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How To Divide Rational Numbers
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Feb 19, 2025
We've already learned how to divide positive fractions and also learned how to multiply rational numbers. Now we'll learn how to divide positive and negative fractions as well as mixed numbers. And then we'll also learn how to divide algebraic expressions Chapters: 00:00 Introduction 00:52 How to find the multiplicative inverse of a number 04:23 How to divide and simplify fractions 10:22 How to divide mixed numbers 13:32 Solving a real world problem 17:44 How to divide algebraic expressions
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0:01
so we are talking about
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dividing rational numbers everybody know
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what a rational number is by now it's a
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fraction right a number that can be
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written as a fraction is called a
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rational number you know that you know
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what whole numbers are you know what
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integers are you know what irrational
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numbers are so you know okay now the
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first thing that we need to learn in
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this new section which I expect that you
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guys are all going to be writing down
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information I don't want to see anyone
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not writing down anything because I
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don't want to have to well I don't
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remember how to do this so you better be
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writing down stuff I better see you with
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your notes better I see you writing down
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things everyone no
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exception yes yeah go ahead yeah right
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so the first concept that we going to
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learn is called the multiplicative
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inverse everybody says that multip
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multiplicative
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multiplicative right so what does that
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mean what is the multiplicative inverse
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of
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1/3 what is that yeah you want to Define
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it yeah the definition of the word all
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right the product of the number and it's
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multiplicative inverse is always equal
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to one what does that mean if a number
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times is multiplicative invers is one
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what does that mean yeah can we just
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call
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mi if you want yeah if you take a
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fraction and then times it by the
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fraction two invers it's always going to
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equal one exactly exactly how you define
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it right so if you take a number and
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flip it and multiply by a flip um self
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you get one right 1/3 * 3 is equal to 1
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so that's what this means here right so
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if I have a b * B A I'm always going to
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get one because guess what by the
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butterfly effect right butterfly
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whatever the A's are going to cancel out
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and the B's are going to cancel out and
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you going to get what one cuz basically
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you technically getting what AB over a
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which is equal to one so if you time the
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number by multiplicative inverse you get
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one yes how didn't know which number to
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flip say it again I don't know which
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number to flip say that again how do you
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know what you oh you will find out we'll
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get right now we just understand the
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concept right and then we going to see
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how this applies to division of rational
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numbers right so for example 2/3 * 32 is
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= 1 because this technically 6 over 6
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right
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now let's try and find some
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multiplicative inverse and then we're
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going to go to div yeah go ahead right
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for example what is the multiplicative
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inverse of 76 7 over 16 what would it be
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16 16 over 7 does that make sense to
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everyone right you just have flipping it
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that's all you doing just you're just
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flipping it right now what is the
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multiplicative of
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three 1/3 does that make sense to
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everyone because three is like
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technically 3 over what one right so if
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you want to find the multiplica inv of 3
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is
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1/3 now we have a mixed fraction here so
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what is the multiplicative inverse of -6
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and 1/3 hold up what do we do first I
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got it yeah I don't know 37 say it
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again so we going to to first six * 3
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right which 18 + 1 19 so we have to turn
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this into improper fraction that is Nega
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what
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193 so therefore the multiplicative
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inverse of9 /3
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is 319 right 319 so that will be the
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multiplicative inverse of this number so
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pretty simple now that we understand
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what a multiplicative inverse is we're
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going to see how that relates to
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fractions right so now dividing
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fractions so when you divide a fraction
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what you're technically doing is this if
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I'm dividing A over B / c d what I'm
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doing is I'm multiplying a b by the
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multiplicative inverse of C
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D okay that's a lot of terms so
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basically a b / c d is technically a b *
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D over C yes um like I think I know this
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is multiplying by
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reciprocal that's why reciprocal right
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and the way you reciprocal is r e c i
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reciprocal so when you divide a fraction
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you what you do is you multiply the left
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hand fraction by uh the reciprocal of
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the right and fractions this is the left
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this is the right so a/ B / C over D is
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basically a b * d c does that make sense
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right so now for example if I have 1/ 9
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and I'm trying to divide that by
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52 first we're going to learn how to
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divide fractions by fractions right 1 9
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/ 52 is 1 9 * 12 5 okay now since we
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learn how to do multiplication yes um
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when you're doing when we dividing um
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the fact
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I the butterfly yes I was going to talk
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about that so you actually ahead of me
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right so since we are
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multiplying when you have a
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multiplication we always think about
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what simplifying the fraction right
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because going forward any fraction
