Reduce Interminate Decimals to a Common Denominator
An
interminate decimal is another word for a repeating
decimal. To find the common denominator there are two cases: pure repeating
decimals and leading digits followed by repeating decimals.
In
the case of pure repeating decimals set out the numbers in their cycles until
the first time the end of the cycles coincide. The common denominator is made
up of the replications of the digit 9; the number of replications is the length
of the common cycle.
Example:
Find
the common denominator of and .
Set
out the repeating decimals in their cycles separating each cycle by  .
974974
565656
The
end of the two cycles coincides after 6 digits. Therefore 999999 is the common
denominator.
In
the case of leading digits that are not repeated find the decimal fraction that
contain the largest number of leading digits that are not repeated. Set out the
repeating digits in their cycles beginning after nonrepeating digits until the
first time that the cycles coincide. The common denominator is made up of
replications of the digit 9 followed by replications of the digit 0. The number
of 9’s is the length of the common cycle and the number of 0’s is largest
number of leading digits that are not repeated.
Example:
Find
the common denominator of , , and .
The
longest length of no repeating digits is 2. Set out the all the numbers with
repeating digits in their cycles after the first 2 digits.
.3 

.04 

.72 
572572572572 
.14 
562456245624 
The
end of the cycles coincides after 12 digits. Therefore 99999999999900 is the
common denominator.