>> We want to simplify the following fractions.
A fraction is in simplest form.
A fraction is considered simplified if the numerator
or denominator do not contain any common factors.
So one method for determining the common factors between 12 and 30,
so one way of identifying the common factors between 12 and 30 would be by looking
at the prime factorization of each, so let's determine the prime factorization of 12.
It would be 2 times 6 and 2 times 3, so the prime factorization
of 12 would be 2 times 2 times 3.
Let's go ahead and do the same for 30.
It would be 3 times 10 and 2 times 5, so we have 2 times 3 times 5
for the prime factorization of 30.
And now we can identify which prime factors the numerator and denominator have in common.
Here we have a 2 over 2 which simplifies to 1,
and here we have a 3 over 3 which simplifies to 1.
So whatever is left is the simplified fraction for 12/30, so 12/30 is equal to 2/5.
Now it is true if you know your multiplication tables really well, and you can look at 12
and 30 and identify the greatest common factor of 6.
It would be much shorter to divide the numerator by 6 and the denominator by 6
to obtain the simplified fraction of 2/5.
But when first learning how to simplify fractions,
I prefer the prime factorization method because it makes it very easy to identify all
of the common factors, and whatever factors remain give us the simplified fraction.
Let's go and try a second example.
We'll start by determining the prime factorization of 36 and 48,
so we'll start with 6 times 6; each of these would be 2 times 3, so we have two factors
of 2 and two factors of 3 for 36.
And it's easier to identify them if you put them in order from least to greatest,
and now let's take a look at 48, and it's 6 times 8, 2 times 3, 4 times 2, and 2 times 2,
so we have 1, 2, 3, 4 factors of 2 and a factor of 3 for the prime factorization of 48.
So 2 times 2 times 2 times 2 times 3, and now let's look for the common factors.
We have 2 over 2 simplifies to 1, 2 over 2 simplifies to 1,
and we also have a 3 over a 3, which simplifies to 1.
So now we can see that 36/48 simplifies to 3/4.
So as the numerator and denominator get much larger,
the prime factorization technique is a great method.
But again, if you know your multiplication tables really well and you can look at 36 and 48
and identify 12 as the greatest common factor, it would be much shorter
to just divide the numerator and denominator by 12 to obtain the simplified fraction of 3/4.
We want to simplify the following fractions.
A fraction is in simplest form if the numerator
and denominator do not contain any common factors.
Sometimes just by looking at a fraction it's hard
to determine what the common factors would be, so I have to use the prime factorization method
which means we'll determine the prime factorization of the numerator and denominator
and then simplify out the common factors.
So we'll start by determining the prime factorization of 24 and 108,
so we could use 4 times 6, so the prime factorization
of 24 would be 2 times 2 times 2 times 3, and now we'll do the same for 108.
It's even so we'll start with 2 times 54, 2 is prime.
How about 9 times 6, 3 times 3, and 2 times 3?
So we have two factors of 2 and 3 factors of 3.
And now once we have the prime factorization it's much easier to identify the common factors
between the numerator and denominator.
Here we have a 2 over 2 that simplifies to 1.
Two over 2 that simplifies to 1, and we also have a 3 over 3 which simplifies to 1.
So whatever factors remain will give us the simplified fraction for 24/108.
So our numerator we have a factor of 2, and our denominator we have 3 times 3,
which is equal to 9, 24/108 simplifies to 2/9.
Now looking at this fraction in prime factored form notice the factors that simplified
out were 2 times 2 times 3, so 24 and 108 had a common factor of 12 meaning
if we knew our multiplication tables really well it would have been a little bit faster
to divide the numerator and denominator by 12 to obtain the simplified fraction of 2/9,
but that does get much more difficult as these numbers get larger and larger.
Let's go and take a look at one more example.
We want to simplify 75/105, so again we'll start by writing out the prime factorization of 75
and 105, and the more you make these prime factorization trees the faster you get.
Seventy-five is equal to 3 times 25, which is 5 times 5, so we have 3 times 5 times 5.
And for 105 it ends in a 5 so how about 5 times 21, and then 21 is 3 times 7,
so we have a factor of 3, a factor of 5, and a factor of 7.
And now for the fun part we get to simplify.
Here's a 3 over 3 that simplifies to 1.
A 5 over 5 simplifies to 1, so 75/105 simplifies to 5/7.
Or again if you're very good with your multiplication tables and you can recognize
that these two have a common factor of 15.
We could divide both the numerator and denominator by 15
to obtain the simplified fraction of 5/7, but again this gets much more difficult
as these numbers get larger and larger,
and the prime factorization method stays pretty much the same. ------------------------------48d4cbd5f12f--