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# Comparing fractions 2 (unlike denominators)

Sal compares fractions by finding a common denominator. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

Use less than,
greater than, or equal to compare the two fractions
21/28, or 21 over 28, and 6/9, or 6 over 9. So there's a bunch
of ways to do this. The easiest way is if they
had the same denominator, you could just compare
the numerators. Unlucky for us, we do not
have the same denominator. So what we could do is we
can find a common denominator for both of them and convert
both of these fractions to have the same denominator
and then compare the numerators. Or even more simply, we could
simplify them first and then try to do it. So let me do that
last one, because I have a feeling that'll be
the fastest way to do it. So 21/28-- you can see that
they are both divisible by 7. So let's divide
both the numerator and the denominator by 7. So we could divide 21 by 7. And we can divide-- so let
me make the numerator-- and we can divide
the denominator by 7. We're doing the same
thing to the numerator and the denominator,
so we're not going to change the
value of the fraction. So 21 divided by 7 is 3,
and 28 divided by 7 is 4. So 21/28 is the exact
same fraction as 3/4. 3/4 is the simplified
version of it. Let's do the same thing for 6/9. 6 and 9 are both divisible by 3. So let's divide
them both by 3 so we can simplify this fraction. So let's divide
both of them by 3. 6 divided by 3 is 2,
and 9 divided by 3 is 3. So 21/28 is 3/4. They're the exact same fraction,
just written a different way. This is the more
simplified version. And 6/9 is the exact
same fraction as 2/3. So we really can
compare 3/4 and 2/3. So this is really
comparing 3/4 and 2/3. And the real benefit of
doing this is now this is much easier to find a common
denominator for than 28 and 9. Then we would have to
multiply big numbers. Here we could do
fairly small numbers. The common denominator
of 3/4 and 2/3 is going to be the least
common multiple of 4 and 3. And 4 and 3 don't share any
prime factors with each other. So their least common
multiple is really just going to be the
product of the two. So we can write 3/4
as something over 12. And we can write 2/3
as something over 12. And I got the 12 by
multiplying 3 times 4. They have no common factors. Another way you could
think about it is 4, if you do a prime
factorization, is 2 times 2. And 3-- it's already
a prime number, so you can't prime
factorize it any more. So what you want to do
is think of a number that has all of the
prime factors of 4 and 3. So it needs one 2,
another 2, and a 3. Well, 2 times 2 times 3 is 12. And either way you
think about it, that's how you would get
the least common multiple or the common
denominator for 4 and 3. Well, to get from 4 to 12,
you've got to multiply by 3. So we're multiplying the
denominator by 3 to get to 12. So we also have to multiply
the numerator by 3. So 3 times 3 is 9. Over here, to get
from 3 to 12, we have to multiply the
denominator by 4. So we also have to multiply
the numerator by 4. So we get 8. And so now when we
compare the fractions, it's pretty straightforward. 21/28 is the exact
same thing as 9/12, and 6/9 is the exact
same thing as 8/12. So which of these is
a greater quantity? Well, clearly, we have the
same denominator right now. We have 9/12 is clearly
greater than 8/12. So 9/12 is clearly
greater than 8/12. Or if you go back
and you realize that 9/12 is the exact
same thing as 21/28, we could say 21/28 is definitely
greater than-- and 8/12 is the same thing as 6/9-- is
definitely greater than 6/9. And we are done. Another way we could
have done it-- we didn't necessarily
have to simplify that. And let me show you
that just for fun. So if we were doing it
with-- if we didn't think to simplify our
two numbers first. I'm trying to find a
color I haven't used yet. So 21/28 and 6/9. So we could just find
a least common multiple in the traditional way
without simplifying first. So what's the prime
factorization of 28? It's 2 times 14. And 14 is 2 times 7. That's its prime factorization. Prime factorization
of 9 is 3 times 3. So the least common
multiple of 28 and 9 have to contain a 2,
a 2, a 7, a 3 and a 3. Or essentially, it's
going to be 28 times 9. So let's over here
multiply 28 times 9. There's a couple of
ways you could do it. You could multiply
in your head 28 times 10, which would be 280,
and then subtract 28 from that, which would be what? 252. Or we could just multiply
it out if that confuses you. So let's just do the second way. 9 times 8 is 72. 9 times 2 is 18. 18 plus 7 is 25. So we get 252. So I'm saying the
common denominator here is going to be 252. Least common
multiple of 28 and 9. Well, to go from 28 to 252,
we had to multiply it by 9. We had to multiply 28 times 9. So we're multiplying 28 times 9. So we also have to multiply
the numerator times 9. So what is 21 times 9? That's easier to
do in your head. 20 times 9 is 180. And then 1 times 9 is 9. So this is going to be 189. To go from 9 to 252, we
had to multiply by 28. So we also have to multiply
the numerator by 28 if we don't want to change
the value of the fraction. So 6 times 28-- 6
times 20 is 120. 6 times 8 is 48. So we get 168. Let me write that
out just to make sure I didn't make a mistake. So 28 times 6-- 8 times 6 is 48. 2 times 6 is 12, plus 4 is 16. So right, 168. So now we have a common
denominator here. And so we can really just
compare the numerators. And 189 is clearly
greater than 168. So 189/252 is clearly
greater than 168/252. Or that's the same
thing as saying 21/28, because that's what
this is over here. The left-hand side is
21/28, is clearly greater than the right-hand side,
which is really 6/9.