Question 1901: How do you convert the recurring decimal 0.23333333 to a fraction. Give your answer in it's simplest form.
Found 4 solutions by khwang, Robee16, Alan3354, greenestamps:
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website!
Let the upper bar means the repeatitive decimals.
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0.23 = 0.2 + 0.03
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= 0.2 + 1/10 * 0.3
(since by geometric series: 0.3 = 0.3/(1-0.1) = 0.3/0.9 = 3/9)
= 0.2 + 1/10 * 3/9
= 1/5 + 1/30
= 7/30
Kenny
Answer by Robee16(18)  (Show Source):
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! How do you convert the recurring decimal 0.23333333 to a fraction. Give your answer in it's [sic] simplest form.
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= 0.2 + 0.033333...
n = 0.03333....
10n = 0.3333...
-1n =-0.0333...
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9n = 0.3
n = 0.3/9 = 1/30
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1/5 + 1/30 = 7/30
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it's = it is
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Here is a variation of the process shown by tutor @alan3354.
(1) n = 0.233333...
(2) 10n = 2.33333...
Subtract (1) from (2):
9n = 2.1
n = 2.1/9 = 21/90 = 7/30
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Here is a different method, without using algebra.
0.2333333... = 2.333333.../10 = (2 1/3)/10 = (7/3)/10 = 7/30.
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And here is a shortcut that is based on the basic algebraic method.
The numerator of the fraction is (a) all the digits up through the first cycle of the repeating digits, minus (b) just the non-repeating digits. In this example, that is 23-2 = 21.
The denominator of the fraction is one 9 for each repeating digit, followed by one 0 for each non-repeating digit. In this example, with one non-repeating digit and one repeating digit, that is 90.
The fraction is then 21/90 = 7/30.
That shortcut is a great time saver if speed is important, as in a math competition. Here are a couple of further examples of this method.
2 non-repeating digits; one repeating:
0.4166666... = (416-41)/900 = 375/900 ( = 5/12)
3 repeating digits, 2 non-repeating:
0.12345345345... = (12345-12)/99900 = 12333/99900 (which can also be simplified, if required)
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