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How To Identify, Classify, And Write Rational numbers As Fractions
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Feb 19, 2025
We've already studied integers, natural and whole numbers. In this lesson, we'll now be introduced to a new set, the set of rational numbers. We'll learn how to write rational numbers as fractions as well as learn how to identify and classify them ( Section 3.2) Chapters 00:00 Introduction 00:24 Introducing Different Numbers' Set 3:11 What Is A Rational Number ( Definition) 03:48 How do you write rational numbers as fractions 05:02 How to write mixed numbers and integers as fractions 06:52 How to write terminating decimals as fractions 13:40 How to write repeating decimals as fractions
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0:00
all right so we're going to start
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section
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3.2 and section 3.2 is rational numbers
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I need you all to be quiet right
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actually sit right there but before we
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talk about I want to sit right rational
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numbers what I want you to do is is I
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want to talk about uh the set of numbers
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to begin with right
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can so go up there go behind no no no
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not before my thing go behind child oh
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yeah that so we have like you need to
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first talk about the sets right does
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anyone need paper me actually never mind
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I my binder I need
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ER I need to my water bottle looks
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crusty and just and then we also have
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decimals
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right so we're going to talk about these
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sets to begin with and then we're going
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to go deep into lecture before we even
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get into that
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so first we need to talk about natural
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numbers right when you hear the word
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natural natural numbers okay natural
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numbers begin at what at one so one 2 3
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4 all the way to Infinity right those
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are called natural numbers natural
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numbers just natural the word natural
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next is whole numbers now whole numbers
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unlike natural numbers they begin at
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zero okay 0 1 2 3 4 all the way to
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Infinity right so those are whole
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numbers okay what's the difference it
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begin at Z whole numbers begin at zero
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natural numbers begin at one okay and
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then we have integers now the integers
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they begin at negative Infinity so we
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have let's say5 -4 -3
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-21 0 1 2 2 3 4 all the way to Infinity
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those are called integers okay those are
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integers okay and then we have decimals
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now decimals are all of these plus
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decimal numbers
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like
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6.55 4 and all of that right 1 2
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3.5 and all that are called decimals
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okay and then the next set after these
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set is the set of what rational numbers
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rational numbers okay now natural
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numbers are the easiest and integers is
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what we've been used to like so far
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we've talked a lot about integers and
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then yesterday or like last week we
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began to talk about fractions okay a lot
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of you guys don't like fractions because
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you don't like to deal with like lowest
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common denominators and things like that
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but it's not that hard okay it's it's
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not it's really not that complicated so
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I'm going to erase this real quick right
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you need
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so now today we're going to talk about
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rational numbers and before we talk
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about it we need to know what is a
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rational
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number rational right what do you hear
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when I say rational what's the word key
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word there rational ration or what else
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ratio ratio ratio a ratio is
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what a ratio is a fraction right so
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basically by
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definition a a rational number is a
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number that can be written as a fraction
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right a number that can be written as a
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fraction is called a rational number
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does that make sense a number that can
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be written as a fraction is called a
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rational number for example one one is a
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rational number right because can one be
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written as a fraction yes or no yeah how
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would you write one one as a fraction
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one one one over one excellent one over
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one what about can you write zero as a
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fraction no no yes we can you just said
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it what is it Z is z no 0 over Z that's
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not that's in determination in calculus
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one Z no 0 over 1 0 over one you know
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what 1/ Z is actually Infinity yes once
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you get the calculus we're going to talk
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about that right what about -3 can you
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write3 as a fraction yeah so what is it
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going to be I don't know probably like
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33 over what over
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what on right now 23 is already as a
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fraction so we don't need to do that now
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0.87 can also be written as a fraction
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and we're going to learn how to do that
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in a minute right we're not going to do
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that just yet but we're going to learn
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how to do that what about one and 1/2
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can you write that as a
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fraction yes or no what is what do we
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call this
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number um you start with an N mixed
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mixed fraction right mixed fraction now
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here's the thing now we're going to
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learn how to write mixed
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fraction as and integers as fractions
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which we just started to do right so for
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example I have
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six and 16 how do I turn this into a
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fraction from a mix what do I
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do
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yes 6 6 * 6 + 1 + 1 is 37 over six right
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so that is 37 over 6 that's excellent
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right what about -23 how would I turn
