A matter of fact a lot of you might might be saying oh what's the LCD? Uh the LCD is 15. Look I know the LCD. It's super easy. I'm like oh very good, very impressive. But what about these numbers? What if I had 1 over 508 and uh let's say 1 over 36? What's the LCD?

[00:02:51]Well, you know, this is where we get these expressions, right? People are like, "Well, okay, listen. I only want to do math when the problems are nice and easy." It doesn't work that way, but this is not that difficult. So, let's go ahead and get into this right now. Okay.

[00:03:03]So, when we do have the the fractions are set up with the denominators being the same. Like in this particular situation, there's no need for the LCD.

[00:03:14]Uh let me just go ahead and just tell you what the LCD stands for. Okay? So, we're going to get into this um in detail, but this stands for the lowest common denominator. So, C D right here, C D is what? Common denominator. The denominators have something in common.

[00:03:32]What would they have in common? Maybe they're the exact same number, right?

[00:03:36]So, when you're looking for the LCD, we want to find the lowest number that uh the denominators can be in common with, i.e. the same. So these uh fractions right here have common denominators. All right? In other words, the denominators are already the same. So this fraction problem is already set up to uh solve.

[00:03:57]Okay? It's like going to be so so easy to do this. So notice that um addition and subtraction work the same way. It's the same procedure. So uh what you need to do is you just keep the denominator.

[00:04:09]So in this case, it's five. And then we're going to perform this operation with numerators. We're going to take this numerator. We're going to subtract it from that numerator. It's very very easy. We're just going to go ahead and do this operation. Uh this will u um be our numerator. So we have 3 - 1 is 2 or 2. And there you go. Okay. So this is we're going to have to get to this stage and do this. Um but we can't do this part in in in this particular problem yet because the denominators are not the same. But as soon as we get those denominators the same, we're basically going to do this step. Okay. All right.

[00:04:45]So, now that you kind of see the big picture, let's go ahead and figure out how to find the lowest common denominator. Now, stylistically, some of you might write fractions this way with little angle fraction bar.

[00:04:58]Perfectly fine. Write a fraction like this. Uh, but this is equivalent to uh writing a fraction with a nice little horizontal uh fraction bar like so. Me personally, you're better off writing uh more I like to write fractions this way, but you also see fractions written this way as well. Especially like if you're, you know, working with let's say carpentry or or you're using a ruler. This is a pretty common. Both are perfectly fine.

[00:05:24]I'm not trying to make any uh judgments here on it. But uh I would suggest uh writing fractions getting used to writing fractions as horizontal fraction bar like this because this is uh basically the way we do this in algebra.

[00:05:37]Okay, we don't really use this notation much. We kind of u stylistically write fractions this way, but they mean the same thing. So let's go ahead and rewrite the fractions this way. Okay.

[00:05:49]And again, I'm looking at the problem.

[00:05:51]I'm saying, "All right, these do not have uh the same number, right? So, we're going to have to figure out what number, okay, does 16 and 20 have in common?" All right. Well, 16 and 20 have uh multiple numbers. Matter of fact, they have infinite numbers in common.

[00:06:10]Now, what we're looking for are multiples. Now, I'm just kind of explaining this. This is not the procedure, but what says we have 16 * 1 is 16. 16 * 2 is 32. Okay. And then we have 20 * 2 is 40. 20 * 3 is uh 60 etc. These are what we call multiples. Okay.

[00:06:32]And I can go on with 16. 16 * 3 16 * 4 etc. These are multiples. So the lowest common denominator is when we have a common multiple. All right. So, uh, if you, you know, you don't really understand this, that's, you know, totally, you know, I I get this because most students don't truly understand the LCD. Another way to think of the LCD, let's kind of go back here. Uh, 1/3 + 1/5. And we know that the LCD, lowest common denominator is 15. I'm assuming most of you out there do know this because this is a pretty easy problem.

[00:07:07]Another way to think of this as uh the LCD as it's the lowest number that both these numbers divide into evenly. Okay, so three and five they divide into 30.

[00:07:20]Okay, so you could divide um uh 30 by three and you could divide 30 by five, no problem. But this is not the lowest number that you could divide um these two numbers into. 15 is the lowest number. That's another way to kind of conceptually think of the LCD. Okay, the LCD is actually the lowest common multiple. Okay, uh it's cousin, the cousin of the LCD is the LCM. Lowest common multiple of these numbers down here in the denominator. Right? So, you know, this is very important stuff because in algebra when you're dealing with fractions with variables, which are called rational expressions, you have to have a real strong understanding of multiples and LCD, etc. So, you know, you're better off learning this right now with arithmetic and you can transfer this knowledge over when you study algebra. Okay, so let's get into this.

[00:08:16]All right, so we want to find the LCD.

