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## Fractions as division

# Creating mixed numbers with fraction division

CCSS.Math:

## Video transcript

We've already seen that
a fraction like 2/9 can be interpreted
as 2 divided by 9. Let me do the 2
in the same color. That if we have a
fraction, it could be interpreted as the numerator
divided by the denominator. And this leads to all sorts
of interesting conclusions, some of which we've already
seen, and some of which are a little bit new. So, for example, if I
had the fraction 7/7, this can be interpreted
now as 7 divided by 7, as our numerator divided
by our denominator. And 7 divided by 7 is
of course equal to 1. And this is consistent with
what we've already seen. 7/7 would get us to
a whole, and a whole is the exact same thing as one. But we could do things a little
bit more interesting as well. We could take something
like 18/6 and realize, wow. This is the same thing
as 18 divided by 6, which we know is equal to 3. And we should do a
little reality check. Does this make sense, that
18/6 should be equal to 3? Well, we could rewrite it. We could rewrite 18/6. Let me make the sixths
that same orange color. That's going to
be the same thing. 18 is 6 plus 6 plus 6. And then all of that over 6,
and then that's the same thing. That's the same thing as
6/6 plus 6/6 plus 6/6. And I could make this
right over here in orange. And we've already seen, or we've
seen many, many videos ago, that 6/6, just like 7/7, these
are each equal to a whole. These are each equal to 1. And we can now view
this as 6 divided by 6, which is the same thing as 1. So this is 1 plus 1 plus 1,
which is, of course, equal to 3. But this starts to raise
an interesting question. This all worked out just fine
because 18 is a multiple of 6. 6 divides evenly into 18. But what happens if we
start having fractions where the denominator does not divide
evenly into the numerator? Let's say we have a
fraction like 23 over 6. Well, we know that
we can interpret this as 23 divided by 6. And if we actually divide
23 by 6-- let's do that. So we divide 6 into 23. We know 6 goes into
23 three times. 3 times 6 is 18. And then when you subtract, you
end up with a remainder of 5. So we might say, hey, 23 divided
by 6 is equal to 5 remainder 3. But that's not that satisfying. What do I do with
this remainder? This really isn't a number here. This is just saying that
we're going five times, and then we have a
little bit left over. What we can do now is
manipulate this a little bit so that we can realize that this
is a number, and in particular, a mixed number. So for example, we could
start with a 23 over 6, and we could divide
it into-- or we could decompose the
numerator into one part that is divisible by 6,
evenly divisible by 6, and the remainder. So for example, 23 over 6 we
can rewrite as 18 plus 5 over 6. Notice I decomposed the
23 into one part that is a multiple of 6, and
it's the largest multiple that essentially
fits into 23, or that is less than or equal to
23, and then the remainder. When you divide 6 into 23,
you get a remainder of 5. You could view it
as, I divided it into the remainder
and everything else. And the reason why
this is interesting is because we know that this is
going to be equal to 18 over 6 plus 5 over 6. Well, we already
know that 18/6 is the same thing as 18
divided by 6, or 3. So this is the same thing as 3. So we know that 23 over 6,
which is the same thing as 18 plus 5 over 6, is the
same thing as 3 plus 5/6. Or, if we want to write
it as a mixed number, we could write it as 3 and 5/6.