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this into a
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fraction put a one under put a one under
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it right just put a one under it
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sometimes the easiest side the hardest
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to to do because it has to be
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complicated no right what about four and
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23 Carly how do I turn this into a
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fraction it's a mixed fraction
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um 4 by that's 12 to two so it's 14 143
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right
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like what about this um Sophie how do I
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turn seven into a fraction so you just
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put a one under put a one under there
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right 7 over one and that's it right so
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to turn mixed numbers into fractions
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that's pretty easy stuff okay it's just
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really not that complicated now it
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becomes a little bit more complicated
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now when you have like decimals right
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and not just any type of decimal here we
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have what you call a terminating decimal
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right do you remember I defined that for
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you guys last week a terminating decimal
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is a number is a decimal that has like
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terminating digit right meaning it's not
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like
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0.645 that it's limited 0.64 is two two
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two digit after the decimal that's a
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terminating decimal meaning it ends
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somewhere right it's number repeating is
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terminating does that make sense what
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does the word terminate means when I say
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terminate what does that
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mean I'm talk the Terminator no terminat
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is to finish right it's the price is to
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finish something so this is a this this
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number this decimal can we call a
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finished number it's finished right
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because the ends is 0.64 terminating
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it's not on ending it's ending it's
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ending somewhere does that make sense
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now how do I turn this
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0.64 into a
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fraction put your thinking caps on and
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figure out how to do this would you take
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it and then you go one two all right and
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then you put the dot there
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uhhuh it's getting there right so 0.64
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right every time I move the decimal what
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I put under the fraction
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one one place is what a 10 and two will
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be what 100 100 right so if I want to
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turn this into a fraction I'm going to
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start from here 0.64 over right I start
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with one and I said okay every time I
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move this I'm going to add a zero every
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time I move this I'm going to add a zero
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so therefore now we have 0.64 turns into
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what turns into 64
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over
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100 does that make sense do we all get
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it any questions no right but
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are we done not yet simplify because
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that's the key word we have to simplify
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because this is saying what write this
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ter decimals in their simplest forms
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that means you have to simplify the
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fraction now that's what you have to now
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remember what goes into 64 that goes
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into 100 two 32 you can start at 3 right
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two and then what do you have 50 32 over
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50 do we stop here no we can go further
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right if you know your multiplication
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table this become
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it walking apart but if you don't then
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you have to go step by step right so
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what goes into 32 that goes into
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50 two two how do we know if the number
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is divisible by two it's 16 and 20 right
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16 and 25 right divide two and that 16
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and 25 can I go any further or do I stop
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here stop here right because there's no
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number that goes into 16 that goes into
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five now here's how you can tell if a
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number is divisible by two any number
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that ends with a zero or a even number
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is divisible by two right anytime you
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see a zero next to a number you know
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that you can divide that by two anytime
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you share a multiple of two you know
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that you can divide that by two or a
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even number any even number is divisible
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by two right for three is easy also for
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three if you have for example how do I
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know if this numbers
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here to know if a number is divisible by
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two by three let this there the easy
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thing that I learned a long time ago I
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was your age probably younger right if
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you add the digit in the number 7 + 2 is
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equal to what uh 10 10 n nine because 9
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is a multiple of three is divisible by
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this number is divisible by two try the
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same here 3 + 1 is 4 4 + 5 is 9 since 9
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is a multiple of three this number is
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divisible by 3 this is how you know and
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if you want to know if a number is
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divisible by five you just have to see
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if it end where zero or five or multiple
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of five okay so if you know these things
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it will help you in solving these
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problems now we have that now how do I
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change this now we have 9.67 5 how do I
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turn this into a
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fraction you got it yeah so you what we
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do you would move the dot to the end we
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do we can do that or what else can we do
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so if you move the dot to the end right
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so if you start here
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9675 what do we do put under 10 10 how
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many times I move this three times three
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times 1 2 three right so a th000 right
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is
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9675 over 1,000 now if you feel like you
