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We're asked, how many inches
are in 4 and 1/2 yards? And we'll do it
a couple of ways. One, we could just say how
many inches are in 4 yards and how many inches
are in a 1/2 of a yard? And this really
4 plus 1/2 yards. Or we could convert this
into an improper fraction first and then convert. But before I even
do that, let's just think about how many
inches there are in a yard. So if I have 1 yard,
we know that there are 3 feet for
every 1 yard, Right? And when you say, why am
I multiplying by 3 instead of saying there's 1
yard for every 3 feet? And the easiest way to
think about it is you're going to have a larger
value over here, and you're also going to want
to have these units right over here cancel out. So yard is canceling
out with yard. So you have 1 yard is
equal to 3 feet, which is kind of what we already knew. I'm just showing you how
the dimensions cancel out. And how many inches
are there per foot? Well, we know that there are
12 inches for every 1 foot. And same logic over here,
inches is a smaller unit of measurement so it makes sense
that we're multiplying by 12. 3 feet is going
to be more inches, so we're multiplying by 12. And also these
units cancel out-- foot in the numerator,
foot in the denominator. 3 times 12 divided by 1
is equal to 36 inches. So you might have
already known it. But this is nice to have
the dimensions cancel out like this. We know that 1 yard
is equal to 36 inches. Or there are 36 inches
for every 1 yard. And so we can now
either break this down, or we can turn this into
an improper fraction. First I'll just break it down
into 4 yards plus 1/2 yards. So we could say that
this is 4 yards. So 4 yards is going
to be equal to-- well, let's just multiply
it times 36 inches, 36 inches for every 1 yard. The yards cancel out. 4 times 36 is 120,
plus 24, so that's 144. So this is equal to 144 inches. That's just the 4 yards. And then if we do the 1/2
yards, so 1/2 of a-- I guess I say 1/2
yard, once again, times 36 inches per yard,
the yards cancel out. 1/2 times 36 is going to
be equal to 18 inches. So 4 and 1/2 yards
is the same thing as 4 yards plus 1/2 yards, which
is the same thing as 144 inches plus 18 inches, which is
going to give us-- let's just add it up over
here on the right, 144 plus 18. 4 plus 8 is 12. 4 plus 1 is 5. You have this 1 up here, so
it's 6, and then we have a 1. So when you add them all
together, you get 162 inches. The other way to
do this would have been to convert this
into an improper fraction and then multiply by the unit. So let's do it that way. If I have 4 and 1/2 of
anything, really-- so let me write 4 and 1/2. I'm trying to find
a suitable color. So if I have 4 and 1/2,
this is the same thing-- 4 is the same thing as 8/2. This is the same thing-- so
let me write it this way. 4 and 1/2 is the same
thing as 4 plus 1/2, which is the same
thing as-- if we want to have the same
denominator as this 2 over here or as
this 1/2 over here, this is the same thing as 8/2. Or you could say 4/1 is
the same thing as 8/2, if we want to have a common
denominator, so 8/2 plus 1/2. Actually, let me write it
that way, just so you really understand what we're doing. 4 is the same thing as 4/1. So it's 4/1 plus 1/2. If we want to find a
common denominator, it's 2. So 4/1 is the same
thing as 8/2 plus 1/2, which is equal to 9/2. Now, I did it this way,
which takes longer, just so you really understand
how we converted it, why it makes-- hopefully
conceptually why it just makes intuitive sense, why 4 and
1/2 is the same thing as 9/2. But if you want a
simple process for it, you could just say,
look, 4 times 2 is 8. 8 plus 1 is 9. And that gives you that 9
right over there, so 9/2. So we have 9/2 yards that we
want to convert to inches. Same process-- times
36 inches per yard. Yard in the numerator,
yard in the denominator. We are left with 9/2 times 36. We could say times 36/1 if we
like, 36 Inches for every 1 yard. 36-- or the number 36
really is the same as 36/1. And then we're left with
just inches in our units. We're just left with inches. And over here there's several
ways that we can simplify it. Probably the easiest
way to simplify it is we can divide both our numerator
and our denominator by 2. So let me write it this way. I don't want to skip steps. So we have 9 times 36 over
2 times 1, or over 2 inches. And we can divide the
numerator and the denominator by 2 to simplify it. They're both divisible by 2. 36 divided by 2 is 18. 2 divided by 2 is 1. So we're really just left
with 9 times 18 inches. We can just multiply 9 times 18. Let me do it over here. 18 times 9. 8 times 9 is 72. 1 times 9 is 9, plus 7 is
16, so we get 162 inches. So all of this simplifies to
162 inches, and we are done.