 Okay, so far we've done some fractions. We've talked about the real number set, rational numbers, whole numbers, integers, and we've talked about irrational numbers, and broken out a couple fractions, and mainly, or the most important thing we've talked about is the prime numbers, okay? Now prime numbers. Integers, as we've said before, are positive and negative whole numbers, and the whole number includes zero, so it starts off with whole numbers, from negative infinity goes all the way to zero, from zero all the way to positive infinity. Now any integer, positive integer, that is not a prime number, is made up of prime numbers, and that is the main thing you have to grasp, which is you have to understand prime numbers to be able to deal with the rational number set, which is really 90% of the numbers you're going to deal with in high school. So if you know the rational numbers, and to understand the rational numbers, you have to know the prime numbers, then if you know your prime numbers, you can deal with most of the numbers you're going to deal with in high school, or you're going to encounter in high school. So prime numbers are super important. So let's break down some more fractions, okay? Bigger ones this time. I'm going to stick with glue. It's darker. The pink really didn't work out too well. This time we added a division sign, right? Right there. The only thing with division sign is, what you do with it is, in a fraction, you change it to multiplication, and you flip this fraction. The first thing you've got to do when you're multiplying and dividing fractions is take care of the division signs. So you're going to change this division sign into a multiplication. And you're going to flip this. Now as we said before, as soon as you do something to another, break it down or flip it or like this, kill the numbers before it. So you don't get confused. Now we're going to start breaking them down through their prime numbers. Eight is two times four. Four is two times two. Yes, I'm going to kill the numbers before it. We no longer have a four. Four became two times two, so it's gone. Eight became these guys. So eight is really two times two times two. Fifteen. Three times five. Twelve. Two times six is five times seven. Seventy-five. Now when you get to the bigger numbers, the more of this you do, the easier it's going to become because you're going to learn tricks how to do things. Now, if you see any number that ends with zero or a five, you can divide it by five. Any number that ends as even, you can divide it by two. Now, with seventy-five, what I'm going to do is divide it by twenty-five because it's a multiple of twenty-five. It makes life easier instead of dividing it by fifteen. So I'm going to take this and say it's three times twenty-five. And twenty-five is five times five. Can you see that? Ah, you can barely see it, but that's good enough. You know what's down there. So kill the numbers after that came before it. One-o-four. One-o-four, it ends with a four. It's an even number, so you can divide it by two. One hundred divided by two is fifty. Four divided by two is two. So one-o-four divided by two is fifty-two. Yes, yes. Fifty divided by two is twenty-five. Two divided by two is one. So fifty-two divided by two is twenty-six. Kill every number that has branched off. If it branches off, you kill the number that is branching off from it. One-o-four was actually two times two times two times thirteen. And you can't break down thirteen anymore because it's a prime number. So that's it. We've broken down fractions, multiple fractions multiplied together, into their prime numbers. Their core essence, what they're made out of. This is dealing with rational numbers. It makes life simpler. The more of this you do, the easier it becomes. This is doing it the long way. Well, not the long way, but this is doing it with every step involved. The more you do, the more steps you can eliminate. But make sure you're doing this enough that you know how to do it. Don't automatically skip steps and try to do it the quick way because you're going to make mistakes. Learn this process first and then start eliminating things. We'll talk about that stuff later, but right now let's deal with it this way. Now with fractions, if you remember what we said is as long as there's no addition or subtraction sign between the numbers, if they're not being added or subtracted from each other, if it's only division, if it's only multiplication and division, then anything from the top can cancel out anything from the bottom. So what do we got? Let's start knocking things off. Anything from the top can cancel out anything from the bottom. Five. So what you want to do is when you get a complicated thing like this that's messy all over the place, you want to go through it and circle the numbers that are left because what you're going to do now is multiply the numbers in the top and multiply the numbers in the bottom because whatever's in the top multiplies each other, whatever's in the bottom multiplies each other. What do we got? Three times seven is 21. 21 times four. What is that? We'll do it on the side. 143 is 13 times 11. So the final answer for this thing is, let's see, where can we write it? We have absolutely no room. The final answer for this thing is 143 over 84. That's your final answer, 143 over 84. Now you know you can't break that down any further because you took it down to its prime numbers and nothing else can cancel each other. So if they give this to you on a test, a question, saying reduce this, that's a trick question. You can't reduce it. You break it down to its prime numbers and nothing cancels each other off. So you're stuck with it. That's your final answer. That's multiplying and dividing fractions. It's super easy. What you have to do is practice it. I'm going to put some of the stuff on my website, write out some problems, and I'm going to have the solutions in the bottom. There's some version of that anyway, so you can practice it. Make sure you go there and practice some of these things. Very important. Once you master prime numbers, you master integers. And since rational numbers are just fraction of integers, you master rational numbers. And since you're in high school, most of the numbers that you're going to deal with in high school are rational numbers. Then you've learned how to manage numbers in high schools, which is 90% of what you're going to do in high school. It's dealing with rational numbers. And grade 12 is a little different. Up to grade 11, you're just dealing basically with rational numbers. Some, well, 10% irrational, 20% irrational, depending on your teacher. If your teacher's really good, they'll give you a lot more rational numbers because they're a lot more fun. Or you've got a lot more power. Let's put it that way. We'll talk later.