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don't want to do this there's multiple
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options right you can also write this as
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what 9 and then
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675 over what 1,000 you can write this
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way as well right this is also a mixed
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fraction last year I remember when we
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when we did the section some people like
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it this way some people like it this way
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I won't penalize you you can choose this
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method or you can choose the other
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method now we don't stop here because
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we're going to try to do what we're
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going to try to simplify right now it's
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easier when you do this to simplify
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because now I'm looking at it how do I
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know what would be the first number that
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goes into 675 that goes into 1,00 five
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because we know that because this ends
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with the zero and this end with five
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they are both divisible by five right so
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I'm going to do that I'm going to divide
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just the fraction by 5 and I have my n
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so 675 over 5 I don't know what that is
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right what by, 5 you guys know
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two what 200 right now what was 675 5 do
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you have a calculator I have it on my
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phone I need some help 675 over 5
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right 675
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be like divided by yeah divided by what
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five that's it's 135 135 right that's
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135 and then I can also div I can divide
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this again by what five five again right
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five on both sides so what do I get I
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get would be 27 27 and this would be 200
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/ 5 is 40 40 right and this is
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37 now can I any further no there's no
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number that goes into 37 that goes into
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40 so therefore 9.6 75 turns out to be
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what 9 and 37 over 40 all right you see
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how we work this out okay so we're going
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to keep moving on now we're going to
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learn how to it's going to get a little
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harder now so now how do you turn
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repeating decimals repeating decimals
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into
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fractions we talk about non repeating
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now we going to have repeating so how do
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I turn this into a fraction last year I
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remember when we did this it was a
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little bit of
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uh so how do you go from that to a
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fraction
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okay now how do I turn this into a
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fraction in the simplest form what do I
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do here what you move the dot over you
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can move it put it under but you'll be a
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little hard to do that right that's
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that's a method to this smartness so put
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so we're going to let like we're going
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to call this like X right x =
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0.6 66 dot dot dot right these and these
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are the same thing aren't they me saying
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this is the same thing it's not no don't
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think this because it's 6660 means
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nothing okay we not we not scared of
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that we not are we scared of it no right
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so 666 like that right we're going to
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let x equals this number are we good on
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that we can can we right so I'm trying
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to turn this into that right these and
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these the same I'm technically saying
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the same thing okay so now what I'm
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going to do is this what if I choose to
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multiply this by
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10 right 10x will give me what 10 x will
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be just me timesing this multiplying
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this number by 10 I'm going to get I'm
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going to move this right it becomes
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6.6 6 6
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6 you agree with me on that right so now
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if I do
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10x - x which means this minus that what
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do I get if I do 10 x - x what is that
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going to give me
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10 it gives you what if you do this
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right watch here pay attention if you do
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that everybody pay attention here
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everyone right if I do 10x minus that
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this portion is going to go away isn't
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it that's going to cancel out that so we
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going to be left what your
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six right
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and that gives me what 10 x - x what 9x
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=
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6 now can I solve for x yeah yes by
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doing what div dividing by 9 right
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divide by 9 divide by 9 so X is 6 over 9
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and I can simplify this what goes into
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six that goes into 9 three three if I
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divide this by three we get
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what two 23 so therefore 0.
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66666 is technically
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23 does that make sense I know this one
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is a little bit complicated but if you
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do more we going to get it yes what
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cancels out the two six oh if I do this
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minus that they have the same link here
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right is a chain the chain is the same
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so this and this chain are the same so
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those are gone so it's basically saying
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60 and that's just six that make sense
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all right okay so now now what I want
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you to do is we're going to work on a
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couple of things here and then we're
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going to close this or we can also learn
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how to
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classify um like numbers to see if a
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number belongs into um let's say
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rational irrational and all of that
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stuff I don't know if I want to do
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irrational numbers that's going to be in
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like the next section so we can stop
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here and then we're going to do some
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work in the book okay no I like
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