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# Finding common denominators

Rewrite 2/8 and 5/6 to have the same denominator. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to rewrite
the following two fractions as fractions with
a least common denominator. So a least common
denominator for two fractions is really just going to be the
least common multiple of both of these denominators over here. And the value of
doing that is then if you can make these
a common denominator, then you can add
the two fractions. And we'll see that
in other videos. But first of all, let's just
find the least common multiple. Let me write it out
because sometimes LCD could meet other things. So least common denominator
of these two things is going to be the same thing
as the least common multiple of the two
denominators over here. The least common
multiple of 8 and 6. And a couple of ways to think
about least common multiple-- you literally could just
take the multiples of 8 and 6 and see what they're
smallest common multiple is. So let's do it that way first. So multiples of six
are 6, 12, 18, 24 30. And I could keep going if we
don't find any common multiples out of this group here with
any of the multiples in eight. And the multiples of
eight are 8, 16, 24, and it looks like we're done. And we could keep
going obviously-- 32, so on and so forth. But I found a common
multiple and this is their smallest
common multiple. They have other common
multiples-- 48 and 72, and we could keep adding
more and more multiple. But this is their
smallest common multiple, their least common multiple. So it is 24. Another way that you could have
found at least common multiple is you could have taken the
prime factorization of six and you say, hey,
that's 2, and 3. So the least common multiple has
to have at least 1, 2, and 1, 3 in its prime factorization
in order for it to be divisible by 6. And you could have said, what's
the prime factorization of 8? It is 2 times 4
and 4 is 2 times 2. So in order to be
divisible by 8, you have to have at least three
2's in the prime factorization. So to be divisible by 6, you
have to have a 2 times a 3. And then to be divisible by 8,
you have to have at least three 2's. You have to have two
times itself three times I should say. Well, we have one 2 and
let's throw in a couple more. So then you have another
2 and then another 2. So this part right over here
makes it divisible by 8. And this part right over
here makes it divisible by 6. If I take 2 times 2 times 2
times 3, that does give me 24. So our least common
multiple of 8 and 6, which is also the least common
denominator of these two fractions is going to be 24. So what we want to do is
rewrite each of these fractions with 24 as the denominator. So I'll start with 2 over 8. And I want to write that
as something over 24. Well, to get the
denominator be 24, we have to multiply it by 3. 8 times 3 is 24. And so if we don't
want to change the value of the
fraction, we have to multiply the numerator and
denominator by the same thing. So let's multiply the
numerator by 3 as well. 2 times 3 is 6. So 2/8 is the exact
same thing as 6/24. To see that a
little bit clearer, you say, look, if I have 2/8,
and if I multiply this times 3 over 3, that gives me 6/24. And this are the same
fraction because 3 over 3 is really just 1. It's one whole. So 2/8 is 6/24 let's do
the same thing with 5/6. So 5 over 6 is equal
to something over 24. Let me do that in
a different color. I'll do it in blue. Something over 24. To get the denominator
from 6 to 24, we have to multiply it by 4. So if we don't want to
change the value of 5/6, we have to multiply the
numerator and denominator by the same thing. So let's multiply the
numerator times 4. 5 times 4 is 20. 5/6 is the same thing as 20/24. So we're done. We've written 2/8 as 6/24 and
we've written 5/6 as 20/24. If we wanted to add them
now, we could literally just add 6/24 to 20/24. And I'll leave you
there because they didn't ask us to
actually do that.