We are continuing our study of fractions.
Today, we'll be looking
at improper fractions and mixed numbers.
Our goals are to convert an improper fraction
to a mixed number and to convert a mixed number
to an improper fraction.
Let's start off by looking at the three types of fractions.
A proper fraction is a fraction
with a smaller numerator than denominator.
Proper fractions are less than one.
A few examples of proper fractions would be,
for example, 2/3, 1/5, 2/9 and so on.
You kind of get the idea.
An improper fraction is a fraction in which
the numerator is grater than or equal to the denominator.
Improper fractions are greater than or equal to one.
So for example, if we have something like 7/5
or 9/2 and so forth, these are considered improper fractions
because the value of them are greater than or equal to one
and also the numerator is either equal to
or greater than the denominator.
Now despite the name improper,
there's nothing improper about improper fractions.
In fact, in many math classes beyond algebra,
improper fractions are actually preferred to mixed numbers,
which brings us to a mixed number,
and it is basically what it sounds like.
A mixed number is a number written as the sum
of an integer and a proper fraction.
So for example, 5 1/2, 23 7/8, and so on.
You kind of get the idea there.
Now, even though it is a sum, we do not put an addition sign
in between the integer and the fraction.
Okay, so what we want to do is we want to be able to convert
from an improper fraction to a mixed number,
and here are our procedures.
Step one is we divide the numerator by the denominator.
We write down the whole number of our quotient.
We place any remainder in the quotient over the denominator,
which gives us our fraction part,
and then if we need to, we need to simplify the fraction.
Let's take a look at an example of this.
Let's say we want to convert 13/5 to a mixed number.
Well, step one says divide, so we're going to take 13
and divide this by five.
Of course, there are two fives in 13.
Two times five would give us 10 with a remainder of three.
This is all the information we need
to convert an improper fraction to a mixed number.
Two represents the whole number of our mixed number
and our remainder of three gives us
the numerator of the fraction part,
and of course, the denominator stays the same.
So 13/5 is equal to 2 3/5.
The model of 13/5 is shown below
and you can easily see that the first 10/5
gives us two whole units with 3/5 remaining.
Now let's take a look at another example.
We want to convert 32/6 to a mixed number,
so step one, we're going to divide 32 divided by six
and we can see that it gives us a remainder of two.
This five gives us the integer part to our mixed number
and the remainder gives us the numerator
to the fraction part, 2/6.
Now, we do notice that 2/6 can be simplified
so our final mixed number would 5 1/3.
I could've simplified this at the beginning,
but I decided to simplify it after.
Now there is a way to verify this on the graphing calculator
so let's go ahead and go over that.
Starting with the original fraction of 32/6,
I'm going to enter that in as 32 divided by six.
Remember, a fraction bar does represent a division sign.
When I hit enter, it'll give me the decimal value of that.
Notice how five is the integer part of our mixed number
and the fraction part is represented as a decimal.
Well, if I subtract out five,
which would leave just the decimal portion,
I can convert this back to a fraction
by hitting Math, Enter, Enter,
and that verifies the fraction part of our mixed number.
Okay, now let's do the opposite.
Let's convert a mixed number into an improper fraction.
The procedures are multiply the denominator
by the whole number.
Take that product and add that
to the numerator of the fraction.
Take that sum and place it over the denominator.
Let's give that a try.
Step one, multiply the denominator times the whole number.
Eight times two would give us 16.
Add the numerator, which is three, and this gives us 19,
so what that tells us is 2 3/8 is equal to 19/8.
Let's take a look at the model of this.
When we multiply eight by two,
we're actually finding out how many eighths
there are in two whole units.
Here are 8/8 and another 8/8, which gives us the 16/8,
plus the 3/8, of course, would give us the 19/8.
Let's try one more.
Convert 7 5/12 to an improper fraction.
So the procedure, we're going to multiply 12 times seven.
That'll give us 84, and we're going to add the numerator,
which is five, which gives us 89.
So we have 7 5/12 is equal to 89/12.
I hope that helps, how to review converting
between improper fraction and mixed numbers.
Thank you and have a good day.