We're asked to write
the fractions in lowest terms.
The directions may also say, simplify the fractions.
The fraction is written in lowest terms
or simplified when the only common factor
between the numerator and the denominator is 1.
And we'll show how to write these fractions
in lowest terms using prime factors as well as division.
So beginning with 60 84ths,
we begin by determining
the prime factorization of 60 and 84.
Well 60 is equal to six times ten,
six equals two times three, which are both prime,
ten equals two times five which are both prime.
So the prime factorization of 60 is equal to
two, times two, times three, times five,
and we should write the prime factorization
in order from least to greatest.
Now for 84 because it's even, we know would be
two times 42, two is prime,
42 is still even, two times 21,
two is prime, 21 equals three times seven, both are prime.
So the prime factorization of 84 is
two, times two, times three, times seven.
The nice thing about the prime factorization method,
is that once we have the prime factors,
we can see all the common factors
between the numerator and denominator.
Notice how there are two twos in common,
as well as a three in common.
And because two over two is equal
to two halves or two divided by two,
which equals one, we can simplify two over two
as one over one here, as well as here,
and because three thirds, or three divided by three
is equal to one, three over three
simplifies to one over one.
And now we look at the remaining factors,
notice how the numerator would be
one times, one, times one, times five,
which is five, and the denominator is seven.
So 60 84ths in lowest terms equals five sevenths.
Now let's do this again, using the division method.
Beginning with the original fraction again.
We identify common factors between 60 and 84
and then we divide the numerator and the denominator
by that common factor, so because they're both even, let's
begin by dividing the numerator and denominator by two,
and we keep doing this until the only common factor
between a numerator and denominator is one.
60 divided by two is equal to 30.
84 divided by two is equal to 42.
And let's say from here we recognize
there's a common factor of six between 30 and 42,
so we divide the numerator by six
and the denominator by six.
Well 30 divided by six equals five,
42 divided by six equals seven.
And once again, we have the same fraction, in lowest terms.
So the division method is also nice because it tends
to be quicker as long as we can identify
the common factors between the numerator and denominator.
Let's look at our second example.
Here we have 120 over 135 using prime factors
we determine the prime factorization of 120 and 135.
For 120 lets begin with 12 times 10.
12 is equal to four times three, three is prime,
four is equal to two times two, both are prime,
and ten equals two times five.
So the prime factorization of 120,
we contain three factors of two,
a factor of three, and a factor of five.
135 ends in five, so it has to be five times some value,
135 is equal to five times 27, five is a prime,
27 is equal to nine times three, three is prime,
and nine is equal to three times three.
So the prime factorization of 135
contains three factors of three, and a factor of five.
Now we can see the common factors
between the numerator and denominator.
They have a three in common,
and a five in common because
three over three is equal to three thirds,
or three divided by three which equals one,
three over three simplifies to one over one,
and so does five over five.
So notice now, we have two,
times two, times two, which is eight,
over three times three, which equals nine.
So the given fraction in lowest terms equals eight ninths.
And let go ahead and show this again
using the method of division.
Course when you're doing your homework,
you want to select the method that works best for you.
So starting with 120 over 135,
let's say we recognize, because 120 ends in a zero,
135 ends in a five, we recognize a common factor of five,
so we divide the numerator by five
and the denominator by five.
120 divided by 5 is equal to 24.
135 divided by 5 equals 27.
And then from here lets say we recognize
there's a common factor of three between 24 and 27.
So we divide the numerator by three,
and the denominator by three.
24 divided by three equals eight.
27 divided by three is equal to nine.
So of course we get the same results,
so again, both methods have their benefits,
as well as their challenges.
Determining the prime factorization does take some work,
but we do see all the common factors
and verify the fraction is in lowest terms,
the division method tends to be a little bit quicker,
as long as it can determine the common factors
between the numerator and denominator.
I hope you found this helpful.