We're going to determine
the length of each blue bar.
The first thing we should notice is that the units
are inches, and when giving a length,
it's important to include the units.
So looking at this first length, if we start here,
we can see we have one, two, three inches,
and then some extra length.
So let's partition the length from three inches
to four inches, into smaller pieces.
So if this is one inch from three to four inches,
if we cut it in half, we divided this inch length
into 1/2 inch partitions.
And now we can see the exact length
would be three and 1/2 inches.
Now over here, I also want to make the connection
between mixed numbers and improper fractions.
Three and a half is a mixed number
because we have a whole number and a proper fraction.
But if we divide this entire length into 1/2 inch partitions
we can also determine the mixed number for this length.
So if we partition this entire length into 1/2 inches
here, here, and here, now I can count.
We have one, two, three, four, five, six, seven
1/2 inch partitions, so we can say this is 7/2 inches.
So three and a half inches is equivalent to 7/2 inches.
Again, this is a mixed number, and this is
an improper fraction because the numerator
is greater than or equal to the denominator.
Now when giving a length, we normally
want to give it as a mixed number, or in this form here.
Let's take a look at our second length.
Again we're going to start here, we'll first count
the number of full inches.
So one, two, three, four, five, and now
we have to partition the length of one inch
from five inches to six inches, into smaller pieces.
So from here to here is one inch, we cut this in half,
we have 1/2 inch partitions.
Notice that still does not give us the exact length.
So now let's partition this one inch
into four equal pieces, so we'll divide it here, and here.
And now we can see the exact length
would be five and 3/4 inches.
Again, let's also express this as an improper fraction.
So we'll divide this entire length into 1/4 inch partitions.
So we have four, eight, 12, 16, 20, 21, 22, 23/4 inches.
23/4 inches is equivalent to five and 3/4 inches.
Again we have mixed number here,
and an improper fraction here.
But normally we just give the length as five and 3/4 inches.
Now let's take a look at our last example.
Again we're going to start here, so we have one inch,
and I need to partition the length from
one inch to two inches, into smaller partitions.
We can first make partitions of 1/2 an inch here,
and then 1/4 of an inch here.
Notice how we still can not determine
the exact length unless we cut each of these
partitions in half.
So we could cut it here, here, here, and here.
Giving us a total of eight equal partitions
for this length of one inch,
so each piece is 1/8 of an inch.
So we have one and one, two, three, four, 5/8 of an inch.
This would be one and 5/8 inches.
Or if we divide this entire length into partitions
of length 1/8 of an inch.
We would have eight, nine, 10, 11, 12, 13/8 inches.
Again, we would give the length as just
one and 5/8 inches.
I hope you found this helpful.