How do you change repeating decimals into fractions?

Question

I'm a quick learner, but I can't seem to remember how to do this. To be more specific, some questions are: -8.4444... , .161616161616, -.13333333, 3.76666666
Please change these into fractions or mixed numbers.
Please answer these few questions and explain how to do it, and i will give you best answer. I am checking soon, so please help!!

Answer 1

I saw this on the same question asked by someone else and this was the best answer! I just copied it from the other answer.



Round it.
You know for example that 1/3=.3333333... So you can change -6.3 to -6 1/3. Or here is is another way. Simply make it -6.3. Don't change it.
If it was 6.666666, you could round up the number to 6.67.
This all depends how much the teacher wants the number to go. If to the hundreds, then 6.67. If to the tens, then 6.7. For many of these annoying repeating decimal numbers, the number can be simplified to something/9.
If it is .11111, then it is 1/9.
If it is .22222, then it is 2/9.
If it is .33333, then it is 3/9 or 1/3.
If it is .44444, then it is 4/9.
If it is .55555, then it is 5/9.
If it is so-on, blah blah.
Do you get it yet?
To change a repeating number to a fraction, simply change put the number over 9 and you'll get the fraction which is equivalent to it

Answer 2

Changing Repeating Decimals To Fractions

Answer 3

There is a quick trick to this.
If you have a repeating decimal such as .33333333, yuou ignore the negative sign and first you take the repeating part of the decimal. In this case it is 3. Then you divide this by n 9's, where n is the number of digits in the repeating part. In this case 1. Thus we have 3/9 or 1/3

1. -8.4444444444. First take out the -8 and we have .44444444. Following our rule, the only repeating digit is 4 and the number of digits repeating is 1. Thus .444444 is 4/9
This makes -8.44444444 into -8 and 4/9

2. .161616161616. This is a little different. This time there are 2 digits repeating, 16. Thus two nines are needed. Thus we have 16/99.

3. First remove the .1. Then we have .03333333333333. 3 is the repeating digit and one 9 is needed. So we have 3/9. But since there is a 0 in front of the 3's, we need to divide by 10. Thus we have 3/90 or 1/30. Adding the 1/10 back gives -4/30 or -2/15

4. 3.766666666. Take the 3.7 out and you are left with .0666666. This is the same as 3, except with a 6. Thus we have 6/9, and since there is a 0 in front, 6/90 or 1/15. Adding this to 3 and 7/10 gives 3 and 23/30

Extra. Say we have .123412341234. What would u do?
Since there is 1234 and 4 repeating digits, it would be 1234/9999. This can be done with more complicated numbers like .123456789012345678901234567890
Try that one your self, and check if it works

I hope this helps

Answer 4

Sharky.mark is right.

Here's -1.333...

Let x = -1.333...

Now, we want to be able to just have whole numbers that we can use in proper fractions and eliminate the repeating part and, as Sharky mark did, what we do is we get a multiple of our x that also has the repeating part after the decimal place. Then, we subtract the two from other, the x and the multiple of x that we made here, and the repeating parts magically cancel out, since they are the same, and we are left with an equation made up of whole numbers that we can then solve for our x in terms of a fraction.

So, let's just pick 10x. We could pick any other multiple of 10, but we are going with 10 here, since it is low and so it keeps things simple later when we need to reduce to lowest terms. The object here is to keep the repeating part on the right hand side of the decimal the same for both x and our multiple of x, which will be 10 x here. Now, if we had a more complicated repeating pattern, say .123123123..., multiplying just by 10 would leave us with a different number on the right hand of the he decimal. That wouldn't work to accomplish our goal, so we'd have to multiply by a higher power of 10 to line things up nicely. Here, we can just multiply by 10.

So, we have

10x = -13.333...
x = -1.333...

So 10x - x = -13.333... - (-1.333...)

Notice that, on the right hand side, the repeating decimals are the same in both terms and so they cancel out, and we just have -13 -(-1) which is -13+1 because subtracting negative something is the same as adding that something, so we get -12.

On the left hand side, we get 10x - x or x(10-1), or 9x.

So, 9x = -12 and x = -12/9

Now, -12/9 is -(3 x 4) / (3 x 3) and the threes cancel out, so we have -4/3.

So -1.333... is the same as -4/3 or -1 1/3.

Answer 5

Repeating Fractions

Answer 6

Divide the repeating fraction by the same number of nines:

0.33333333; '3' repeats, so fraction is 3/9 = 1/3

0.44444444; '4' repeats, so fraction is 4/9

0.011111111
  = 0.111111111 / 10; '1' repeats, so fraction is 1/9; divided by 10 is 1/90

3.72727272
  = 3 + 0.72727272; '72' repeats, so fraction is 72/99 = 8/11; 3-8/11

3.76666666
= 3.7 + 0.06666666 = 3.7 + 0.6666/10; '6' repeats, so 6/9 = 2/3
  Divide by 10 to get 2/30. Add (3 and 7/10) or (3 and 21/30) to get
2/30 + 3-21/30 = 3-23/30

Answer 7

Case 1: If the "repeating" part starts immediately after the decimal point (like 49.161616...):

Take the repeating part, and put it over some 9's (like "9" or "99" or "999", etc.) The number of 9's should be the same as the number of digits in the repeating part.

Examples:
49.161616... = 49 + 16/99
0.217217217... = 217/999

Case 2: If the repeating part does not start immediately after the decimal point (like: 5.123777777...):

The secret here is to rewrite the number as a "Case 1" decimal divided by some power of 10. For example:

5.12377777... = (5123.77777...) / 1000

Once you've got the number in "Case 1" form, just apply the "Case 1" rules:

5.12377777...
= (5123.77777...) / 1000
= (5123 + 7/9) / 1000

(From that point, you can use regular fraction addition, division, etc. to reduce it to lowest terms).

Answer 8

I'll show you a few examples...

Let's take -8.4444444
So let x = -8.4444444
So then
10x = -84.444444

Now subtract the two equations:
10x-x = -84.444444 - (-8.444444) = -76
so,
9x = -76
x = -76/9

Here's another example:
x = .161616161616...
100x = 16.1616161616
100x-x = 16.16161616 - .1616161616 = 16
So,
99x = 16
x = 16/99.

You need to set "x =" your number, multiply both sides by 10, 100, 1000, etc (by 10^number of digits repeating), subtract the two to get rid of the repeating digits, and voila!

Do a few and you will get the hang of it pretty quick....

good luck!

Answer 9

You would put the # over 10,100... if it wan't repeating like this: -8 4/10, 16/100, -13/100, 3 76/100 but since it is repeating you put it over 9, 99... like this: -8 4/9, .16/99, -13/99, 3 76/99 I hope this helps!!

Answer 10

OK round off to the nearest decimal... For example
1.3333333333333
to
1.3
then convert to a fraction the way you normally do...
Here's another method...
Multiply a number by the recurring decimal
eg:100*1.33
that will give you 133
now divide 133 by 100(or the number you chose to multiply the decimal by)
express the divsion you just did as a fraction..
133/100 ... Which is a fraction
The rounding up method is better