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How To Multiply Rational Numbers & Evaluate Algebraic Expressions With Fractions
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
0:02
rational numbers okay so basically is
0:05
multiplying
0:06
fractions that's what it is fractions
0:09
yes it's right here so how do you
0:12
multiply fractions it's pretty simple
0:14
stuff not very complicated the only
0:17
thing that gets complicated is when you
0:18
have to um simplify the fractions but to
0:21
multiply fractions and I'm expecting
0:23
everyone to write this thing down right
0:25
so if you want to multiply fractions you
0:28
multiply the numerators together and the
0:30
denominators together you know what a
0:31
numerator is right as the top part of
0:34
the fraction and the denominator is the
0:37
bottom right so to multiply a fraction
0:40
if I have a b right times C over D this
0:45
is a fraction it becomes a * C over B *
0:50
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
1:03
me so make sure you write this down
1:05
because it's going to help you right
1:08
especially when we get to like
1:10
simplifying
1:11
fractions so for example if I have 3
1:16
over4 you
1:17
two I need you to write this down 3 over
1:21
4 right * 1/
1:24
2 that
1:26
becomes that becomes 3 * 1
1:30
right over 4 * 2 which is technically 3
1:35
8 this is how you multiply fraction you
1:39
put the denominators and you multiply
1:41
them together and the numerators you
1:42
multiply them together right so very
1:45
simple step now we're going to first
1:48
talk about how to
1:50
multiply positive fractions positive
1:54
okay so all are positive so if I have 1
1:58
6 * 2 over 3 that becomes what 2 over
2:05
18 right because is 1 * 2 6 * 2 18 now
2:11
what do I need to do now I have 2 18 am
2:14
I
2:16
done what needs to be
2:20
done simplify right simp simplify so
2:23
that becomes now part of our cabulary we
2:27
have to simplify all fractions must be
2:29
simplified because once you get like
2:31
calculus or pre-calculus if you don't
2:32
simplify your fractions your teacher
2:35
assumes that you didn't do it when you
2:36
were in the lower stages of your
2:38
mathematic career and it gives you it
2:40
Tak some points off so you got to make
2:41
sure you simplify every fraction right
2:44
so how do I simplify this fraction I
2:45
have 2 over
2:47
18 by two right so each number here is
2:50
divisible by two so I get what if I
2:52
divide this by
2:53
two and divide it by two it gives me 1
2:57
over what 1/ 9 so that becomes 1 over 9
3:01
okay 1 over 9 so 2 over 18 is basically
3:05
1 over 9 all right now I want you to try
3:08
this here 1 12 * 410 and I want you to
3:12
tell me what you get when you simplify
3:14
that
3:15
fraction 1 2 * 4
3:24
10 did you shift all the desk for did I
3:28
shift all the I feel it's closer I
3:31
didn't shift
3:40
anything yeah
3:43
is that's uh you did 1 * 4 is what four
3:46
right yeah four and what 20 right so if
3:49
you that's okay and then you could
3:52
do two what's the biggest number that
3:56
goes into four and 20 four right you
3:58
divide by four by four oh yeah you get
4:01
what 1 over 5 right 1 15 so that should
4:05
give you 1
4:06
15 all right now what about
4:10
52 * 6
4:21
over2 5 over 12 * 6
4:25
12 so what's a good thing to do here
4:27
before you even do this fraction
4:30
what can you do to make it simple before
4:32
you even
4:34
like divide this by what what goes into
4:37
six into
4:39
12 oh three three well let's go a little
4:43
further right six so I make it sense
4:46
like so if I divide this by six and this
4:49
by six I get 5 over 12 still times 12
4:56
right so that way I don't have to deal
4:57
with gigantic numbers I'm I'm trying to
5:00
really make it easy before I even like
5:02
proceed to the uh to multiplying it
5:05
right so that
5:06
becomes 5 over
5:10
24 all right does that make sense no
5:15
why how you were dividing it I'm not
5:19
dividing I I divided here 6 over 12
5:23
before you multiply I want I want to
5:25
simplify this fraction okay I have 6
5:27
over 12 right before I even do it I know
5:30
that I can divide it by six and this by
5:33
six so I get
5:35
one2 that way I have a smaller fraction
5:39
and then I can go 52 * 1 2 that make
5:42
sense now all right all right now now
5:47
how do you multiply this you can
5:49
simplify before and after before
5:52
sometimes it's easier exactly you can
5:55
multiply and then simplify or simplify
5:58
before you multiply you have have
5:59
multiple options again I don't impose
6:02
anything whichever you're comfortable
6:04
doing okay now look here now we going to
