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How To Multiply Rational Numbers & Evaluate Algebraic Expressions With Fractions
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Feb 19, 2025
In our previous lessons, we learned how to multiply integers and positive fractions, now we'll first learn how to multiply positive and negative fractions, then we'll learn how to evaluate algebraic expressions with fractions Chapters: 00:00 Introduction 00:45 How to multiply fractions 01:555 How to multiply and simplify positive fractions 06:12 How to multiply and simplify negative fractions 12:41 How to multiply mixed numbers
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0:00
so we are talking about multiplying
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rational numbers okay so basically is
0:05
multiplying
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fractions that's what it is fractions
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yes it's right here so how do you
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multiply fractions it's pretty simple
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stuff not very complicated the only
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thing that gets complicated is when you
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have to um simplify the fractions but to
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multiply fractions and I'm expecting
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everyone to write this thing down right
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so if you want to multiply fractions you
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multiply the numerators together and the
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denominators together you know what a
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numerator is right as the top part of
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the fraction and the denominator is the
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bottom right so to multiply a fraction
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if I have a b right times C over D this
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is a fraction it becomes a * C over B *
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d right how come the only one that's
0:54
writing something down is like Dexter
0:56
and Kenzie nobody else get stuff open
1:00
they looking at
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me so make sure you write this down
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because it's going to help you right
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especially when we get to like
1:10
simplifying
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fractions so for example if I have 3
1:16
over4 you
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two I need you to write this down 3 over
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4 right * 1/
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2 that
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becomes that becomes 3 * 1
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right over 4 * 2 which is technically 3
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8 this is how you multiply fraction you
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put the denominators and you multiply
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them together and the numerators you
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multiply them together right so very
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simple step now we're going to first
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talk about how to
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multiply positive fractions positive
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okay so all are positive so if I have 1
1:58
6 * 2 over 3 that becomes what 2 over
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18 right because is 1 * 2 6 * 2 18 now
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what do I need to do now I have 2 18 am
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I
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done what needs to be
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done simplify right simp simplify so
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that becomes now part of our cabulary we
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have to simplify all fractions must be
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simplified because once you get like
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calculus or pre-calculus if you don't
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simplify your fractions your teacher
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assumes that you didn't do it when you
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were in the lower stages of your
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mathematic career and it gives you it
2:40
Tak some points off so you got to make
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sure you simplify every fraction right
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so how do I simplify this fraction I
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have 2 over
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18 by two right so each number here is
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divisible by two so I get what if I
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divide this by
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two and divide it by two it gives me 1
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over what 1/ 9 so that becomes 1 over 9
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okay 1 over 9 so 2 over 18 is basically
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1 over 9 all right now I want you to try
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this here 1 12 * 410 and I want you to
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tell me what you get when you simplify
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that
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fraction 1 2 * 4
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10 did you shift all the desk for did I
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shift all the I feel it's closer I
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didn't shift
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anything yeah
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is that's uh you did 1 * 4 is what four
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right yeah four and what 20 right so if
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you that's okay and then you could
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do two what's the biggest number that
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goes into four and 20 four right you
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divide by four by four oh yeah you get
4:01
what 1 over 5 right 1 15 so that should
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give you 1
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15 all right now what about
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52 * 6
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over2 5 over 12 * 6
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12 so what's a good thing to do here
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before you even do this fraction
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what can you do to make it simple before
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you even
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like divide this by what what goes into
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six into
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12 oh three three well let's go a little
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further right six so I make it sense
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like so if I divide this by six and this
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by six I get 5 over 12 still times 12
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right so that way I don't have to deal
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with gigantic numbers I'm I'm trying to
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really make it easy before I even like
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proceed to the uh to multiplying it
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right so that
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becomes 5 over
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24 all right does that make sense no
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why how you were dividing it I'm not
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dividing I I divided here 6 over 12
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before you multiply I want I want to
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simplify this fraction okay I have 6
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over 12 right before I even do it I know
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that I can divide it by six and this by
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six so I get
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one2 that way I have a smaller fraction
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and then I can go 52 * 1 2 that make
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sense now all right all right now now
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how do you multiply this you can
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simplify before and after before
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sometimes it's easier exactly you can
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multiply and then simplify or simplify
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before you multiply you have have
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multiple options again I don't impose
