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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 2 divided by 2/3 2/3 I could write this as I could write this as 2 over 3 2 divided by 2/3 so this expression here sometimes we might say hey this is 2/3 think of it as a fraction but you could also view this as 2/3 so if we want the other way around if someone were to write 4 over actually let me just do in a different color if someone were to say 4 4/9 4 4 over 9 we could interpret this this could be interpreted as 4/9 4/9 in fact if we wanted to if we wanted to convert this into a decimal that's exactly what we would do we would calculate what 4/9 is we would divide 9 into 4 4/9 so this is all review so let's just use this knowledge to not think about more more complex fractions so if someone were to say actually let me just take let me just take this example right over here someone were to say 2 over instead of saying 2 over 3 if they said 2 over I don't know let me say two over 2/3 2 over 2/3 well what would this simplify to well we could go the other way around this is the same thing as 2 divided by 2/3 so let's write it like that this is the same thing as 2 divided by 2 divided by 2/3 2 divided by 2 3 which is the same thing as dividing by a fraction the same thing as multiplying by its reciprocal so this is going to be the same thing as 2 times 2 times the reciprocal of 2/3 which is we swap the denominator in the numerator 2 times 3 over 2 so 2 times 3 halves if I have three halves twice well that's just going to be equal to that's going to be equal to 6 halves and 6 halves you could view that is well you know two halves as a whole so this is going to be three wholes or you say 6 divided by 2 well that's just going to be equal to 3 you could view it either way and then that is equal to 3 let's do a few more of these unless let's keep making them a little bit more complicated just so we 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 9 over over I'll do this in a tan color over let's say 3 over 7 what is this can you simplify this this complex fraction can you simplify this expression over here well once again we can view this as negative 16 nine divided by three 7s so this could be rewritten as and I could write it either as negative 16 over nine or I could rewrite it as negative 16 ninths so I could put the negative in front of the hole in front of the whole fraction like that or I could say that it's negative 16 over nine or I could even write this as 16 over negative 9 those would all be equivalent but right now I'm writing is negative 16 ninths and I'm going to now divide we can interpret this complex fraction is dividing it by three 7s dividing it by three sevenths and so this is going to be equal to this is going to be equal to negative 16 ninths negative 16 over done actually let me just rewrite it as negative 16 over 9 negative 16 over 9 just to show that we can do it times the reciprocal of this so times 7/3 I just swapped the numerator and the denominator times do that same brown color x times 7/3 and now what am I going to get in the numerator have negative 16 times seven let me think about it ten times seven is 76 times 7 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 9 times 3 9 times 3 is 29 times 3 is 27 and there you have you could view this as negative 112 over 27 or you could put the negative out front and you could say hey this is the same thing as negative 112 112 27th 112 27 and just to make everything come full circle you could view this as negative 1 or 12 divided by 27 or negative 112 27 either way