Multiplication and Division of Rational Numbers Video Lecture - Class 7

we've added and subtracted rational
numbers we've multiplied and divided
integers now it's time to take the next
step and multiply and divide rational
numbers let's go
you
you
welcome to anywhere math I'm Jeff
Jacobson and today we're going to learn
how to multiply and divide rational
numbers let's get started right away
with example 1 negative 2/7 times 1/6
now before we get going one common
question that I get asked by a lot of
students is negative 2/7 does that mean
it's negative 2 over 7 or 2 over
negative 7 or are they both negative
negative 2 over negative 7 and really
either of these would work negative 2
over 7 that's the same thing as negative
2/7 or it could be 2 over negative 7
that's also the same thing as negative
2/7 but this is not negative divided by
negative would give actually give me 2/7
that's a positive right so that's not
that isn't that doesn't work but you can
think of it negative 2/7 as either of
these I think it helps if you're
consistent and just a common thing to do
is just consider the numerator the
negative okay but also remember that if
you have something like this negative 2
over 7 you can rewrite that just like
this okay negative 2/7 and that's
typically what you'll see in class and
in math test textbooks they have the
negative pulled out to the front of the
fraction so now that we've got that
taken care of let's actually do the
problem the same rules apply negative
times negative is a positive negative
times positive is a negative right same
signs give you a positive different
signs give you a negative and when we're
multiplying fractions we always try to
simplify first so if I'm thinking if I'm
thinking of this as negative 2 over 7
times 1 6 is there anything I can
simplify well negative 2 & 6 have a
common factor of 2 so I'm going to
simplify that way so that becomes just
negative 1
that becomes three right they have a
factor of 2 negative 2 divided by 2 is
negative 1 6 divided by 2 is 3 and now
nothing else is simplified so I'm just
going to multiply straight across
negative 1 times 1 is negative 1 7 times
3 is 21 and then here's what I'm talking
about at the end I'm just going to
rewrite it pull the negative out front
so it's negative 1 over 21 and there's
my final answer let's do another example
example 2 now we are dividing mixed
numbers and hopefully you divided mixed
numbers before or multiply them before
the only difference now is that this is
a negative mixed number but we approach
it the same way first thing we're going
to do is change them both to improper
fractions if you want to pause and go
ahead and try to do this yourself go for
it but here we go
5 times 5 is 25 plus 1 is 26 so that is
26 over 5 and that's negative 26 over 5
divided by 3 times 2 is 6 plus 1 is 7
denominator stays the same that's 3 now
dividing by a fraction is the same thing
as multiplying by its reciprocal and one
way to remember that is keep change flip
you keep the original fraction that
doesn't change 26 over 5 you change the
operation division becomes
multiplication and you flip it the
reciprocal of 7/3 is 3/7 now it's just
we're multiplying fractions always try
to simplify first before you multiply
but in this case there is nothing to
simplify so we're just going to have to
multiply straight across I'm going to
keep this negative with the 26 so
negative 26 times 3 well 3 times 20 is
60 plus 3 times 6 is 18 so that's
negative 78 right negative times
positive is negative 5 times 7 is 35 and
then
this is an improper fraction so my last
step is to change it back to a mixed
number 35 into 78 goes to whole times
that would give me 70 and I have 8 left
over 8 over 35 and I have to remember
that this is negative because negative
divided by a positive is going to be
negative so I'm going to give myself a
little bit more room negative 2 and 8
thirty-fifths ok let's try another
example example 3 I'm still multiplying
with fractions
now the only difference is I have 3 okay
3 rational numbers and I'm multiplying
together the fact that they're all
multiplying means that I can change the
order the commutative property says that
the order doesn't matter with addition
or multiplication as long as it's all
multiplication or all addition right 2
times 3 is the same as 3 times 2 so here
I can change the order and in this case
it would really help me simplify some
things so hopefully you notice you've
got negative 7 and negative 1/7 okay
those are reciprocals so what I'm going
to do is change it to negative 1/7 times
negative 7 and I'm going to rewrite that
whenever we're multiplying fractions
with whole numbers sometimes it helps to
write it as a fraction so make it look
like a fraction and we do that by just
putting it over 1 so I've got negative 7
over 1 I just flip that here change the
order and then I still have times the
4/5 and here always try to simplify
before you multiply well 7 and negative
7 have a common factor of 7 so that
becomes 1 negative 7 divided by 7 is
negative 1 don't forget the negative and
that's all I can simplify so now I'm
just going to multiply
I'll just bring it down here so negative
1 times negative 1 is 1 1 times 1 is
one so I'm taking care of that I still
have the times for fists well 1 over 1
is just 1 times 4/5 is 4/5 so
rearranging it and simplifying really
made this problem a lot easy excuse me a
lot easier let's try one more example
example 4 we've multiplied and divided
with fractions and mixed numbers we
can't forget about decimals there are
also rational numbers so here's our last
example negative two point five seven
times three point six we approach it the
same way really the only difference is
the negative you should have already
learned how to multiply decimals so this
hopefully is just kind of a review when
you're multiplying decimals we don't
have to line up the decimal my
suggestion is pretend the decimals
aren't there set up your problem and
then worry about the decimals after that
so instead of negative two point five
seven I'm just going to think of it as
negative 257 times 36 and I would set
that up just like that okay imagine the
decimals aren't there now that I've set
up my problem I'm going to put them back
just so I don't forget and now it's just
going to we're just going to multiply I
don't really need to worry too much
about the negative I just know that
negative times positive will give me a
negative so as long as I remember a
negative at the end I should be fine so
now it's just multiplying 7 times 6 is
42 carry the 4 30 plus 4 is 34 carry the
3 there's 12 right so don't get too hung
up on the negative 6 times 2 is 12 plus
3 is 15 I'm not I'm not thinking well 6
times 2 is or 6 times negative 2 is
negative 12 plus 3 is negative 9 I'm not
thinking that way ok so really you could
even just kind of get rid of that if you
want as long as you remember it at the
end
now I need to go at a zero cost those
off 3 times 7 is 21 2 15 16 17 6 7 Adam
up to 512 9 now again don't forget I
know my answer is negative so I'm going
to put that in there and then the last
step hopefully you remember you got to
figure out where to place the decimal
well two point five seven has two
decimal places as 23.6 has one right
here I had to so my my product my answer
should have three so I'm going to go one
two three and my decimal point should go
right there
so my product negative two sorry
negative nine point two five two and if
I want to kind of estimate see if that
makes sense I could round that to
negative 3 times 4 is negative 12
yeah that's pretty close okay there's
our last example here's something to try
on your own
thank you so much for watching and
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