Converting Repeating Decimals into Fractions

Date: 06/25/2008 at 13:02:11
From: Sarah
Subject: Converting repeating decimals into fractions

How do you convert repeating decimals (in which only one number at the
end is repeating) into fractions?

ex. 0.91666666666666...
ex. 0.38888888888888...
ex. 0.84733333333333...

I know how to do a single number repeating decimal like 0.6666... by
multiplying by 10 and subtracting:

10N = 6.6666...
  N = 0.6666...
---------------
 9N = 6.0000...
  N = 6/9 or 2/3

But I am not able to convert repeating decimals where there are other
numbers before the part that repeats.  There is only one repeating
number, so I thought you would have to multiply it by 10, but it
doesn't give the right answer.

0.9166666 x 10 = 9.16666, then I don't know what to do with a decimal
(how to put it as a numerator)

or

0.9166666 x 1000 = 916

916
--- , but that's the wrong answer.
999

Date: 06/25/2008 at 14:57:33
From: Doctor Achilles
Subject: Re: Converting repeating decimals into fractions

Hi Sarah,

Thanks for writing to Dr. Math, and for explaining your thinking so
clearly.  You are right about how to do a single-digit repeating
decimal by multiplying by 10.  

So to solve

  0.7777777....

you know how to come up with 7/9.  Let's look at a couple of similar
problems.  If you had

  6.7777777.....

you could think of it as 6 + 0.7777... so you would have

  6.7777... = 6 + 0.7777... = 6 + 7/9 = 6 7/9 or 61/9

depending on if you wanted a mixed number or an improper fraction.

Can you use similar thinking to try these two:

  0.0777777...

and

  0.6777777...     ?

Can you please give that a shot and write back and show me your work?  

Good luck!

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/29/2008 at 19:16:26
From: Sarah
Subject: Converting repeating decimals into fractions

1) 0.077777777....
Convert 0.77777 to a fraction:
Multiply by 10. (7.7777)
Subtract, making sure to keep both sides balanced.

10 N = 7.7777...
   N = 0.7777...
----------------
 9 N = 7

Divide both sides by 9 and 0.777777 as a fraction is 7/9.

Because the repeating numbers start in the hundredths place (second
number to the right of the decimal, the denominator must be 2 digits.
Therefore, you add a 0 to the end. (7/90)
Answer: 7/90

2) 0.6777777
   0.6 + 0.0777777

From before: 0.07777 as a fraction is 7/90 so

6/10 + 7/90
54/90 + 7/90 = 61/90

Date: 06/30/2008 at 13:36:00
From: Doctor Achilles
Subject: Re: Converting repeating decimals into fractions

Hi Sarah,

Great work, you did those perfectly!  Now, can you solve the problems
you wrote in with?

  0.91666666666666...

  0.38888888888888...

  0.84733333333333...

If you're stuck on any of those or if you want me to double check an
answer, please write back.

As an extra challenge, you can try these out if you want:

  0.7676767676767676...

  55.7676767676767676...

  0.557676767676767676...

  0.12341234123412341234...

Let me know what you get.  If you're stuck and you want help with any
of those, or if you have any other questions about repeating decimals,
please let me know.

Best wishes.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/01/2008 at 00:16:34
From: Sarah
Subject: Converting repeating decimals into fractions

Can you please make sure that these problems are all correct, 
especially the challenge problems.  Thanks. 

1) 0.91666666666666...
   0.91 + 0.0066666

Convert 0.66666 to a fraction and get 6/9 or 2/3.

Repeating numbers start in the thousandths place (third number right
of the decimal), so denominator must be three digits so 2/300.

 91/100 + 2/300
273/300 + 2/300 = 275/300
Simplify 275/300 = 11/12
Answer: 11/12

2) 0.38888888888888...
   0.3 + 0.08888
Convert 0.8888 to a fraction and get 8/9.

Repeating numbers start in the hundredths place (second number right
of the decimal), so denominator must be two digits so 8/90.

 3/10 + 8/90
27/90 + 8/90 = 35/90
Simplify 35/90 = 7/18
Answer: 7/18

3) 0.84733333333333...
   0.847 + 0.0003333
Convert 0.3333 to a fraction and get 3/9 or 1/3.

Repeating numbers start in the ten-thousandths place (fourth number
right of the decimal), so denominator must be four digits so 1/3000.

 847/1000 + 1/3000
2541/3000 + 1/3000 = 2542/3000
Simplify 2542/3000 = 1271/1500
Answer: 1271/1500

Extra Challenge:

1) 0.7676767676767676...
Multiply by 10 to the power of two (100) (76.76767676767676)
Subtract, making sure to keep both sides balanced.

100 N = 76.767676...
    N =  0.767676...
--------------------
 99 N = 76

Divide both sides by 99.
Answer: 76/99

2) 55.7676767676767676...
   55 + 0.767676...
As shown above, 0.767676... as a fraction is 76/99
Answer: 55 76/99

3) 0.557676767676767676...
   0.55 + 0.0076767676

As shown above, 0.767676 as a fraction is 76/99
Repeating numbers start in the thousandths place, so denominator
must add two digits so 76/9900.

  55/100  + 76/9900
5445/9900 + 76/9900 = 5521/9900
Answer: 5521/9900

4) 0.12341234123412341234...
Multiply by 10 to the power of 4. (10,000) 1234.123412341234
Subtract making sure to keep both sides balanced.

10,000 N = 1234.12341234...
       N =    0.12341234...
---------------------------
 9,999 N = 1234

Divide both sides by 9,999
Answer: 1234/9999

Date: 07/01/2008 at 11:55:01
From: Doctor Achilles
Subject: Re: Converting repeating decimals into fractions

Hi Sarah,

Excellent work, it looks like you've got the hang of this.  You've
done all the problems including the challenges perfectly.  If you have
any other questions, please let me know.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/01/2008 at 11:55:01
From: Doctor Riz
Subject: Re: Converting repeating decimals into fractions

Hi Sarah -

Nice work with Dr. Achilles!  I just wanted to show you a similar but
slightly different way to do these problems, which builds in your step
of changing the denominator depending on how many places there are
before the repeating portion of the decimal starts.

You've shown that you understand the idea of rewriting the number by
multiplying by powers of 10, and that's the key to this approach.

Let's look at 0.677777... again.  My goal is to produce two versions
of that number that both have the repeating 7 starting right after the
decimal.  Clearly one way to do that is to multiply by 10:

     N = 0.67777...
  10 N = 6.77777...

Now let's multiply the original number again.  Since one number is
repeating, we need one more power of 10 this time, or 100:

      N =  0.6777...
  100 N = 67.7777...

Now we can subtract the two equations that we made:

  100 N = 67.7777...
   10 N =  6.7777...
  ------------------
   90 N = 61

Dividing by 90,

  N = 61/90

Can you see how this method ensures the proper denominator without
having to count the places and adjust the denominator as you did 
above?  The key is to get the repeating parts to match up after the
decimal.

So, for a number like 

  0.12787878...

we'd multiply by 10^2 or 100 to get the 78 to start repeating right
after the decimal.  Then, since two numbers are repeating, we'd
multiply again by two more powers of 10, so 10^4 or 10,000:

         N = 0.127878...

  10,000 N = 1278.7878...
     100 N =   12.7878...
  ----------------------------
   9,900 N = 1266
         
Dividing by 9,900, we have N = 1266/9900, which reduces to 211/1650.

You can decide which method you prefer.  I just wanted to include this
second approach for your consideration.

Again, great work on the problems you did for Dr. Achilles!
 
- Doctor Riz, The Math Forum
  http://mathforum.org/dr.math/