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that's not simplified is not totally
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complete right in my algebra 2 when they
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don't simplify I take some points off
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and they why I said well because you
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have you have to simplify it's a rule in
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mathematics that you have to simplify
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any fraction right every fraction so now
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I have 1 9 and 12 5 so how do I simplify
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this
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fraction um
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mhm by
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what if I look at these two does is
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there a relationship between one and
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five none right what about 9 and
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12 what goes into n that goes into 12 th
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right so if you divide this you get four
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divide this you get three so that's
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basically 1 over 3 * 4 5 right and then
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now to multiply you multiply
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across okay so this becomes 4 over 15 so
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this is how you're going to do this WR 4
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over 15 that make sense now what am I
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going to do here I want you all to try
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this 37 / 8 and tell me how you did it
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so how would you divide this fraction 37
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/
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8 what 37 and you're trying divide that
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by eight okay so tell me how you first
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set it up and then divide it and then
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finish it so
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first I want you to do it on your own
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first
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by you can use
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it okay I'll give you one hold
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on thank you
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if you got the tell what you
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got Havey what you
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got did you do
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it all right so what you
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got I started by taking the eight and
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turning turning that into a fraction and
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then I multip by the
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recipal and so I got 3 over7 * 1 over8
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okay and I got 356 356 because there's
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no number I remember that goes into
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eight right so therefore it's 3 over 56
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are we clear on that wait simplify that
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you can't because there's no
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relationship between three and there's
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nothing I mean like 3 56 because b six
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this 56 it's not a it's not 5 * 6 it's
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56 three does not go into 56 right
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yeah um how come
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swap
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only cuz we don't swap this one we only
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swap the one on the right hand side
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right the only one that's always swapped
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or flipped is the one on the right hand
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side does that make sense okay now let's
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try this 4 five right times divided
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by
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um 20 so go ahead and do that 4 five by
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20 and see what you
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get next what you got um 12 so tell me
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how you did it um so I turn the 20
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into 12th okay
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and then you multiply
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right M right I M
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cross the 4 20
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into and and that's 125 125 that's
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correct 125th right because the four
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goes into 25 so you got 5 * 5 that's 25
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12 okay we're going to move on from this
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and now we're going to talk about how do
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you divide and mix number right mixed
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number now we got -4 and then 2/3 / 3
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and 1 you guys try it on your own and
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see what you
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get4 2/3 and then divid by 3 and then 1
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n
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okay for
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all right so how do youall do it yes um
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Abby so then
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I so first it was negative I made the
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fraction into a uh improper yes improper
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fraction that's
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1443 okay
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289 and then I 28 over 9 right everybody
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knows how she got that no no no no sorry
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yeah 28 over 9 right I'm not M and then
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oh
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yeah I did um I switched the n
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and symb
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into
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14 143
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* um 9 * 9 28
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okay and
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then
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meod
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14 24
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tce can go into three
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times okay so I
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got 3 two that's right3 23 over2
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right oh you can you can leave it like
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this so you can turn this back into a
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mixed number which is one one2 right you
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have two option you can leave it like
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this there's nothing wrong with this or
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this one all right now let's try to work
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on the word problem which I know you
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guys love so much right
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yeah go ahead and make sure so I don't
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know about you but my dogs don't like
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dog food they hate it I have to cook for
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them maybe 3 days they won't eat any dog
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K yeah that's spoil dog
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absolutely they're not
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spoiled you know why because dark food
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first of all is processed it's like you
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EA living on McDonald's every day would
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you do that no yes no is James a boy or
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a girl with this James is a boy James
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feeds her dog oh well let's call it
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Jamie
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then if he if
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she right so Jamie feeds her dog three
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and three quarter cups of dog food per
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day if she buys a bag of food way too
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specific that contains 165 cups how many
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days will it last I want you all to try
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to solve this problem right so Jamie
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feeds her dog with three and three cups
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a day and she buys a bag that had about
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165 cups so how many days do you think
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that would last so how would you do this
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problem how would you that depends on
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how much the dog eat no you have the
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information and she Fe how much does the
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dog eat every day a lot a lot too much
