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Converting decimals to fractions
By Hazel S. Powlesland on Monday, November 11, 2002 - 08:31 pm:

I know who to change recurring decimals to fractions but what about ones like 0.6434343....... or others of the same sort

By Demetres Christofides on Monday, November 11, 2002 - 08:47 pm:

You can understand this better with an example.
Suppose you have x = 3.2451451451...
The trick is to multiply x by appropriate powers of 10.
We have 10x = 32.451451451451....
and 10000x= 32451.451451451451.....
Now 9990x = 10000x - 10x = 32419
So x = 32419/9990

Can you write now 0.643434343... as a fraction ?

Demetres

By Emma McCaughan on Monday, November 11, 2002 - 10:15 pm:

Don't forget that you can often simplify at the end:

x=0.818181818...
100x= 81.818181818...
99x=81
x = 81/99 = 9/11

By Kerwin Hui on Monday, November 11, 2002 - 10:47 pm:

If you just want a quick rule and are happy to add fractions, you have a way of writing down the answer immediately (well, as a sum of two fractions):

Suppose we have the number 0.52567567567567567567...

(1) Identify the repeating sequence (e.g. in this case it is 567).
(2) Write down the same number of 9's as you have in the repeating sequence (i.e. 999)
(3) Add 0's after the 9 in (2), the number of zeros to add is precisely the number of digits after the decimal point before your sequence starts (in this case, we have 99900)
(4) Put your repeating sequence on the numerator, and what you have in (3) in the denominator (so we have 567/99900=21/3700).
(5) Now write down the fraction that represents everything in front of your repeating sequence (i.e. 0.52=52/100=13/25).
(6) Add the result from (5) and (6) together. This is the fraction you are looking for (in this case it is 389/740).

Kerwin

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