Math Antics - Simplifying Fractions

In a previous video, we learned all about factoring whole numbers, and now we're gonna learn how factoring can help us when working with fractions. We're gonna learn how to simplify fractions. Simplifying a fraction means rewriting the fraction using the smallest top and bottom numbers we can without changing the value of the fraction. To help us understand what simplifying a fraction really means, let's take a look at the simplest fraction I can think of… 1 over 2. Now this is already as simple as it can possibly be, so let's go the other way and complicate it by dividing our rectangle here up into more parts. The amount of our rectangle that's shaded is still the same, but now the numbers for our fraction are 3 over 6. The numbers are bigger because our rectangle is now divided into more parts. The fraction we have now, 3 over 6, is equivalent to our original fraction, 1 over 2. That means they have the same value. They represent the same amount. So what if someone gives you the fraction 3 over 6, like three-sixths of a candy bar? Well, we know from our picture that that means they're really giving you one-half. But how can we show that using math and not pictures? Well, that's where factoring comes in. Let's take our complicated fraction, 3 over 6, and factor both the top and bottom numbers. Now the bottom number 6 can be factored into 2 times 3. The top number, 3, is a prime number. Its only factors are 1 and itself, so we can write that as 1 times 3. There, we've re-written our fraction using factoring, and now it kind of looks like two fractions being multiplied together. 1 over 2 times 3 over 3. Of course, 3 over 3 is what I like to call a whole fraction since its value is equal to 1. Now here's the interesting part. Since 3 over 3 equals 1, and multiplying by 1 has no effect on a number, we can just get rid of that 3 over 3. Basically, the 3 on the top and the 3 on the bottom cancel each other out, and once they're gone, we're left with a fraction 1 over 2. So that means that the fraction 3 over 6 simplifies to 1 over 2. Another way of thinking about it is that we're trying to find any whole fractions that are hiding in the fraction we're trying to simplify. And if we find any, we can just get rid of them, and the fraction we're left with is simpler than the one we started with. Now that we know the basics, let's learn the procedure for simplifying fractions. First, replace the top and bottom numbers of the fraction with their prime factors. Next, look to see if any of the factors are the same on the top and bottom. If they are, then we call them common factors because they're something that both the top and bottom have in common. If you find a pair of common factors, you can cancel them out. Just draw a line through them like this. And last, once all the common factors have been cancelled, we need to re-multiply any factors that are left over on the top or bottom. This makes sure that you end up with only one number on the top and bottom of your simplified fraction. Oh, and there's one important thing to remember. If you're ever able to cancel out all of the factors on the top or bottom of a fraction, don't be tempted to write in a zero. Put a 1 in there instead. The reason you can write in a 1 is because 1 is always a factor of any number, it's just we usually don't write it in. For instance, if you're going to factor the number 15, you just say that it's 5 × 3. But you could also say that it's 5 × 3 × 1. In fact, you could even say it's 5 × 3 × 1. See why there's always a 1 left over when you're canceling common factors? Alright, so that's the basic idea behind simplifying fractions. And once you know the procedure, it's really not that hard. But you might want to re-watch this video just to make sure you've got the idea. Now there aren't any exercises for this video cause it's really just an introduction. But in part 2, we'll see a couple more examples of how you can use the procedure to simplify fractions and then you'll get plenty of exercises to do as homework. Oh yeah! Welcome to part 2 of Simplifying Fractions. In part 1, we learned the procedure for simplifying fractions. Basically, you just take the top and bottom numbers and factor them down to their prime factors. And then you see if there's any factors that are the same on the top and bottom. We call those common factors. And if there are, you just cancel them out. And once you've cancelled out all the common factors, you re-multiply whatever's left over to get your final answer. In this video, we're gonna see a couple examples of how we can use that procedure to simplify fractions. Let's start with an easy one. Let's simplify the fraction 5 over 15. Step 1 is to factor the top and bottom numbers. So, we know that 15 factors in to 5 × 3. And 5 is a prime number, that means its only factors are 1 and itself. But 1 is always a factor, so we don't need to write that down. Step 2 is to look for common factors and cancel them. We can see that there's a 5 on the top and there's a 5 on the bottom. They're not directly over each other, but that doesn't matter. They still form a common factor pair, and so we can cancel them out like this. Step 3 is to re-organize our answer. Now we don't have any factors that need to be re-combined by multiplying. We just have a 3 on the bottom and we don't have any factors left over on top, but you'll remember that there's always a factor of 1. So 5 over 15 simplifies to 1 third. Alright, I think we need to see another example, but a harder one this time. Let's simplify the fraction 30 over 36. The procedure is the same. Step 1 is we factor the top and bottom numbers all the way down to their prime factors. Let's do the top number first. 30 factors into 5 × 6. 5 is prime, but 6 can be factored into 2 × 3. So our 30 on top becomes 5 × 2 × 3. Now the bottom number. 36 can be factored into 6 × 6, and each of those 6's can be factored into 2 × 3. So our bottom number becomes 2 × 3 × 2 × 3. Well, it looks like we do have some common factors. There's a 2 on both the top and bottom that will cancel each other out. And even though there's more than one 2 on the bottom, we can only cancel one of them out because there's only one 2 on top. Remember, you always have to cancel common factors as pairs. Now we can see that there's another pair we can cancel. There's a 3 on both the top and bottom, so we can just cross those out. Okay, that's all the common factors we can cancel. So now all we have to do is see what's left over. We have a 5 on the top and a 2 × 3 on the bottom. We don't want to leave our problem looking like this, so we need to recombine any factors that didn't cancel. That means multiplying together our 2 and 3 on the bottom, which gives us 6. There… we're left with the fraction 5 over 6. That's the simplified form of the fraction 30 over 36. They both have the same value, but the simplified one is written using the smallest numbers possible. Now some of you may have been taught that the way to simplify fractions is to find the greatest common factor of the top and bottom numbers and just cancel that. Basically, that's what we are doing when we cancel all of the common factors using our procedure. In fact, if you multiply all of the common factors together, you'll get the greatest common factor, or GCF as I like to call it… you know… to sound cool. Alright, so that's how you simplify fractions. But I'll bet some of you are wondering, why would we even want to simplify fractions? That's a good question. Basically, it's to make life simpler. Well, at least for your teacher who has to grade all your homework. For you, it just makes life more complicated. No, just kidding. Simplified fractions make your life easier too, cause usually simplified fractions are much easier to work with. For example, if your friend said to you, Here, you can have 27 fifty-fourths of my sandwich. It would have been much easier if they had just said that you could have one-half of their sandwich, since one-half is the simplified form of 27 fifty-fourths. So now, whenever you see a fraction, you can ask yourself, Hmmm… could that be any simpler? And if so, you'll know just what to do. So get on out there, work on those exercises, and start making the world a simpler place for us all. Learn more at www.mathantics.com