Adding fractions with unlike denominators

- [Voiceover] Let's say that
we have the fraction 9/10,

and I want to add to
that the fraction 1/6.

What is this, what is this going to equal?

So when you first look at this, you say,

"Oh, I have different denominators here.

It's not obvious how I add these."

And you'd be right and the way to actually

move forward is to find
a common denominator,

to convert both of these fractions into

fractions that have a common denominator.

So how do you think about
a common denominator?

Well, a common denominator's
gonna have to be

a common multiple of these two
denominators of 10 and six.

So what's a common multiple of 10 and six?

And it's usually simplest to
find the least common multiple,

and a good way of doing that
is start with the larger

denominator here, 10, and say,
okay is 10 divisible by six?

No. Okay, now, is 20 divisible by six?

No. Is 30 divisible by six?
Yes. 30 is divisible by six.

So I'm just going through
the multiples of 10

and saying, "Well what is
the smallest multiple of 10

that is divisible by six?"
And that's going to be 30.

So I could rewrite both of these fractions

as something over 30.

So nine over 10. How would I write that as

something over 30? Well I multiply

the denominator, I'm multiplying
the denominator by three.

So I've just multiplied
the denominator by three.

So if I don't want to change
the value of the fraction,

I have to do the same
thing to the numerator.

I have to multiply that by three as well

because now I'm just multiplying
the numerator by three

and the denominator by three,
and that doesn't change

the value of the fraction.

So nine times three is 27.

So once again, 9/10 and 27/30

represent the same number.

I've just written it now
with a denominator of 30,

and that's useful because
I can also write 1/6

with a denominator of 30. Let's do that.

So 1/6 is what over 30?

I encourage you to pause the video

and try to think about it.

So what did we do go from six to 30?

We had to multiply by five.

So if we multiply the denominator by five,

we have to multiply the
numerator by five as well,

so one times five, one times five is five.

So 9/10 is the same thing as 27/30,

and 1/6 is the same thing as 5/30.

And now we can add, now we can add

and it's fairly straightforward.

We have a certain number of 30ths,

added to another number of 30ths,

so 27/30 + 5/30, well that's going to be

27, that's going to be 27 plus five,

plus five, plus 5/30,

plus 5/30, which of course

going to be equal to 32/30.

32 over 30, and

if we want, we could try
to reduce this fraction.

We have a common factor of 32 and 30,

they're both divisible by two.

So if we divide the numerator
and the denominator by two,

numerator divided by two is 16,

denominator divided by two is 15.

So, this is the same thing
as 16/15, and if I wanted

to write this as a mixed
number, 15 goes into 16 one time

with a remainder one.

So this is the same thing as 1 1/15.

Let's do another example.

Let's say that we wanted
to add, we wanted to add

1/2 to

to 11/12, to 11 over 12.

And I encourage you to pause the video

and see if you could work this out.

Well like we saw before, we wanna find

a common denominator.

If these had the same denominator,

we could just add them immediately,

but we wanna find a common denominator

because right now they're not the same.

Well what we wanna find is a multiple,

a common multiple of
two and 12, and ideally

we'll find the lowest common
multiple of two and 12,

and just like we did before,
let's start with the larger

of the two numbers, 12.

Now we could just say
well 12 times one is 12,

so that we could view that
as the lowest multiple of 12.

And is that divisible by two? Yeah, sure.

12 is divisible by two.

So 12 is actually the least
common multiple of two and 12,

so we could write both of these

fractions as something over 12.

So 1/2 is what over 12?

Well to go from two to
12, you multiply by six,

so we'll also multiply
the numerator by six.

Now we see 1/2, and 6/12,
these are the same thing.

One is half of two, six is half of 12.

And how would we write
11/12 as something over 12?

Well it's already written
as something over 12,

11/12 already has 12 in the denominator,

so we don't have to change that.

11/12, and now we're ready to add.

So this is going to be equal to six,

this is going to be equal to six plus 11,

six plus 11 over 12.

Over 12. We have 6/12 plus 11/12,

it's gonna be six plus 11 over 12,

which is equal to, six plus 11 is 17/12.

If we wanted to write
it as a mixed number,

that is what, 12 goes
into 17 one time with

a remainder of five, so 1 5/12.

Let's do one more of these.

This is strangely fun. Alright.

Let's say that we wanted to add,

We're gonna add 3/4 to,

we're gonna add 3/4 to 1/5.

To one over five.

What is this going to be?

And once again, pause the video and

see if you could work it out.

Well we have different denominators here,

and we wanna find, we wanna rewrite these

so they have the same denominators,

so we have to find a common multiple,

ideally the least common multiple.

So what's the least common
multiple of four and five?

Well let's start with the larger number,

and let's look at its
multiples and keep increasing

them until we get one
that's divisible by four.

So five is not divisible by four.

10 is not divisible by four,
or perfectly divisible by four

is what we care about.

15 is not perfectly divisible by four.

20 is divisible by four, in
fact, that is five times four.

That is 20. So what we
could do is, we could write

both of these fractions as
having 20 in the denominator,

or 20 as the denominator.

So we could write 3/4
is something over 20.

So to go from four to
20 in the denominator,

we multiplied by five.

So we also do that to the numerator.

We multiply by three times five to get 15.

All I did to go from four
to 20, multiplied by five.

So I have to do the same
thing to the numerator,

three times five is 15.

3/4 is the same thing
as 15/20, and over here.

1/5. What is that over 20?

Well to go from five to 20,
you have to multiply by four.

So we have to do the same
thing to the numerator.

I have to multiply this
numerator times four to get 4/20.

So now I've rewritten this
instead of 3/4 plus 1/5,

it's now written as 15/20 plus 4/20.

And what is that going to be?

Well that's going to be
15 plus four is 19/20.

19/20, and we're done.