Welcome to a lesson on simplifying fractions. In this lesson we will define a simplified fraction
and also simplify fractions.
A simplified fraction is a fraction has no common factors other than one
between the numerator and denominator.
looking at the basic fraction of one half shown here.
With a rectangle represents one whole.
One half is a simplified fraction because
the only common factor between one and two
is one.
We also know that one half is equivalent to two fourth shown here.
We know their equivalent because the same amount is shaded
but for two fourth,
were using smaller parts or pieces of the whole
to represent the same amount.
Two fourth is not simplified,
because notice two and four
share a common factor other than one
they share a common factor of two.
For some additional examples, seven eighth
is a simplified fraction
or a fraction written in lowest terms because
the only common factor between seven and eight is one.
And the fraction of four sixth is not simplified
or written in lowest terms because
four and six have a common factor of two.
So in general to simplify a fraction
we remove all common factors
other than one
from the numerator and denominator.
And we'll be discussing two methods for simplifying fractions in this lesson.
The first method is the prime factorization method.
So to simplify a fraction
using the prime factorization method step one.
We factor the numerator and denominator into prime factors.
And then step two
we simplify out the common factors.
The second method is divisibility method.
To apply this method is to simplify a fraction.
We divide the numerator and denominator
by the common factors
until there are no longer any common factors other than one.
We will be showing both methods for our examples.
Let's go back to that fraction of four sixth.
Which again if we define the rectangle as one whole
would be model here.
Looking at the second model we can see that
two third is equivalent to four sixth
because again the same amount of the whole is shaded
but for this model were using larger part or larger pieces
to represent the amount.
So if we wanted to simplify four sixth using the prime factorization method.
We would write the prime factorization of four
which is two times two
and then the prime factorization of six
which is two times three.
Once we have both the numerator and denominator in prime factor form
we can see the common factors that will simplify out or simplified to one.
Notice how four and six share a common factor of two.
Two over two simplifies to one
or we can think of this as two divided by two
either way these two's simplify to one's.
Noticed now the numerator is one times two or two
and the denominator is one times three or three
giving us the simplified fraction of two third.
The second method is the divisibility method where
starting with four sixth
we just look at four and six
and notice they have a common factor of two.
So we divide both the numerator and denominator by two
which gives us a simplified fraction of two third.
Either of these two methods can be used
to simplify any fraction.
Let's look at another example.
We ask to write the fraction in simplest form or lowest terms.
We have the fraction three fifteenth.
So if we wanted to model three fifteenth
but the rectangle is one whole.
We would cut our petition the rectangle into 15 equal parts or pieces
and then because the numerator is three we would shade three of them
so this is a model for three fifteenth.
Because three and five share a common factor other than one
this fraction will simplify
which means we can express three fifteenth
as an equivalent fraction
using larger parts or pieces of a whole as we see here.
We can tell by the model of these two fractions are equivalent
because the same amount is shaded
three fifteenth is equivalent to one fifth.
And because one and five only have a common factor of one
one fifth
is the simplified fraction of three fifteenth.
Let's also show it using the prime factorization method
as well the divisibility method.
So we need to find the prime factorization of three and 15.
Well three is a prime number so the numerator stays three.
And 15 is equal to three times five.
Noticed how we can see the common factor of three
between the numerator and denominator.
Three over three or three divided by three simplifies to one.
So these three's simplified to one.
So were left with a one in the numerator.
And one times five or five in the denominator.
One fifth is the simplified fraction of three fifteenth.
So this is the prime factorization method.
Now let's also show the divisibility method.
So starting with three fifteenth if we recognized that
three and 15 share a common factor of three.
We can divide the numerator and denominator
by three
which would give us an equivalent fraction
and also the simplified fraction
for three fifteenth.
Three divided by three is equal to one.
15 divided by three equals to five.
Let's look at one more example that more challenging.
We ask to write the fraction seventy-two ninetieth
in simplified form or lowest terms.
So using the prime factorization method
the first step is to find the prime factorization of 72
as well as 90.
So 72 is equal to eight times nine.
Eight is equal to two times four where two is prime.
Four is equal to two times two both are prime.
And nine is equal to three times three both are prime.
So the prime factorization of 72
is two, times two, times two, times three, times three.
And now for the prime factorization of 90 we have nine times ten.
Nine is equal to three times three both are prime.
Ten is equal to two times five both are prime.
It's best to put the prime factorization in order from least to greatest so we have
two times three
times three times five.
Now that we have the prime factorization.
We can easily identify the common factors between the numerator
and denominator.
Two over two simplifies to one over one.
So does three over three.
And three over three.
Now we multiply.
So the numerator is two times two which is four.
The denominator is just five.
So four fifth
is the simplified fraction for seventy-two ninetieth.
So this is the prime factorization method.
Which I prefer because you can actually see the common factors.
The other method is the divisibility method so starting with seventy-two ninetieth.
Let's say we recognize there's a common factor of two.
So we could divide the numerator and denominator by two to begin.
72 divided by two is equal to 36.
90 divided by two is equal to 45.
And now from here.
Let's say we recognize there's a common factor of nine.
So now we divide the numerator and denominator by nine.
And I think we have it now.
36 divided by nine is equal to four.
45 divided by nine is equal to five.
Of course both method yield the same results
but using the prime factorization method
we can actually see the common factors.
Using the divisibility method
we must easily be able to identify
any common factors.
I hope you found this helpful.