There's a second representation that can go hand in hand with an improper fraction.
So, just to remember an improper fraction is one where the top, the numerator
is bigger than the bottom, the denominator. So, that improper fraction represents a number bigger than one.
So, the other representation that goes along with that is a mixed number.
So, looking at the example below we have
an improper fraction of nine over four. So, what this looks like in as a fraction
is four over four plus four over four plus one over four and that's nine over four.
However, for using mixed numbers we can then take the four over fours and make them just solid ones because they're a whole.
So, then you can have one plus one plus one over four and that gives you two wholes
and one over four or if you're saying it as a mixed number, it's common to say
two and one over four and that's a mixed number or a mixed fraction.
Now, because mixed numbers go hand in hand with improper fractions
it's important to have a way to go in between the two.
So, let's start with this mixed number of three and two over six.
So, we have three holes and two pieces of six.
So, that looks like six over six, plus six over six, plus six over six,
plus two over six. Now, when we're adding that together
we can actually see that we have these three whole pieces
but that means we have six pieces each time we have a whole.
So, six plus six is twelve plus six more is eighteen. So, those three holes really come down to
being eighteen pieces and plus the other two pieces we have
from the fourth hole gives us a total of twenty pieces.
We keep the bottom the denominator the same because that was the overall subdivision
that we were working with, six. So, we can get a formula here to convert
from a mixed number to an improper fraction.
So, what we do is we start with the number up front, in this case, the three
and we multiply by how many pieces made up a whole so, in this case, six.
So, we say three times six which goes in the top and that gives us eighteen
and then we add the two that was already in the top of a fraction.
So, three times six plus two gives us the twenty and then we keep the bottom the same as
it originally was, in this case, 6. So, if you do that same process each time
you want to go from a mixed number to an improper fraction
this will always work and you don't have to draw out a picture or a diagram to help you figure out what that is.
So, just to go through that formula one more time with a different example,
let's take a look at two and one over three. So, we start by taking the two out front
and multiplying by the three in the bottom and then adding the one that was already there.
So, two times three plus one in the top and then we keep the denominator the same.
So, two times three is six plus one is seven and then we keep the three and
that answer matches what we see with the picture on the right.
Now, to go the other way around if you're going from an improper fraction to a mixed number
we have to use a different process so the first thing we want to do
is divide the twenty-nine by the eight. So, there's a couple of ways to do this,
the first way is to use long division and the long division looks like this so
twenty-nine divided by eight. We ask ourselves how many times does eight go into twenty-nine?
It goes in three times. So then we write three times eight
is twenty-four and then we do the subtraction to figure out what's leftover.
So, twenty-nine minus twenty-four leaves us with a remainder of five.
So, now that we've finished the long division, what we call the quotient,
the whole number part of our division becomes the whole number part of the
mixed number and then the bottom part the remainder
of five becomes the numerator of the fraction that goes along with it
and then the eight remains as the denominator.
So, this becomes three and five over eight.
If you find the long division algorithm a little bit confusing here
you can also do this by counting up multiples of eight. You need to get the
closest multiple of eight to twenty-nine but that doesn't go over.
So, we'd have eight, sixteen, twenty four so, that's three eighths.
So then, what's left from twenty-four to twenty-nine is five
and so we get the same idea of quotient and remainder but we don't have to use
a long division algorithm if you're not comfortable with that.