Hello. I'm Professor Von Schmohawk, and welcome to Why U.
In the previous lecture, we saw how to add or subtract fractions with different denominators
by creating a "common denominator".
We saw that the easiest way to do this is to multiply each denominator
by the denominator of the other fraction.
However, sometimes this creates an unnecessarily large common denominator.
We will see in this lecture how to find the smallest possible common denominator,
otherwise known as the "Least Common Denominator".
"Prime Factorization" is the key to finding the least common denominator
of two or more fractions.
Recall from our lecture on factoring, that this involves finding all the prime numbers
which, when multiplied together, produce the number that we wish to factor.
Prime factorization allows us to break down each denominator
into its basic building blocks of prime factors,
so that we can see how to create the smallest possible common denominator.
As an example, let’s say that we want to add one-sixth and one-thirtieth.
We must make the denominators of these fractions the same, so that we can add their numerators.
If we factor each denominator, we can write each of them as a product of prime numbers.
Since our goal is to make these two denominators the same, let's see what is making them different.
The first denominator is missing one factor of five that the second denominator has.
We can remedy this situation by
multiplying the numerator and denominator of the first fraction by five.
This doesn't change the value of the fraction,
since it’s the same as multiplying the entire fraction by one.
However, its denominator will gain the additional factor of five that it needs.
The two fractions will then have a common denominator of thirty.
This is the smallest denominator that will work for both fractions,
so it is the least common denominator.
Now, adding the two numerators, we get a sum of six-thirtieths.
As another example, let’s say that we wish to add one-sixth, plus one-eighteenth.
We first factor each denominator.
The first denominator has only one factor of three,
while the second denominator has two.
We can make the two denominators the same,
by multiplying the top and bottom of the first fraction by three.
The two fractions will then have a common denominator of eighteen,
which is their least common denominator,
and we can now add their numerators.
Let’s try another example, adding one-thirtieth, plus one-forty-second.
Factoring each denominator, we see that the first denominator needs a factor of seven
that the second denominator has,
and the second denominator needs a factor of five that the first denominator has.
So, we multiply the top and bottom of the first fraction by seven,
and the top and the bottom of the second fraction by five.
The two fractions now have a least common denominator of two-hundred-ten,
and we can now add their numerators.
To see the advantage of using the least common denominator,
let’s take the example we looked at in the beginning of the lecture,
seven-twelfths plus five-eighteenth,
and find a common denominator as we did in the previous lecture,
multiplying the top and bottom of each fraction by the other fraction’s denominator.
Using this technique, we end up with a common denominator of two-hundred-sixteen,
and a sum of one-hundred-eighty-six two-hundred-sixteenths.
Now let’s try it again, using the least common denominator.
We’ll start with the same fractions, and factor the denominators.
Now we can see that the first denominator only needs an additional factor of three,
and the second denominator only needs an additional factor of two.
So we end up with a much smaller common denominator of thirty-six,
and a sum of thirty-one thirty-sixths,
which is a lot simpler that the result we obtained by multiplying the two denominators.
The power of using the least common denominator
is especially apparent when we add or subtract more than two fractions.
For example, let’s calculate three-eighths, plus one-sixth, minus five-twelfths.
We first factor each denominator.
We can see that to make these denominators the same,
we need to add a factor of three to the first denominator,
two factors of two to the second denominator,
and one factor of two to the third denominator.
This gives us a least common denominator of twenty-four,
and a total of three twenty-fourths.
We have seen that in order to add or subtract fractions with different denominators,
we need to create a common denominator.
We have also seen that these calculations can sometimes be greatly simplified
if we use the least common denominator.
However, even when we use the least common denominator,
the result of our addition or subtraction
may be a fraction that can still be written in a simpler form.
In the next lecture we will see how to convert any fraction to its simplest form.