Converting Decimals to Fractions
TERMINATING DECIMALS: Put the decimal’s digits in the numerator. In the denominator, the number of zeros equals the number of digits behind the decimal. Example: 0.079 = 79/1000
SIMPLE REPEATING DECIMALS: Put the decimal’s repeating digits in the numerator. In the denominator, the number of nines equals the number of repeating digits. Example: 0.7979797979… = 79/99.
COMPLEX REPEATING DECIMALS: Subtract the non-repeating digits from the digits after the decimal point that include one set of the repeating decimals. Put this number in the numerator. In the denominator, the number of nines equals the number of repeating digits and the number of zeros equals the number of non-repeating digits. Example: 0.12379797979… = (12379 - 123) / 99000 = 12256/99000 (which can then be simplified).
SIMPLE REPEATING DECIMALS: Put the decimal’s repeating digits in the numerator. In the denominator, the number of nines equals the number of repeating digits. Example: 0.7979797979… = 79/99.
COMPLEX REPEATING DECIMALS: Subtract the non-repeating digits from the digits after the decimal point that include one set of the repeating decimals. Put this number in the numerator. In the denominator, the number of nines equals the number of repeating digits and the number of zeros equals the number of non-repeating digits. Example: 0.12379797979… = (12379 - 123) / 99000 = 12256/99000 (which can then be simplified).
posted by ASL at 6:22 AM
6 Comments:
William’s Rule (why this works without using algebra): Split the decimal into two parts, one part is the repeating decimal and one part is the terminating decimal. Example: 0.12379797979… = 0.123 + 0.00079797979… The repeating decimal can be further simplified: 0.001(.79797979…). Change all the decimals into fractions: 123/1000 + 1/1000(79/99). Then combine the fractions: 123/1000 + 79/99000 = 99(123)/99(1000) + 79/99000 = (12300-123+79)/99000 = 12256/99000
I like this one - seems there is always some new approach (at least new to me) in math - esp. since converting "non-factors of ten" decimals into fractions has never been a strong point in the past for me and while I could follow the text's example, I didn't readily extend it to complex repeating decimals. Because I followed William's approach, I hadn't felt the discussion had been prematurely cut off - but I really appreciate the e-mail about handling a situation arising in class for which one isn't prepared! This is certainly helpful in all situations!
Thank you for posting this andrea, I am good with the common decimals but other than that I seem to have some problems. So thank you!!!
If you have a decimal number like 1.232323231111... my method can be adapted to work for separating different repeting decimal expansions in the same number.
this is the same as 1 + 23/99 - 23/9900000000 + 1/9000000000
Example #2, 3.44444555666...
this is the same as 3 + 4/9 - 4/900000 + 5/900000 - 5/900000000 + 6/900000000
Finish by placing these portions of the sum under common denominators, add them up, and then simplify to lowest terms or a mixed number
I wrote down both methods, thank you Andrea and William. I honestly don't even remember learning how to convert repeating decimals in highschool math or my previous college math class.. This is so brilliant!
Thank for all posting the rules and steps to refresh ones memory. It was needed very much! Again thank you! It was a very big help!
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