How to solve one-step multiplication and division equations with fractions and decimals

Let's get some practice
solving some equations,

and we're gonna set up some equations

that are a little bit hairier than normal,

they're gonna have some
decimals and fractions in them.

So let's say I had the
equation 1.2 times c

is equal to 0.6.

So what do I have to multiply

times 1.2 to get 0.6?

And it might not jump out
immediately in your brain but

lucky for us we can think about this

a little bit methodically.

So one thing I like to do is say okay,

I have the c on the left hand side,

and I'm just multiplying it by 1.2,

it would be great if this just said c.

If this just said c instead of 1.2c.

So what can I do there?

Well I could just divide by 1.2

but as we've seen multiple times,

you can't just do that
to the left hand side,

that would change, you no longer could say

that this is equal to that if
you only operate on one side.

So you have to divide
by 1.2 on both sides.

So on your left hand
side, 1.2c divided by 1.2,

well that's just going to be c.

You're just going to be left with c,

and you're going to have
c is equal to 0.6 over 1.2

Now what is that equal to?

There's a bunch of ways
you could approach it.

The way I like to do
it is, well let's just,

let's just get rid of the decimals.

Let's just multiply the
numerator and denominator

by a large enough number so
that the decimals go away.

So what happens if we multiply

the numerator and the denominator by...

Let's see if we multiply them by 10,

you're gonna have a 6 in the numerator

and 12 in the denominator,
actually let's do that.

Let's multiply the numerator
and denominator by 10.

So once again, this is the
same thing as multiplying by

10 over 10, it's not changing
the value of the fraction.

So 0.6 times 10 is 6,
and 1.2 times 10 is 12.

So it's equal to six
twelfths, and if we want

we can write that in a
little bit of a simpler way.

We could rewrite that
as, divide the numerator

and denominator by 6, you get 1 over 2,

so this is equal to one half.

And if you look back at
the original equation,

1.2 times one half, you could
view this as twelve tenths.

Twelve tenths times one half

is going to be equal to six tenths,

so we can feel pretty good

that c is equal to one half.

Let's do another one.

Let's say that we have 1 over 4

is equal to y over 12.

So how do we solve for y here?

So we have a y on the right hand side,

and it's being divided by 12.

Well the best way I can think of

of getting rid of this
12 and just having a y

on the right hand side is
multiplying both sides by 12.

We do that in yellow.

So if I multiply the
right hand side by 12,

I have to multiply the
left hand side by 12.

And once again, why did I pick 12?

Well I wanted to multiply by some number,

that when I multiply it by y over 12

I'm just left with y.

And so y times 12 divided by 12,

well that's just going to be 1.

And then on the left hand
side you're going to have

12 times one fourth,
which is twelve fourths.

So you get 12 over 4, is equal to y.

Or you could say y is equal
to 12 over 4, y is equal to,

let me do that just so you
can see what I'm doing,

just flopping the sides, doesn't
change what's being said,

y is equal to 12 over 4.

Now what is twelve fourths?

Well, you can view this
as 12 divided by 4,

which is 3, or you could
view this as twelve fourths

which would be literally, 3 wholes.

So you could say this would be equal to 3.

Y is equal to 3, and you can check that.

One fourth is equal to 3 over 12,

so it all works out.

That's the neat thing about equations,

you can always check to see
if you got the right answer.

Let's do another one, can't stop.

4.5 is equal to 0.5n

So like always, I have my n
already on the right hand side.

But it's being multiplied by 0.5,

it would be great if it just said n.

So what can I do?

Well I can divide both sides,
I can divide both sides

by 0.5, once again, if I do
it to the right hand side

I have to do it to the left hand side.

And why am I dividing by 0.5?

So I'm just left with an
n on the right hand side.

So this is going to be,
so on the left hand side,

I have 4.5 over 0.5, let me just,

I don't want to skip too many steps.

4.5 over 0.5, is equal to n,

because you have 0.5 divided by 0.5,

you're just left with an n over here.

So what does that equal to?

Well 4.5 divided by 0.5,

there's a couple ways to view this.

You could view this as forty-five tenths

divided by five tenths,
which would tell you

okay, this is going to be 9.

Or if that seems a little bit confusing

or a little bit daunting, you
can do what we did over here.

You could multiply the
numerator and the denominator

by the same number, so that
we get rid of the decimals.

And in this case, if you multiply by 10

you can move the decimal one to the right.

So once again, it has to be multiplying

the numerator and the
denominator by the same thing.

We're multiplying by 10 over
10, which is equivalent to 1,

which tells us that we're not

changing the value of this fraction.

So let's see, this is going to be 45

over 5, is equal to n.

And some of you might say wait wait wait,

hold on a second, you just
told us whatever we do

to one side of the equation,
we have to do to the other side

of the equation and here you are,

you're just multiplying the
left hand side of this equation

by 10 over 10.

Now remember, what is 10 over 10?

10 over 10 is just 1.

Yes, if I wanted to, I could
multiply the left hand side

by 10 over 10, and I could
multiply the right hand side

by 10 over 10, but that's
not going to change the value

of the right hand side.

I'm not actually changing
the values of the two sides.

I'm just trying to
rewrite the left hand side

by multiplying it by 1 in
kind of a creative way.

But notice, n times 10 over
10, well that's still going

to just be n.

So I'm not violating this principle

of whatever I do to the left hand side

I do to the right hand side.

You can always multiply one side by 1

and you can do that as
many times as you want.

Like the same way you can add 0

or subtract 0 from one side,

without necessarily having to show

you're doing it to the other side,

because it doesn't change the value.

But anyways, you have n
is equal to 45 over 5,

well what's 45 over 5?

Well that's going to be 9.

So we have 9 is equal to,
why did I switch to green?

We have 9 is equal to n, or
we could say n is equal to 9.

And you could check that:

4.5 is equal to 0.5 times
9, yup half of 9 is 4.5

Let's do one more, because
once again I can't stop.

Alright, let me get some space here,

so we can keep the different
problems apart that we had.

So let's do, let's have
a different variable now.

Let's say we have g
over 4 is equal to 3.2.

Well I wanna get rid
of this dividing by 4,

so the easiest way I
can think of doing that

is multiplying both sides by 4.

So I'm multiplying both sides by 4,

and the whole reason is 4
divided by 4 gives me 1,

so I'm gonna have g is equal
to, what's 3.2 times 4?

Let's see 3 times 4 is
12, and two tenths times 4

is eight tenths, so it's
gonna be 12 and eight tenths.

G is going to be 12.8, and
you can verify this is right.

12.8 divided by 4 is 3.2.