[00:08:17]We want to find uh the lowest common multiple both of these numbers um go into. Or another way to think about it is what is the lowest number that both of these numbers can go into? Well, we can find that. That is called the lowest common denominator. So, what we need to do is find the prime factors of each of these denominators. Okay? So, we're going to have to do some work here. Not that difficult of work. We're going to have to find the prime factors of this number and this number. And then whatever prime factors we have between this number and this number. And by the way, if we had other denominators, um they're all your denominators, all the prime factors. We have to have each unique prime factor represented in our LCD. So what we're going to do is we're going to find each prime factor. Okay, unique prime factor. And then we're going to multiply all those unique prime factors together. The product of all those unique prime factors will be the LCD. Now, this may seem a little bit confusing. you know the way I explained it I think when you see this in action it will be uh much much more understandable if that's even grammatically correct the way I said that but let's move on okay you don't want to learn English for me uh math is my specialty okay so 916us 3 20ths okay so again we're going to need the LCD which means we need to find the prime factors of the respective denominators and the easiest way to find the prime factor is prime factors of numbers is to build a factor tree. Okay, so if you've never seen a factor tree, this is what they look like. Super easy.

[00:09:51]Let me show you. Let's just focus in here on 16 and then I'll go through 20 and then we'll talk about the rest of this. So, what you want to do is figure out two numbers that basically two factors of 16 other than 1 and 16. Now, you could go, oh, 16 is 4 * 4. That's perfectly fine. You could start that way. Doesn't make a difference how you start your factors. In other words, like 20 I could go 2 * 10 or 4 * 5. Your final answers will be the same. Okay, your factor the way you factored out your um your way your factor tree looks might look different than say this, but your final answer will be the same. So don't worry about that. So here we go.

[00:10:30]So 16 is 8 * 2. Now these are factors on 16. So you have to look at the factors and you have to ask yourself um do we have any prime factors? Well, two is a prime number. So, what you want to do is circle two. 8 is not a prime number.

[00:10:48]Okay? In other words, I could factor this down further. So, let's go ahead and factor uh eight further. So, that would be 4 * 2. And look, oh, we have another prime factor here. So, circle that. And so, we look at each factor and we keep factoring each factor that can be factored. So, four can be factored as 2 * 2. So, we're going to circle all the prime factors. So 16 is going to be equal to 2 * 2 * 2 * 2. All right, these are the prime factors of 16. But you always want to express uh prime factors if you have more than one as a power.

[00:11:24]It's very very important and I'm going to show you why here in a second. So 16 is equal to 2 * 2 * 2 * 2 or 2 to the 4th power. Okay. All right. Now let's go to take a look at 20. So 20 is 2 * 10. 2 is pr uh prime. 10 is not. So we can continue to factor. So 10 is the same thing as 2 * 5. And these numbers are prime. So 20 is equal to 2 * 2 * this five. All right. And again we can write 2 * 2 as 2^ 2ar. All right. So this is really really really important. And then we have five. Okay. So we have the prime factors of all the numbers um in the denominators. In this case, we only have two numbers. We're only adding two or subtracting two numbers, but we had if you were subtracting or adding three fractions. You could have another number over here. Let's say it was 70. And you would do the same thing as well. Okay?

[00:12:21]But here is a prime factor. Okay? Uh here 2^2 is another prime factor and five is another prime factor. Okay? So now let's go ahead and build our LCD.

[00:12:35]So this is where if you don't follow this little procedure I'm telling you, you can get uh confused in terms of your final answer. So your lowest common denominator is going to be your prime factors time your your uh other prime factors when it comes to um the factors of the numbers in the denominator. So let's go ahead and uh do the obvious one. Five is a prime factor. So we're going to need a five represented in our LCD. Okay, that is a prime factor.

[00:13:05]So now we have 2 to the 4th and 2^2ar.

[00:13:09]So the question is well 2 is certainly a prime factor. So should we write two?

[00:13:16]Okay uh well or uh should we write 2 to the 4th and then another 2^ squared?

[00:13:24]Which is correct? What should we do here? Because we have to represent two.

[00:13:28]Well here is what you do. Okay. All right. So here this is 2 to the 4th and here is 2 to the 2 power. The deal is when you're building out your LCD you always take the highest power of the number. Okay? So we need a two represented but this is 2 to the 4th.

[00:13:46]This is 2 ^2 you always take the highest power. So that's 2 to the 4th and then we don't have to worry about that 2^2.

[00:13:52]So this is the LCD. Okay. So what does this mean? Well, uh, the LCD, the lowest common denominator is going to be 5 * 2 4th power. And what is that equal to?

[00:14:04]Well, let's go ahead and figure this out now. So, the LCD is 2 4th * 5. So, 2 4th is 16. 16 * 5 is 80. Okay. Now, a lot of you might have come up with this answer.