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what is it what's the number how much is
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dog eat three and 3/4 so we know how
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many cups did she buy I need another
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marker 165 right so how would you set
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this up so far so good I need another
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marker
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please all these markers are bad are you
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like literally like drinking the ink
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those markers how you finish things
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that all right does
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anybody I want to know how you set it up
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first before you even go so yes nor how
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you set it up 3 and
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34
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165 over one all right good we way to
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set it up so it's 165 right divided by
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what three and 34 right so now what we
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do from here so we mhm um switch it and
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make it um we switch the we put
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65 we switch we switch we switch this
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guy right first we have to turn this
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into what mix uh improp fraction which
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is what four 15 4 15 it's 15 over four
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but 15 over four right and then after
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that let's go step I said it's 3 * 4
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which is 12 well I'm slow so let me go
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little by little right 3 * 4 is 12 12 +
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3 is 15 15 over
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4 right that's what I'm saying I'm very
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slow so now we have 165 over one and
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then times 4 over 15 now what we do from
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here
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do I got it
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butterfly um 15 and 165 15 goes into 165
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11 times 11 times is that right no I use
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my calculator I don't think that's right
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cuz 4 * 15 is 60 so it goes by five time
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let's do at least five we know this is I
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don't know and this will be what's 165
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of five it says 11 11 here wait hold 165
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16 15 oh so this is 11 all right so that
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will be 11 yeah is 11 so this is 11 1 *
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4 over 1 which is what 44 right so how
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long will food last 44 days 44 days
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those dogs arey
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has 150 cups of dogs well if you have
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big dogs you know that's that's good
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yeah dog e a lot
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what it's called gig all right now Emily
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you skipped the word problem just in
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time okay now we're going to learn how
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to um I know now we're going to learn
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how to use this to uh solve um what we
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call algebraic expressions okay great so
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now we're going to talk about algebraic
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expressions which I know you guys also
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love
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so how do you divide them so if I have 5
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over 3 a right and I'm trying to divide
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that by um
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15 over ABC so now I'm going to try to
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solve this in front of you guys real
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quick right so look I have letters I
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have variables right it does not change
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it does not change the problem that much
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it's the same principle that we using
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Okay so let's let's pause for a second
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it's easy so it's going to be 55 over 3
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a
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right gab I can hear you from
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here I can hear your
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voice don't move him back there with
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Emily Emily can't do that I will
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actually fight
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somebody I'm not saying that way I'm
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just saying like are we following here
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right right for a second so we have 5
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3ab time abc 15 right yeah
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so now if I'm going to simplify this I
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need to go numbers and numbers and then
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variables and variables right so now
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five and 15 five goes into 15 three
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times right and now a b c watch this A
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and A are going to cross up B and B are
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going to cross up correct so I'm only
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left upstairs with what c c is the only
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one left up stairs and then downstairs I
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have what 3 * what 3 which is 9 so the
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final result is C over9 so a lot of
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people get intimidated when they see
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letters you shouldn't be right now let
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me twist it up a little bit and see if
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you guys can figure this out now what if
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I had 3 a 2 over 5 times
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um noid by doing exponent one second one
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second
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I'm just the number A
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B let's see if you guys can use your
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brain to figure this out right it's not
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that hard right just
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because do you guys know expon yes or
19:53
no no all right let's this
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here take number
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I have hiup which is a
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goodi
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yeah 2 power three right let me explain
20:10
this to you real quick pay
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attention 2 3 power is literally 2 * 2 *
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2 hey hey hey hey hey hey hey that's
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enough right so a to the^ two is a * a
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that's all it is so it's not
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intimidating so now I'm going to go 3 *
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a * a over 5 right and I'm going to flip
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this guys so that becomes a * B over 10
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now I can simplify this can I yeah right
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actually I was let me change this let me
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let me do something different here I
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should have done it you can the a yeah a
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over 10 let's do it this way right so
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now I'm going to do this so I'm going to
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have 3 * a * a over 5 right
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times 10
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time
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right so now can I simpli this yes right
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the a are going to
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what out right 5 is two so now 38 * 2
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which is what 68 and then over what have
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anything down say is over one so it's
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basically
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68 right so that's what I'm
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saying and they get kind of this is so
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hard really simple stuff all right okay
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so this is the today's lecture now we're
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going to work on some
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problems and I'm going to give you stuff
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to work on it's not homework is it
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