Adding Mixed Numbers with Unlike Denominators

U02_L3_T1_we4 Adding Mixed Numbers with Unlike Denominators

Add.

Simplify the answer and write
as a mixed number.

And we have three mixed numbers
here: 3 and 1/2 plus

11 and 2/5 plus 4 and 3/15.

So we've already seen that we
could view this as 3 plus 1/12

plus 11 plus 2/5-- let
me write that down.

This is the same thing as 3
plus 1/12 plus 11 plus 2/5

plus 4 plus 3/15.

The mixed number 3 and 1/12
just literally means 3 and

1/12 or 3 plus 1/12.

And since we're just adding
a bunch of numbers, order

doesn't matter, so we
could add all the

whole numbers at once.

So we have 3 plus 11 plus 4,
and then we can add the

fractions: the 1/12 plus
2/5 plus 3/15.

Now, the blue part's pretty
straightforward.

We're just adding numbers.

3 plus 11 is 14 plus 4 is
18, so that part right

there is just 18.

This will be a little bit
trickier, because we know that

when we add fractions, we have
to have the same denominator.

And now we have to make all
three of these characters have

the same denominator and that
denominator has to be the

least common multiple
of 12 and 5 and 15.

Now, we could just do it kind
of the brute force way.

We could just look
at the multiples.

We could pick one of these guys
and keep taking their

multiples, and then figuring
out whether those multiples

are both divisible
by 5 and 15.

Or the other way we can do
it is take the prime

factorization of each of these
numbers, and just say that the

least common multiple has
to contain the prime

factorization each of these
guys, which means it contains

each of those numbers.

So let me show you what
I'm talking about.

If we take the prime
factorization of 12, 12 is 2

times 6, 6 is 2 times 3, so 12
is equal to 2 times 2 times 3.

That's the prime factorization
of 12.

Now, if we do 5, prime
factorization of 5, well, 5 is

just 1 and 5, so 5 is
a prime number.

It is the prime factorization
of 5.

There's just a 5 there.

This 1 is kind of useless.

So 5 is just 5.

And then 15, let's do 15.

Actually, when I did the prime
factorization of 5, I should

have said, look, 5 is prime.

There's no number larger than
1 that divides into it, so

it's actually silly to even
make a tree there.

And now let's do 15, 15's
prime factorization.

15 is 3 times 5, and now both
of these are prime.

So we need something that has
two 2's and a 3, so let's look

at the 12 right there.

So our denominator has to have
at least two 2's and a 3, so

let's write that down.

So it has to be 2
times 2 times 3.

It has to have at least that.

Now, it also has to have
a 5 there, right?

Because it has to be a
common multiple of 5.

5's another one of those prime
factors, so it's got to have a

5 in there.

It didn't already have a 5.

And then it also has to
have a 3 and a 5.

Well, we already have a 5.

We already have a 3 from the
12, and we already have a 5

from the 5, so this number will
be divisible by all of

them, and you can see that
because you can see it has a

12 in it, it has a 5 in it,
and it has a 15 in it.

So what is this number?

2 times 2 is 4.

4 times 3 is 12.

12 times 5 is 60.

So the least common multiple
of 12, 5 and 15 is 60.

So this is going to be plus.

We're going to be over 60.

So all of these are going
to be over 60.

All of these three fractions
are over 60.

Now, to go from 12 to 60,
we have to multiply the

denominator by 5, so we also
have to multiply the numerator

by 5, so 1 times 5 is 5.

5/60 is the same
thing as 1/12.

To go from 5 to 60 in the
denominator, we have to

multiply by 12, so we
have to do the same

thing for the numerator.

12 times 2 is 24.

The last one, 15 to 60, you have
to multiply by 4, so you

have to do the same thing
in the numerator.

4 times 3 is 12.

And now we have the
same denominator.

We are ready to add.

So let's do that.

So this is going to be 18 plus,
and then over 60, we

have 5 plus 24, which is 29.

29 plus 12, let's see, 29
plus 10 would be 39

plus 2 would be 41.

It would be 41.

And as far as I can tell,
41 and 60 do not

have any common factors.

41 actually looks prime to me.

So the final answer
is 18 and 41/60.