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Topic B: Multiplication and division of integers and rational numbers

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- What I hope to do in this video is emphasize the relation, the connection, between fractions and division and then using that knowledge to help us simplify some hairy looking fractions. So, let's say, let's just do a little bit of a review. So, if I say two divided by, two divided by three, two divided by three, I could write this as, I could write this as two over three. Two divided by, two divided by three. So, this expression here, sometimes we might say, "Hey, this is 2/3. Think of it as a fraction." But you can also view this as two divided by three. So, if we went the other way around. If someone were to write four over, actually, let me just do it in a different color, if someone were to say four, 4/9, four, four over nine, we could interpret this, this could be interpreted as four divided by nine. Four divided by nine. In fact, if we wanted to, if wanted to convert this into a decimal that's exactly what we would do. We would calculate what four divided by nine is. We would divide nine into four. Four divided by nine. So, this is all review. So, let's just use this knowledge and I'll think more complex fractions. So, if someone were to say, actually, let me just take this example right over here, if someone were to say two over, instead of saying two over three. If they said two over, I don't know, let me say two over 2/3. Two over 2/3. Well, what would this simplify to? Well, we could go the other way around. This is the same thing as two divided by 2/3. So, let's write it like that. This is the same thing as two divided by, two divided by 2/3, two divided by 2/3, which is the same thing as, dividing by a fraction's the same thing as multiplying by its reciprocal. So, this is going to be the same thing as two times, two times the reciprocal of 2/3, which is, we swap the denominator and the numerator, two times three over two. So, two times three halves, if I have three halves twice, well, that's just going to be equal to, that's going to be equal to six halves, and six halves, you can view that as, well, two halves is a whole, so this is gonna be three wholes. Or you can say six divided by two. Well, that's just gonna be equal to three. You can view it either way. And then that is equal to three. Let's do a few more of these. And let's keep making them a little bit more complicated. Just let me get some good practice. And like always, pause the video. You should get excited when you see one of these things. And pause the video and see if you can do it on your own. All right, let's do something really interesting. Let's say negative 16 over nine over, over, I'll do this in a tan color, over, let's say, three over seven. What is this? Can you simplify this complex fraction? Can you simplify this expression over here? Well, once again, we can view this as negative 16/9 divided by 3/7. So, this could be rewritten as, and I can write it either as negative 16 over 9 or I could rewrite it as negative 16/9. So, I can put the negative in front of the whole, in front of the whole fraction like that, or I could say that it's negative 16 over nine, or I can even write this as negative 16 over negative nine. Those would all be equivalent. But right now I'm writing at negative 16/9 and I am going to now divide, we can interpret this complex fraction as dividing it by 3/7. Dividing by 3/7. And so, this is going to be equal to, this is going to be equal to negative 16/9, negative 16 over nine. Actually, let me just rewrite it as negative 16 over nine. Negative 16 over nine just to show that we can do it. Times the reciprocal of this. So, times 7/3. I just swap the numerator and the denominator. Times, do that same brown color, times, times 7/3, and now what am I going to get? In the numerator, I have negative 16 times seven. Let me think about it. 10 times seven is 70. Six times seven is 42. 70 plus 42 is 112. So, this, so this part right over here, this part right over here would be negative 112. Negative 112 over nine times three. Nine times three is 20, nine times three is 27. And there you have it. You can view this as negative 112 over 27 or you can put the negative up front and you can say, "Hey, this is hinting "as negative 112, 112/27." 112/27 and just to make everything come full circle you can view this as negative 112 divided by 27 or negative 112/27. Either way.