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# Comparing and ordering fractions

## Video transcript

What I want to do in this
video is order these fractions from least to greatest. And the easiest way
and the way that I think we can be sure we'll
get the right answer here is to find a common denominator,
because if we don't find a common denominator,
these fractions are really hard to compare. 4/9 versus 3/4 versus
4/5, 11/12, 13/15. You can try to estimate
them, but you'll be able to directly
compare them if they all had the same denominator. So the trick here, or at
least the first trick here, is to try to find that
common denominator. And there's many ways to do it. You could just pick
one of these numbers and keep taking its multiples
and find that multiple that is divisible by all the rest. Another way to do it is look
at the prime factorization of each of these numbers. And then the least
common multiple of them will have to have at least all
of those prime numbers in it. It has to be composed
of all of these numbers. So let's do it that second way. And then let's verify that
it definitely is divisible. So 9 is the same
thing as 3 times 3. So our least common
multiple is going to have at least
one 3 times 3 in it. And then 4 is the same
thing as 2 times 2. So we're going to also
have to have a 2 times 2 in our prime factorization
of our least common multiple. 5 is a prime number. So we're going to need
to have a 5 in there. And then 12-- I'm going
to do that in yellow. 12 is the same thing
as 2 times 6, which is the same thing as 2 times 3. And so in our least
common multiple, we have to have two 2's. But we already have two 2's
right over here from our 4. And we already have
one 3 right over here. Another way to think about it
is something that is divisible by both 9 and 4 is going
to be divisible by 12, because you're going
to have the two 2's. And you're going to have
that one 3 right over there. And then, finally, we need
to be divisible by 15's prime factors. So let's look at
15's prime factors. 15 is the same
thing as 3 times 5. So once again, this number
right over here already has a 3 in it. And it already has a 5 in it. So we're cool for 15,
for 12, and, obviously, for the rest of them. So this is our least
common multiple. And we can just
take this product. And so this is going to be
equal to 3 times 3 is 9. 9 times 2 is 18. 18 times 2 is 36. 36 times 5, you could do that
in your head if you're like. But I'll do it on the
side just in case. 36 times 5, just so
that we don't mess up. 6 times 5 is 30. 3 times 5 is 15 plus 3 is 180. So our least common
multiple is 180. So we want to rewrite
all of these fractions with 180 in the denominator. So this first fraction,
4/9, is what over 180? To go from 9 to 180, we have to
multiply the denominator by 20. So let me do it this way. So if we do 4/9, to get the
denominator of 9 to be 180, you have to multiply it by 20. And since we don't
want to change the value of the
fraction, we should also multiply the 4 by 20. So we're just really
multiplying by 20/20. And so 4/9 is going to be
the same thing as 80/180. Now, let's do the
same thing for 3/4. Well, what do we have to
multiply the denominator by to get us to 180? So it looks like 45. You could divide 4 into
180 to figure that out. But if you take 4 times
45, 4 times 40 is 160. 4 times 5 is 20. You add them up. You get 180. So if you multiply
the denominator by 45, you also have to multiply
the numerator by 45. 3 times 45 is 120 plus 15. So it's 135. And the denominator here is 180. Now, let's do 4/5. To get our
denominator to be 180, what do you have
to multiply 5 by? Let's see. If you multiply 5 by
30, you'll get to 150. But then you have another 30. Actually, we know
it right over here. You have to multiply it by 36. Well, then you have to multiply
the numerator by 36 as well. And so our denominator
is going to be 180. Our numerator, 4
times 30 is 120. 4 times 6 is 24. So it's 144/180. And then we have
only two more to do. So we have our 11/12. So to get the
denominator to be 180, we have to multiply 12
by-- so 12 times 10 is 120. Then you have 60 left. So you have to multiply it
by 15, 15 In the denominator, and 15 in the numerator. And so the denominator
gives us 180. And 11 times 15. So 10 times 15 is 150. And then you have one more 15. So it's going to be 165. And then, finally,
we have 13/15. To get our denominator to be
180, have to multiply it by 12. We already figured out
that 12 times 15 is 180. So you have to
multiply it by 12. That will give us 180
in the denominator. And so you have to also
multiply the numerator by 12, so that we don't change
the value of the fraction. We know 12 times 12 is 144. You could put one
more 12 in there. You get 156. Did I do that right? 12 plus 144 is going to be 156. So we've rewritten
each of these fractions with that new common
denominator of 180. And now, it's very
easy to compare them. You really just have to
look at the numerators. So the smallest
of the numerators is this 80 right over here. So 4/9 is the smallest. 4/9 is the least
of these numbers. So let me just
write it over here. So this is our ordering. We have 4/9 comes first, which
is the same thing as 80/180. Let me write it
both ways-- 80/180. Then the next the
smallest number looks like it's this
135 right over here. I want to do it in
that same color. The next one is going to
be that 135/180, which is the same thing as 3/4. And then the next one is
going to be-- let's see, we have the 144/180. So this is going to
be the 144/180, which is the same thing as 4/5. And then we have two more. The next is this 156/180. So then we have
our 156/180, which is the same thing as 13/15. And then we have one
left over, the 165/180, which is the same thing-- I
want to do that in yellow. We have our 165/180, which
is the same thing as 11/12. And we're done. We have finished our ordering. So if you're doing the Khan
Academy module on this, this is what you would input
into that little box there.