6:07
learn how to multiply negative fractions
6:11
right negative fractions here I have 34
6:14
* -7 over 9 right so what is that going
6:18
to give
6:20
me what do you think 3 over 4 * -7 / 9
6:31
yeah
6:33
um for the numerators
6:37
be7 M which would be negative my bra's
6:42
not working today 21- 21 all right and
6:46
then in the denominator 4 * 9 right
6:48
which
6:49
is 36 if you know your multiplication
6:52
table right and then when I get here yes
6:55
for the N I just do 10us the thing that
6:59
you're m
7:05
again
7:07
yeah
7:09
why why in your thing is there half
7:12
cursive half not cursive I don't care I
7:15
just chose to do however I feel like it
7:18
there's no Rhyme or Reason I just do
7:20
whatever I want right so I have 21 over
7:24
36 do I leave this like this oh no no
7:28
I'm trying to simplify that right what
7:29
goes into 21 that goes into 36 seven
7:33
seven no no what goes into three right
7:38
three how do you know if a number is
7:39
divisible by three remember I told you
7:41
you add the digit right 2 + 1 is what
7:43
three so therefore it's divisible by 3 3
7:46
+ 6 is 9 so you know 3 goes into 21 and
7:49
3 also goes into 26 right all right so
7:52
now you go divide this by three you get
7:55
-7 because it's a negative don't forget
7:57
that and then you get 12 here right so
8:00
it should be - 712 now let me ask you a
8:04
question if you were to do do the same
8:07
thing with this fraction before you even
8:09
multiply it right what would you want to
8:12
do to make this a lot simpler I have 9
8:15
12 * -23 I always want to make
8:18
sure can I simplify this fraction first
8:21
I even multiply can I do that yes or no
8:25
yes right so what goes into 9 that goes
8:27
into 12 press three right three so if I
8:31
divide this by three what do I get three
8:35
and so it becomes therefore negative I
8:38
don't forget my negative right 3 over 4
8:42
* -2 over 3 right now if you were in
8:47
calculus I would have thought you you
8:48
could just simplify this like this but
8:50
right now I don't want to overwhelm you
8:52
right you can simplify this and you left
8:54
-2 or four you can do that but let's go
8:57
step by step okay so now now what do I
8:59
do here -3 * -2 that equals what -6 -3 *
9:05
-2 that
9:07
equal remember a negative * a negative
9:09
becomes a positive right so I get 6 over
9:15
12 and I can simplify it by what six
9:19
right divide by six again so I
9:23
get2 all right now you guys can go ahead
9:26
and try this one 6 over 9 * 3 over
9:47
11 mhm
10:04
you got it sides um both vers you can if
10:10
but can you simplify this one you think
10:12
you can no no right but can you do this
10:15
one though
10:18
yeah right 6 over 9 we can simplify that
10:21
can we 2 over three 2 over three right
10:23
it because three goes into six three
10:26
goes into 9 so you have
10:28
23 time * -3 11 what does that give
10:33
you 6 33 negative or POS 6 over 33 and
10:39
then what what goes into six that goes
10:42
into 33 three right so you get -2 over
10:48
11 right again let me show you something
10:51
else that you could do and if you are
10:53
comfortable with it you do it that way
10:55
if you're not just do it the simple way
10:57
right now if I had the same fraction I
10:59
have 6 over 9 * -3 11 right here what I
11:03
would have done right I know that three
11:09
goes into this right I got two here and
11:11
then three goes into nine I have three
11:14
are we good on that I also know that
11:17
this three and these three are going to
11:19
what cancel out right three and I'm left
11:23
what one here I'm left what one so now I
11:26
have 2 * -1 over what 11 so this is2 11
11:32
now some people absolutely hate this
11:34
because this is too much for my head I
11:36
don't like it you don't have to do it
11:38
this way if you like the traditional way
11:40
you want to just go your step by step
11:42
I'm fine but if you feel like this is
11:44
more comprehensible to you and you you
11:46
want to do this by all means go ahead
11:49
okay so I don't impose anything if you
11:52
like this do this way if you like this
11:54
way do that way all right now we're
11:57
going to learn how to multiply mix
11:58
fraction actions isn't that fun no no I
12:04
know you don't like mixed fractions
12:06
you're so
12:07
[Music]
12:09
mean your your laughs are a little like
12:12
creepy really it's a little suspicious
12:15
how is you suspicious because I'm
12:18
laughing like that you laugh like that
12:21
like that so how should I laugh
12:24
like is that
12:26
better it's a little bit better oh no
12:29
certainly not that one laugh like that
12:33