[00:14:16]You're like, "Oh, I know it was 80. You know, I didn't need to listen to you ramble for five minutes explaining this." But again, all right, what if I made the denominators like crazy? What if I made it like 1608 and uh 20,182?

[00:14:34]Okay, if these were your denominator, I can tell you right now, you would not be a happy camper. You might be looking like this and like, okay, okay, you made your point. Listen, when it comes to easy numbers, sometimes when you you're thinking about the LCD, we don't even know why we know the LCD. We're just like, it's this. I just know it's this is the answer. Okay, but we need a procedure, right? and you need to have a good understanding of what the LCD is.

[00:14:58]So there you go. So that's the lowest common denominator. It is 80. Okay. So what does this mean now? Well, let me show you. Okay. This is the next phase of doing this problem. So some of you out there is thinking to yourself, this is kind of a lot of work to do this problem. Well, yeah, you know, when you truly want to understand it, I'm going to say it's a lot of work, but there is a lot of concepts that we need to make sure we understand. So we have two fractions here. the denominators are not the same. So, we're going to uh write a new brand new denominator for both of these fractions. And that brand new denominator is going to be 80. So, how do we do that? Well, easy. We need to just rewrite these fractions differently. For example, if I had the fraction 1/2, right? And I said, hey, uh let's make up a new fraction um with a new denominator that's still equal to this fraction. Well, I could write 5 over 10. I can write three over six. I can write 4 over8. I could write 7 over4.

[00:15:57]Doesn't make a difference. All of these fractions right here are equivalent to 1/2. Uh the only deal is is they have different denominators. Okay? So you can rewrite a fraction with a brand new denominator. You just have to adjust the numerator. Okay? So, it's this top number we're going to have to make some adjustments to uh if we want the bottom number, the denominator to be 80. So, let's go ahead and figure out how to do that right now. Okay, so of course, we want to make uh uh the denominator 80.

[00:16:28]Okay, so let's take a look at our fractions right here. Let's focus in on this one first. So, how do I make a 16 into an 80? Easy. Just multiply it by five. 16 * 5 is 80. If I multiply the denominator here by five, I got to multiply the numerator by five. That's how we adjust this fraction or rewrite this fraction such that it has a denominator of 80. So 5 * 9 is 45. So 45 over 80 is the new and improved fraction uh with this denominator that we want.

[00:17:01]Okay, 45 over 80. If I said, hey, reduce or simplify this fraction, you would go back and reduce it that way. Okay. So, how do we turn 8 through 20 into an 80?

[00:17:12]Easy. Multiply it by four. So, we have to multiply that numerator by four. So, that would be 12. Okay. So, we have 12 over 80. So, this fraction now right here is equivalent to 12 over 80. And now we are ready to finish up this problem. Finally, finally, we're going to uh we have two fractions with the same denominator. So, this is going to be easy. All we need to do is keep one of those denominators. That's 80. And go ahead and find the difference of the uh numerator. So that' be 45 - 12 right there. So 45 - 12 is 33 over 80. And of course you always want to make sure that you have a fully simplified fraction.

[00:17:53]But this is as far as you can go. You can't reduce this uh any further. And that is it. So, uh, at least in my videos, okay, if you're you're, you know, for new to, uh, my channel or if you're new to my videos, what I do is I take one problem and I break things down like super, you know, clear and understand, at least I try. Okay, the whole point of doing that is so you could have a comprehension of what's going on. You know, if I just did a bunch of math problems, like here, I'm going to uh do a lot of problems all day long. But if you don't really truly understand what I'm doing, you might pick up the pattern, but that's not the same. You might be like basically duplicate the pattern, but that's not the same as truly synthesizing this material, truly comprehending this. You know, I've been doing this for a long long long long time because I love teaching mathematics. But I found out a long time ago that the obvious best way to teach math is to really focus on the concepts. You know, build your skills up. Not just to say, "Hey, here's how you do this problem, da da da da da."

[00:18:59]You know, if that's the way you're approaching mathematics, you're probably struggling. Okay? So, take a deep breath, learn one skill at a time, but once you learn how to do one problem, all the other problems become much much uh more attainable, much easier. Yeah.

[00:19:13]you know, you might have more difficult problems. Even if I gave you more um uh digits, you know, in terms of the denominator here, you would just simply do the work. You know, a lot of these more challenging problems are more work, but at least you're not going to be lost. Okay. So, if you need additional help with fractions, again, uh three suggestions. One, I have ton more uh tons of uh videos on my YouTube channel on fractions. Uh that's my first recommendation. Second thing is my math foundation course or my pre-alggebra course. All right, so with that being said, I definitely wish you all the best in your mathematics adventures. Thank you for your time and have a great day.