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anything whichever you're comfortable
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doing okay now look here now we going to
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learn how to multiply negative fractions
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right negative fractions here I have 34
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* -7 over 9 right so what is that going
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to give
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me what do you think 3 over 4 * -7 / 9
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yeah
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um for the numerators
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be7 M which would be negative my bra's
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not working today 21- 21 all right and
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then in the denominator 4 * 9 right
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which
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is 36 if you know your multiplication
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table right and then when I get here yes
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for the N I just do 10us the thing that
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you're m
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again
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yeah
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why why in your thing is there half
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cursive half not cursive I don't care I
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just chose to do however I feel like it
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there's no Rhyme or Reason I just do
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whatever I want right so I have 21 over
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36 do I leave this like this oh no no
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I'm trying to simplify that right what
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goes into 21 that goes into 36 seven
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seven no no what goes into three right
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three how do you know if a number is
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divisible by three remember I told you
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you add the digit right 2 + 1 is what
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three so therefore it's divisible by 3 3
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+ 6 is 9 so you know 3 goes into 21 and
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3 also goes into 26 right all right so
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now you go divide this by three you get
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-7 because it's a negative don't forget
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that and then you get 12 here right so
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it should be - 712 now let me ask you a
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question if you were to do do the same
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thing with this fraction before you even
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multiply it right what would you want to
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do to make this a lot simpler I have 9
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12 * -23 I always want to make
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sure can I simplify this fraction first
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I even multiply can I do that yes or no
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yes right so what goes into 9 that goes
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into 12 press three right three so if I
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divide this by three what do I get three
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and so it becomes therefore negative I
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don't forget my negative right 3 over 4
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* -2 over 3 right now if you were in
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calculus I would have thought you you
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could just simplify this like this but
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right now I don't want to overwhelm you
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right you can simplify this and you left
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-2 or four you can do that but let's go
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step by step okay so now now what do I
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do here -3 * -2 that equals what -6 -3 *
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-2 that
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equal remember a negative * a negative
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becomes a positive right so I get 6 over
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12 and I can simplify it by what six
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right divide by six again so I
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get2 all right now you guys can go ahead
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and try this one 6 over 9 * 3 over
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11 mhm
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you got it sides um both vers you can if
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but can you simplify this one you think
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you can no no right but can you do this
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one though
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yeah right 6 over 9 we can simplify that
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can we 2 over three 2 over three right
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it because three goes into six three
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goes into 9 so you have
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23 time * -3 11 what does that give
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you 6 33 negative or POS 6 over 33 and
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then what what goes into six that goes
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into 33 three right so you get -2 over
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11 right again let me show you something
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else that you could do and if you are
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comfortable with it you do it that way
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if you're not just do it the simple way
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right now if I had the same fraction I
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have 6 over 9 * -3 11 right here what I
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would have done right I know that three
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goes into this right I got two here and
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then three goes into nine I have three
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are we good on that I also know that
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this three and these three are going to
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what cancel out right three and I'm left
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what one here I'm left what one so now I
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have 2 * -1 over what 11 so this is2 11
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now some people absolutely hate this
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because this is too much for my head I
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don't like it you don't have to do it
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this way if you like the traditional way
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you want to just go your step by step
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I'm fine but if you feel like this is
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more comprehensible to you and you you
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want to do this by all means go ahead
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okay so I don't impose anything if you
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like this do this way if you like this
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way do that way all right now we're
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going to learn how to multiply mix
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fraction actions isn't that fun no no I
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know you don't like mixed fractions
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you're so
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[Music]
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mean your your laughs are a little like
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creepy really it's a little suspicious
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how is you suspicious because I'm
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laughing like that you laugh like that
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like that so how should I laugh
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like is that
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better it's a little bit better oh no
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certainly not that one laugh like that
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no way all right so let's try this
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two 1/3 * 2 57 right so if I want to
