Welcome to a lesson
on adding fractions.
To add fractions with the same
or like denominators,
we will add the numerators,
keeping the denominator the same,
and then we'll simplify, if possible.
So, for 1/5 plus 2/5,
the denominator's going to stay five,
and then we add the numerators
and, since one plus two is equal to three,
the sum is 3/5.
If we take a look at the model for this,
here's 1/5, here's 2/5.
If we add these pieces together,
notice how they're the same size,
and we'll have three pieces,
where each piece represents 1/5.
Therefore, the sum is 3/5.
It's important to understand this idea, because
to add fractions with different
or unlike denominators,
the first step is to find the
least common multiple of the denominators.
This is called the least common denominator, or LCD.
Once we identify the LCD,
we're going to create equivalent fractions
with the common denominator,
and then once we have the same denominators,
we'll add the numerators,
keeping the denominators the same.
And, as always, simplify, if possible.
So, if we have 1/2 plus 1/4,
the first step is to identify
the least common multiple of two and four,
which is the smallest number
divisible by both two and four,
which would be four.
So, we want both fractions
to have a denominator of four.
Well, this one already has a denominator of four,
this one has a denominator of two,
and since two times two is equal to four,
we can multiply this denominator by two
as long as we do the same to the numerator.
Multiplying the numerator and denominator by two
is equivalent to multiplying by one,
but now, notice how we have a common denominator of four.
We would have 2/4
plus 1/4.
So, the denominator stays four,
and then we add the numerators.
Two plus one is equal to three.
The sum is equal to 3/4.
It's important that we understand what's happening here,
so let's take a look at a common error
when adding fractions,
and then we'll explain why we have to have
a common denominator.
A common mistake when adding fractions
is to add the numerators and add the denominators,
as we see here, and this is incorrect,
and let's see why.
Here's a model for 1/2,
and here's a model for 1/4.
Using this idea,
we're saying if we add these two pieces together,
we'll get these two pieces,
which represents 2/6.
Notice that 2/6 is actually
less than 1/2,
so there's no way we could add these two pieces
and get a smaller amount
and that's why this is incorrect.
Let's take a look at what's really happening
when we obtain a common denominator.
If we take the fraction of 1/2 modeled here,
but divide it into four equal pieces,
rather than just two,
notice how now, instead of 1/2,
we can write this as 2/4.
Here's a model for 1/4,
and notice using these two models,
the pieces are the same size,
and therefore, if we add these,
we'll have three pieces of the same size,
each piece representing 1/4.
Therefore, the sum is 3/4.
So, when we obtain the
least common denominator,
what we're doing is actually dividing the fractions
into equal-size pieces.
Let's take a look at a few more examples.
Here, we have 3/10 plus 1/5.
The first step is to identify the
least common multiple of 10 and five,
which will be our LCD.
So, if we can look at this,
and recognize that the smallest number
divisible by both 10 and five
is 10, that's great.
If we can't, that's not a problem.
We can list multiples of 10,
list multiples of five,
to identify the LCD.
If we count by 10s, we'd have
10, 20, 30, and so on.
If we count by fives, we'd have
five, 10, 15, and so on.
Notice how the least common multiple,
which is our LCD, would be 10.
So, we want both of these fractions
to have a denominator of 10.
Well, the first one already has
a denominator of 10, so we'll leave that the same,
and then, for the second fraction of 1/5,
since five times two is equal to 10,
we'll multiply both the
numerator and denominator by two,
which is the same as multiplying by one.
Notice how this gives us 3/10
plus, this is now 2/10,
which is equal to 5/10.
But we're not done here,
because 5/10 can simplify.
We could write five as one times five,
and 10 as two times five,
so the fives simplify to one,
giving us a sum of 1/2.
Now, there are a couple ways
to identify the LCD,
but listing multiples will always work.
Next, we have 3/4 plus 1/6.
Again, the first step is to identify the
least common multiple of four and six,
which will be our LCD.
That would be the smallest number
divisible by bother four and six.
So, again, we can save ourselves some time
if we recognize the LCD would be 12,
but if we can't, that's not a problem.
We can always list multiples of four
and multiples of six
to identify the LCD.
So, for four, we'd have four, eight,
12, 16, and so on,
and for six, we'd have six, 12, 18, and so on.
And, again, notice how the least common multiple,
or the LCD, is 12.
So, we want to rewrite both of these fractions
with a denominator of 12.
So, we have 3/4
plus 1/6,
and since four times three is equal to 12,
we'll multiply the top and bottom by three.
And then, since six times two is equal to 12,
we'll multiply the top and bottom by two.
So, this gives us 9/12
plus, this'd be 2/12,
and now the denominator stays the same.
Add the numerators,
nine plus two is equal to 11.
And this does not simplify.
Let's take a look at a couple more examples.
Here, we have three plus 3/5.
We want this three to be in fraction form.
Let's go ahead and put this over one.
So, now we have 3/1
plus 3/5.
Well, the least common multiple of one and five
will just be five.
So, we'll have to multiply 3/1 by 5/5
to have our common denominator, five.
So now, we'll have 15/5
plus 3/5.
Common denominator is five,
add the numerators, 15 plus three is equal to 18.
Again, these don't have any common factors
other than one,
so this is in simplest form.
Next, we have 5/12 plus 3/8.
First step is to determine the LCD,
or least common multiple of 12 and eight,
so that would be the smallest number
divisible by both 12 and eight.
So, if we recognize this to be 24, that's great.
If we don't, that's okay, too.
We can always list multiples of 12,
list multiples of eight,
to identify the LCD.
So, if we count by 12s, we'd have
12, 24, 36, and so on.
If we count by eights, we'd have
eight, 16, 24,
32, and so on,
and notice how the
smallest number they have in common is 24.
So, we want both fractions
to have a denominator of 24.
So, a 5/12
plus 3/8,
and since 12 time two is 24,
multiply the top and bottom by two,
and then, since eight times three
is equal to 24,
we'll multiply both the top and bottom by three.
So now, we'll have 10/24
plus, this is now 9/24,
and the denominator is the same,
so now we add the numerators.
Since 10 plus nine is equal to 19,
we have a sum of 19/24,
which, again, does not simplify.
We will take a look at some more examples
in the next few videos,
but hopefully this gives you a good foundation on
why we have to have a common denominator
and how we can identify the
least common denominator.
While there are other ways to identify the
least common denominator,
listing multiples will always work.
I hope you found this helpful.