no way all right so let's try this
12:36
two 1/3 * 2 57 right so if I want to
12:43
multiply this mixed
12:44
fractions uh let's figure out what we're
12:46
going to do here and then we're going to
12:48
try this one too 3
12:52
38 time 2
12:55
13 all right yep do you need your V to
12:59
do I need my book your book tell you
13:01
what to teach tell you the ansers um
13:08
yes she's she's making you want to teach
13:11
the class sure maybe thank you all so
13:14
she won't do evil laugh so all right so
13:17
there you go
13:19
again so how do I do this here right I'm
13:22
going to turn this into what a improper
13:26
fraction first right
13:30
this is not 25 by the way this is two
13:33
right so I want to turn this into a
13:35
improper fraction yeah can I do 2 * two
13:40
and then multiply the
13:42
fractions um let's try to see if that
13:45
works too okay right let's try so let's
13:48
do it this way 2 * 3 + 1 correct that is
13:53
7/3 right and then 2 * 7 14 + 5 19 over
13:58
7 right
14:00
now look here I like easy stuff I don't
14:04
want to multiply these two first and
14:06
then do cuz I don't want to do that
14:07
right can I cancel out the sevens yes
14:10
yes I can right so I'm going this way
14:13
I'm left what 19 over three does that
14:18
make sense I know the answer right 19
14:20
over three all right and you can turn
14:22
this into a mix fraction if you want you
14:24
don't have to now let's try if you can
14:26
do it the other way 2 13 where she said
14:29
I don't know if you can but we'll see
14:31
right she said 2 * 2 is what four right
14:36
and then 1 and five this is 5 over 21 uh
14:41
let's see if you do it this way 4 * 21
14:43
that's 80 uh 84 + 5 that's 89 that's a b
14:49
over 21 no we can't do that way you
14:51
can't simplify no we again so I hope
14:55
hopefully nobody watches this and feel
14:57
like what is he teaching that's there's
14:58
no way we can do do that way all right
14:59
we can put it t get in a little bit so
15:02
you have to multiply them you have to
15:04
turn them into improper fractions first
15:06
and then simplify all right so let's try
15:09
this here 3 38 and then 2
15:13
1/3 3 * 8 is 24 right 24 + 3 27 over 8 2
15:20
* 3 is 6 6 + 1 7 so 7/3 right now look
15:26
at here can I simplify this again before
15:28
I multiply it can I yes three goes into
15:32
what 27 right how many
15:36
times nine times right three goes into
15:40
27 nine
15:42
times like that way we can do this way
15:45
it'll be easier right 9 * 7 if you know
15:48
your multiplication is
15:51
63 63 over this is one here this is
15:54
eight and then you're done because
15:56
there's no number that goes into eight
15:58
that goes into 63 is there no right so
16:02
this is it okay so this is how you
16:04
multiply mixed fraction you first have
16:06
to turn them into improper fraction and
16:09
then multiply now what we're going to do
16:11
is now we're going to learn how to
16:12
evaluate um rational expression using
16:15
multiplication
16:18
so I want you to
16:23
evaluate 12 time um AB
16:31
if a
16:33
=
16:37
67 and
16:40
bal4 9 all right so how do you think we
16:43
going to solve this problem I want you
16:45
to evaluate what am I saying when I say
16:47
evaluate what does that mean solve to
16:49
solve it right to find the expression so
16:52
we know that the Expressions were 12 a
16:55
what does that mean 12 a
16:59
times times right the multiplication so
17:01
I'm going to substitute a by what 6 67
17:05
and B
17:07
by4 49 so we have it's going be 12 * 6 7
17:13
* 4 9 how many fractions do I have here
17:17
three three three fractions right so
17:20
we're going to do them two at like I'm
17:23
going to do these two together and then
17:25
whatever whatever I get here I'm going
17:26
to combine it with this one correct
17:29
right so now two right if I just single
17:32
out these two here for now right two
17:34
goes into what six mhm right three times
17:39
so we're left with 1 over 1 * 3 over 7
17:42
so that is 1 1 * 3 is 3 1 * 7 is 7 so
17:48
now it's 37 * -4 over 9 right now again
17:54
can I simplify this again before I
17:55
multiply yes it's a fraction negative
17:57
just no the well the the whole fraction
18:00
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
18:08
preferably people like to keep it in the
18:10
middle okay but it doesn't matter where
18:12
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
18:21
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
18:37
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
18:57
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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