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multiply this mixed
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fractions uh let's figure out what we're
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going to do here and then we're going to
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try this one too 3
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38 time 2
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13 all right yep do you need your V to
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do I need my book your book tell you
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what to teach tell you the ansers um
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yes she's she's making you want to teach
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the class sure maybe thank you all so
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she won't do evil laugh so all right so
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there you go
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again so how do I do this here right I'm
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going to turn this into what a improper
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fraction first right
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this is not 25 by the way this is two
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right so I want to turn this into a
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improper fraction yeah can I do 2 * two
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and then multiply the
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fractions um let's try to see if that
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works too okay right let's try so let's
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do it this way 2 * 3 + 1 correct that is
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7/3 right and then 2 * 7 14 + 5 19 over
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7 right
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now look here I like easy stuff I don't
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want to multiply these two first and
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then do cuz I don't want to do that
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right can I cancel out the sevens yes
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yes I can right so I'm going this way
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I'm left what 19 over three does that
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make sense I know the answer right 19
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over three all right and you can turn
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this into a mix fraction if you want you
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don't have to now let's try if you can
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do it the other way 2 13 where she said
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I don't know if you can but we'll see
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right she said 2 * 2 is what four right
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and then 1 and five this is 5 over 21 uh
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let's see if you do it this way 4 * 21
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that's 80 uh 84 + 5 that's 89 that's a b
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over 21 no we can't do that way you
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can't simplify no we again so I hope
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hopefully nobody watches this and feel
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like what is he teaching that's there's
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no way we can do do that way all right
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we can put it t get in a little bit so
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you have to multiply them you have to
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turn them into improper fractions first
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and then simplify all right so let's try
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this here 3 38 and then 2
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1/3 3 * 8 is 24 right 24 + 3 27 over 8 2
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* 3 is 6 6 + 1 7 so 7/3 right now look
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at here can I simplify this again before
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I multiply it can I yes three goes into
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what 27 right how many
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times nine times right three goes into
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27 nine
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times like that way we can do this way
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it'll be easier right 9 * 7 if you know
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your multiplication is
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63 63 over this is one here this is
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eight and then you're done because
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there's no number that goes into eight
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that goes into 63 is there no right so
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this is it okay so this is how you
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multiply mixed fraction you first have
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to turn them into improper fraction and
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then multiply now what we're going to do
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is now we're going to learn how to
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evaluate um rational expression using
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multiplication
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so I want you to
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evaluate 12 time um AB
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if a
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=
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67 and
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bal4 9 all right so how do you think we
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going to solve this problem I want you
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to evaluate what am I saying when I say
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evaluate what does that mean solve to
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solve it right to find the expression so
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we know that the Expressions were 12 a
16:55
what does that mean 12 a
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times times right the multiplication so
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I'm going to substitute a by what 6 67
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and B
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by4 49 so we have it's going be 12 * 6 7
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* 4 9 how many fractions do I have here
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three three three fractions right so
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we're going to do them two at like I'm
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going to do these two together and then
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whatever whatever I get here I'm going
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to combine it with this one correct
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right so now two right if I just single
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out these two here for now right two
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goes into what six mhm right three times
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so we're left with 1 over 1 * 3 over 7
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so that is 1 1 * 3 is 3 1 * 7 is 7 so
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now it's 37 * -4 over 9 right now again
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can I simplify this again before I
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multiply yes it's a fraction negative
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just no the well the the whole fraction
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is negative so even if the numerator is
18:02
negative it still makes the entire
18:04
fraction negative it doesn't matter what
18:05
you can put in the middle you can put it
18:07
on top you can put it in the bottom but
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preferably people like to keep it in the
18:10
middle okay but it doesn't matter where
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you put the negative it's still a
18:13
negative fraction now if I have 3 7 * -4
18:18
9 right what do I do here can I simplify
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the 3 and the 9 yeah right I can do that
18:24
right the three is gone so I'm left with
18:25
one and here I'm left with three right
18:28
so now one * -4 which
18:31
is -4 and then 7 * 3 that is 27 2 one
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right and then this is my final result
18:40
you got it so we can do the butterfly
18:42
method too say it again we can do the
18:44
butterfly method too butterfly method
18:46
which is it's that the one that I just
18:49
did yeah we learned that last year okay
18:51
yeah you can do that you call that
18:52
butterfly method all right I didn't know
18:54
that it had a name butterfly method all
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right
18:58
all right it like all right so what I
19:01
want to do now is I'm going to give you
19:02
some some stuff to work on on your own
19:05
okay
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