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In the last video,
we did some examples

where we had one digit
repeating on and on forever,

and we were able to convert
those into fractions.

In this video, we want to tackle
something a little bit more

interesting, which is
multiple digits repeating

on and on forever.

So let's say I had
0.36 repeating, which

is the same thing as 0 point--
since the bar's over the 3

and the 6, both of
those repeat-- 363636.

And it just keeps going on and
on and on like that forever.

Now the key to doing
this type of problem

is, so like we did
in the last video,

we set this as equal to x.

And instead of just
multiplying it by 10-- 10

would only shift
it one over-- we

want to shift it over enough
so that when we line them up,

the decimal parts will still
line up with each other.

And to do that we,
want to actually shift

the decimal space
two to the right.

And to shift it
two to the right,

we have to have multiplied by
100 or 10 to the second power.

So 100x is going to
be equal to what?

We're shifting this two
to the right-- one, two.

So 100x is going to be equal
to-- the decimal is going

to be there now, so it's
going to be 36.363636

on and on and on forever.

And then let me
rewrite x over here.

We're going to subtract
that from the 100x.

x is equal to 0.363636
repeating on and on forever.

And notice when we multiplied by
100x, the 3's and the 6's still

line up with each other
when we align the decimals.

And you want to
make sure you get

the decimals lined
up appropriately.

And the reason why
this is valuable

is now that when we
subtract x from 100x,

the repeating parts
will cancel out.

So let's subtract.

Let us subtract
these two things.

So on the left-hand side,
we have 100x minus x.

So that gives us 99x.

And then we get, on
the right-hand side,

this part cancels
out with that part.

And we're just left with 36.

We can divide both
sides by 99, and we

are left with x is
equal to 36 over 99.

And both the numerator and the
denominator is divisible by 9,

so we can reduce this.

If we divide the
numerator by 9, we get 4.

The denominator
by 9-- we get 11.

So 0.363636 forever and
forever repeating is 4/11.

Now let's do another
interesting one.

Let's say we have the number
0.714, and the 14 is repeating.

And so this is the same thing.

So notice, the 714
isn't going to repeat.

Just the 14 is going to repeat.

So this is 0.7141414,
on and on and on and on.

So let's set this equal to x.

Now you might be tempted to
multiply this by 1,000x to get

the decimal all the
way clear of 714.

But you actually
don't want to do that.

You want to shift it just enough
so that the repeating pattern

can be right under itself
when you do the subtraction.

So again in this
situation, even though we

have three numbers
behind the decimal point,

because only two of
them are repeating,

we only want to multiply
by 10 to the second power.

So once again, you want
to multiply by 100.

So you get 100x is equal to--
we're moving the decimal two

to the right, one,
two-- so it's going

to be 71.4141, on
and on and on and on.

So it's going to be 71.4141414
and on and on and on.

And then let me rewrite
x right below this.

We have x is equal to 0.7141414.

And notice, now the
141414's, they're

lined up right below each other.

So it's going to work
out when we subtract.

So let's subtract these things.

100x minus x is 99x.

And this is going to be equal
to-- these 1414's are going

to cancel with those 1414's.

And we have 71.4 minus 0.7.

And we can do this in our head,
or we can borrow if you like.

This could be a 14.

This is a 0.

So you have 14 minus 7
is 7 and then 70 minus 0.

So you have 99x
is equal to 70.7.

And then we can divide
both sides by 99.

And you could see all of
the sudden something strange

is happening because we
still have a decimal.

But we can fix
that up at the end.

So let's divide
both sides by 99.

You get x is equal
to 70.7 over 99.

Now obviously, we
haven't converted this

into a pure fraction yet.

We still have a decimal
in the numerator.

But that's pretty easy to fix.

You just have to multiply the
numerator and denominator by 10

to get rid of this decimal.

So let's multiply the numerator
by 10 and the denominator

by 10.

And so we get 707/990.

Let's do one more
example over here.

So let's say we had something
like-- let me write this way--

3.257 repeating, and we want to
convert this into a fraction.

So once again, we
set this equal to x.

And notice, this is
going to be 3.257257257.

The 257 is going to
repeat on and on and on.

Since we have three
digits that are repeating,

we want to think about 1,000x,
10 to the third power times x.

And that'll let us
shift it just right

so that the repeating
parts can cancel out.

So 1,000x is going
to be equal to what?

We're going to shift
the decimal three

to the right-- one, two, three.

So it's going to
be 3,257 point--

and then the 257
keeps repeating.

257257257 keeps going on
and on and on forever.

And then we're going to
subtract x from that.

So here is x. x is equal to 3.

You want to make sure you
have your decimals lined up.

It's 3.257257257 dot, dot,
dot-- keeps going on forever.

And notice, when we
multiplied it by 1,000,

it allowed us to line up the
257's so that when we subtract,

the repeating part cancels out.

So let's do that subtraction.

On the left-hand side,
1,000 of something

minus 1 of that
something-- you're

left with 999 of that something.

This part is going to
cancel out with that part.

It's going to be equal to--
let's see, 7 minus 3 is 4.

And then you have the
5, the 2, and the 3.

So you get 999x
is equal to 3,254.

And then you can divide
both sides of this by 999.

And you are left with x
is equal to 3,254/999.

And so obviously, this
is an improper fraction.

The numerator is larger
than the denominator.

You could convert this to a
proper fraction if you like.

One way, you could
have just tried

to figure out what to the 0.257
repeating forever is equal to

and just had the 3
being the whole number

part of a mixed fraction.

Or you could just
divide 999 into 3,254.

Actually, we could do that
pretty straightforwardly.

It goes into it three times,
and the remainder-- well,

let me just do it, just
to go through the motions.

So 999 goes into 3,254.

It'll go into it three times.

And we know that because
this is originally 3.257,

so we're just going
to find the remainder.

So 3 times 9 is 27.

But we have to add
the 2, so it's 29.

3 times 9 is 27.

We have a 2, so it's 29.

And so we are left
with, if we subtract,

if we regroup or borrow or
however we want to call it,

this could be a 14.

And then this could be a 4.

Let me do this in a new color.

And then the 4 is still
smaller than this 9,

so we need to regroup again.

So then this could be a 14,
and then this could be a 1.

But this is smaller
than this 9 right

over here, so we regroup again.

This would be an 11,
and then this is a 2.

14 minus 7 is 7.

14 minus 9 is 5.

11 minus 9 is 2.

So we are left with--
did I do that right?

Yep-- so this is going to
be equal to 3 and 257/999.

And we're done.