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### Course: 8th grade > Unit 1

Lesson 1: Repeating decimals- Converting a fraction to a repeating decimal
- Writing fractions as repeating decimals
- Converting repeating decimals to fractions (part 1 of 2)
- Converting repeating decimals to fractions
- Converting repeating decimals to fractions (part 2 of 2)
- Converting multi-digit repeating decimals to fractions
- Writing repeating decimals as fractions review
- Writing fractions as repeating decimals review

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# Converting repeating decimals to fractions (part 2 of 2)

Repeated decimals can be converted into fractions by shifting the decimal to the right and subtracting the decimals. To do this, multiply the number by 10 to the second power, then subtract. For example, 0.363636 repeating is 4/11 and 0.7141414 repeating is 707/990. Another example is 3.257257257 repeating, which is 3257/999. This calculation can be done in the head or by borrowing. After the subtraction, the numerator and denominator can be reduced and the fraction can be simplified. Created by Sal Khan.

## Want to join the conversation?

- what is 0.333333333333333 in a fraction(35 votes)
- Here's a little table of repeating decimals. Notice that they all follow a pattern:

1/9 = 0.111111111111111...

2/9 = 0.222222222222222...

3/9 = 0.333333333333333...

4/9 = 0.444444444444444...

5/9 = 0.555555555555555...

6/9 = 0.666666666666666...

7/9 = 0.777777777777777...

8/9 = 0.888888888888888...

Because 3/9 = 1/3 and 6/9 = 2/3, the following are also true:

1/3 = 3/9

1/3 = 0.333333333333333...

2/3 = 6/9

2/3 = 0.666666666666666...

Hope this helps!(123 votes)

- Why would the repeating decimal 0.714141414... which equals x be multiplied by 100 instead of 1000 or 10?(44 votes)
- so you will have 71 left over and you will get rid of the other numbers that are after the decimal.(21 votes)

- How would you solve a number like 5.345555555...?(17 votes)
- You can do it in this way too...

x=5.345555555...

10x=53.455555...

100x=534.5555...

1000x=5345.555...

Since the decimal part is same, we can subtract 100x from 1000x.

So,

1000x=5345.555...

- 100x=-534. 555...*________________________*

900x=4811*________________________*

x=4811/900

Since x=5.345555...

5.345555...=4811/900(20 votes)

- In the second example, where Sal gets a decimal numerator, I found it easier to use 1000x and 10x instead. Why 1000x and 10x? You make a subtraction where the minuend (top part) is the number multiplied by 10 as many times as you need to move the decimal point the to the
*right*side of the repeating part. For 0.714 with 14 repeating, you multiply by 1000 to get to 714.14 with the last 14 repeating.

For the subtrahend (bottom part of a subtraction), you multiply the number by 10 as many times as you need to get the decimal point to the*left*side of the repeating part. For 0.714 with 14 repeating, use 10x to get to 7.14 with 14 repeating. Now the repating part is directly after the decimal point in both minuend and subtrahend, so they cancel out nicely.(22 votes)- Even I was confused as to why was he complicating it. I found using 1000x and 10x easier as well.(3 votes)

- 2 Minutes into studying and my brain abandoned skull(16 votes)
- when will i use any of this in the future?(14 votes)
- I don't know they just make school students learn the bases of math some you will use a lot some you might never use and some that make absolutely no sense whatsoever(1 vote)

- do not vote for me(9 votes)
- At least you are original, trying to use reverse psychology to try and get votes without breaking the rules of not soliciting votes.(8 votes)

- But, do you have to divide for every single problem? Is there a simpler way?(3 votes)
- Good question! Yes, there’s an alternative method. For this answer, we will consider just repeating decimals between 0 and 1 (if the repeating decimal is greater than 1 or negative, we can convert the part after the decimal point to a fraction and so make a mixed number, negative fraction, or negative mixed number.)

To create the denominator, we use a digit 9 for every digit in the repeating group, then we add a digit 0 to the right for every digit after the decimal point not part of the repeating group (if any).

To create the numerator, we subtract the number formed by decimal digits not part of the repeating group (if any), from the number formed by the decimal digits up to and including the last digit of the first occurrence of the repeating group.

Then we reduce the fraction as needed.

Example: convert 0.4136767... (where 67 repeats) to a fraction.

Two digits (67) repeat, and three digits after the decimal point (413) are not part of the repeating group. So we use 99000 for the denominator.

We use 41367 - 413 = 40954 for the numerator.

So the answer is 40954/99000, which reduces to 20477/49500.(13 votes)

- why sometimes we do 100-1 and sometimes 100-10(7 votes)
- What about if you have a number like 0.67 with the sevens repeating?(4 votes)
- You do it the same way, this time though you have to times it by 10. Because there is only one number recurring.

So, it will be:

10x = 6.77777... recurring.

-x = 0.67777 (By the way the decimal point should be lined up don't know why it's not.)

9x = 6.10000

Then you divide both sides by 9:

9x/9 = 6.1/9

And you can cancel out the 9x/9, leaving 6.1/9.

Finally, you times 6.1/9 both by ten, leaving 61/90.

So, there's your answer, x = 61/90!

Hope this helps!(7 votes)

## Video transcript

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.