Addition of Rational Numbers

Let's add some rational
numbers.

And I'm using that word because
that's the word that

this book uses, but in more
popular terminology we'll be

adding fractions.

So let's just go through all
of these, actually, just to

see all of the examples.

So first we're going to
have 3/7 plus 2/7.

Our denominators are the same,
so we can just add the

numerators.

So our denominator is
7, 3 plus 2 is 5.

That is a.

Let me do every other.

It would take forever
to do all of them.

Not forever, but just more time
than I want to spend.

So c is 5/16 plus 5/12.

Our denominators are
not the same.

We have to find a common
denominator, which has to be

the least common-- it actually
could be any common multiple

of these, but for simplicity
let's do the

least common multiple.

So what's the smallest number
that's a multiple

of both 16 and 12?

So let's see, 16 times 2
is 32, not there yet.

Times 3, 48.

That seems to work.

12 goes into 48 four times.

So let's use 48 as our
common denominator.

So we had to multiply 16 times 3
to get to 48, so we're going

to have to multiply
this 5 times 3.

We're just multiplying the
numerator and the denominator

by the same number, so we're
not really changing it.

So 5 times 3 is 15.

And then to get from this 12 to
this 48 right there, we had

to multiply times 4.

So then to get to 5 to this
numerator over here, we have

to multiply times 4.

5 times 4 is 20.

Now we have the same
denominator.

So this is going to be equal
to, our denominator is 48.

And so we can add 15 plus
20, which is 35.

And can we reduce this?

Let's see, 5 does
not go into 48.

7 does not go into 48.

It looks like we're all done.

Let's do e right there.

8/25 plus 7 over 10.

Once again, we don't have
a common denominator.

But we can solve that.

Let's make, let's see, 50 is the
smallest number that both

of these go into.

25 times 2, so that's 50.

8 over 25, to go to 50
we multiply by 2.

So the 8, we're going to
have to multiply by 2.

So it's going to
be 16 over 50.

And then the 7 over 10,
we're going to want

to put it over 50.

We multiply the 10 times
5, so we have to

multiply the 7 times 5.

So it's going to
be 35 over 50.

Now that our denominators are
the same, we have it over 50.

16 plus 35, what is that?

10 plus 35 is 45,
plus 6 is 51.

So it is 51 over 50.

Problem g.

Let me do it in a new color.

Problem g.

So here we have 7 over 15-- I'll
write the second one in a

different color--
plus 2 over 9.

Once again, the denominators
are different.

Find a common denominator.

What is the smallest number that
both 15 and 9 go into?

Let's see, 15 times 2 is 30.

Nope, not divisible by 9.

15 times 3 is 45, that works.

45 is divisible by 9.

So we use 45.

15 times 3 is 45, so
7 times 3 is 21.

These two fractions
are equivalent.

Plus we're going over 45.

To get from 9 to 45, we have
to multiply times 5.

So to get our numerator
over here, we have to

multiply it times 5.

So 2 times 5 is 10.

2/9 is the same thing
as 10/45.

So now we can add.

We're adding fractions of 45.

21 plus 10 is 31,
and we are done.

Let's do one more problem down
here, a word problem.

Nadia, Peter and Ian are pooling
their money to buy a

gallon of ice cream.

Nadia's the oldest and gets
the greatest allowance.

She contributes 1/2 the cost.
So Nadia is contributing 1/2

the cost. So that is
Nadia right there.

Ian is next oldest and
contributes 1/3 of the cost.

So Ian contributes 1/3.

That is Ian.

Peter, the youngest, gets the
smallest allowance and

contributes 1/4 of the cost.
So Peter gives 1/4 of the

cost. Peter contributes
1/4 of cost.

They figure that this will
be enough money.

When they get to the checkout,
they realize that they forgot

about sales tax and
worry there will

not be enough money.

Amazingly, they have exactly
the right amount of money.

What fraction of the cost of
ice cream was added as tax?

Well, let's see, if we add 1/2
plus 1/3, plus 1/4 of the

cost, let's see what we get.

So we have to find a common
denominator, some number that

is the least common multiple
of 2, 3, and 4.

And let's see, 4, it would
have to be 12, right?

12 is divisible by 2, it's
divisible by 3, and it's

divisible by 4.

So 1/2 is the same
thing as 6/12.

2 times 6 is 12.

1 times 6 is 6.

These are equivalent.

6 is 1/2 of 12.

1/3, if we use 12 as a common
denominator, to go from 3 to

12 you have to multiply by 4.

So you take that 4 and
you multiply it by 1.

4/12 is the same thing as 1/3.

And then 1/4, if you use your
denominator 12, to go from 4

to 12 you have to multiply by
3, so multiply the numerator

by 3 as well, you get 3.

So let's add these.

So 6/12 plus 4/12, plus 3/12 is
going to be equal to-- our

denominator's going to be 12--
it's going to be 6 plus 4,

plus 3, which is equal to 6 plus
4 is 10, plus 3 is 13.

So it's going to be
equal to 13/12.

And this is as an improper
fraction.

Or we could say that this is the
same thing, this is equal

to 12/12 plus 1/12, or we could
say the same thing as

12/12 is just 1, right?

12 divided by 12 is 1.

So this is 1 and 1/12.

So when they pool their money,
they get 1 and 1/12 of the

price of the ice cream that
they want to buy.

So they say what fraction of
the cost of ice cream was

added as tax?

This is the exact amount that
they needed to pay.

So clearly, 1 is the non-taxed
price of the ice cream, so

this 1/12 was the amount
added as tax.

So the answer to the question
is 1/12 of the price

was added as tax.