Hi, let's take a look at simplifying fractions.
The goals of this video will be to simplify fractions
and also to verify and simplify fractions
on the graphing calculator.
All of the following fractions are equal or equivalent
to 1/4.
1/4, 2/8, 3/12, and so on, you can see the pattern.
Let's take a look at it geometrically.
There's of course 1/4, 2/8,
looks like 5/20.
And notice how in each case the shaded region is equal.
We say 1/4 is in simplest form since the numerator
and denominator do not have any common factors.
When giving an answer in fraction form,
it is standard to always give the fraction
in simplified form.
We will discuss two ways to simplify fractions by hand.
Number one, and my preferred method,
is the prime factorization method.
We factor the numerator and denominator into primes
and then simplify out the common factors.
Or, if you're really good at you multiplication tables,
you can probably use the divisibility method,
and in that method you just divide
the numerator and denominator by the common factors
until there are no longer any common factors.
Let's take a look at how this works.
Let's say you want to simplify 18/24.
Well, first thing we will do using the primer factor method
is to make a factor tree for 18 and 24.
You remember that a factor tree,
you break it down into two numbers that multiplied together
they give you 18.
Two times nine.
Two is prime so I'll circle it.
Nine would be three times three, and those are both prime.
So we break down 18 into its primes.
Two times three times three.
Now we do the same for 24.
Again, it doesn't matter how we start,
like I could start by multiplying
two times 12 or four times six.
Two is prime.
12 breaks down into two times six.
Two is prime.
And, of course, six is two times three.
So the prime factors of 24 would be
two times two times two times three.
And I always like to write these in order
from least to greatest just in order to make
the simplifying easier.
Now we have common factors,
we have a two in the numerator, two in the denominator,
that simplifies to one.
Three over three, that simplifies to one.
And whatever's left is our fraction in simplified form.
In the numerator we have three,
in the denominator is two times two which is four.
Pretty straightforward.
If you use the divisibility method
basically what you do is you ask yourself
what is some number that you can divide
evenly into 18 and 24.
And, of course, you could start by dividing
both the top and the bottom by two,
but the problem with this is it does not simplify
it in one step.
Now you have 9/12.
You've got to think of something else,
and they both are divisible by three.
Now, if you know the multiplication tables
well enough to know that by the way
you could divide it, both the top and the bottom, by six
you could have simplified this to 3/4.
Let's go to the graphing calculator
and I'll show you how you can verify these answers.
If we go to the original fraction
and type in 18 divided by 24,
this will convert the fraction to a decimal.
To convert the decimal back to a fraction
we just have to hit Math
and the fraction command is already selected,
so we hit Enter, Enter,
and that will verify our answer.
Let's take a look at another one.
Now, this one looks like it might be
a little more challenging,
and this is why I like the prime factors method.
I'm going to make a factor tree for 84 and also for 96.
For the sake of time I've already done it.
84 works out to two times two times three times seven.
And 96 breaks down into
two times two times two times two
times two times three.
Now I can easily simplify this looking for common factors
between the numerator and denominator.
Look like we have two twos,
that will all simplify to a one.
We also have a three and a three.
That looks like we have everything.
That's what's nice about this method,
is you can really see that you have simplified the fraction
all the way.
Our numerator is seven, our denominator will be eight.
Again, using the divisibility method
you really have to know your multiplication tables well.
If you're going to look at both of these and say,
"Oh, I know they're both divisible by 12",
it will be faster, and the result,
of course, will be the same.
Going back to the graphing calculator for a moment.
I'll type in 84 divided by 96, hit Enter.
That gives me the decimal.
Now if I hit Math, Enter, Enter,
that verifies my answer as 7/8.
Now, I like the prime factor method, again,
because when we introduce variables into fractions
it's just as easy.
So, for example, on this problem,
I know that 25 is five times five.
I know that x to the second power mean x times x.
And I know y to the third means y times y times y.
Looking at the denominator 55 would be five times 11.
x to the third, of course, is x times x times x.
And then we have a factor of y as well.
We can simplify the variables the same way.
Five over five simplifies to one.
x over x, x over x.
We also have a y over y.
The remaining factors represents our simplified fraction.
So in the numerator we have five y-squared.
And in the denominator we have 11x.
I hope that helps you to simplify fractions.
Thank you, and have a good day.