How do you convert 0.78023 (023 repeating) as a fraction?

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Answer 1 · 2016-03-07T15:04:44

$x=77945/99900$

Given: $" "0.78bar023$

'~~~~~~~~~~~~~~~~~~~~~~~~
Note about formatting: using the 'hash' key

write: hash key 0.78bar023 hash key and you get : $" "0.78bar023$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let $" "x=0.78bar023$

Then $1000x=780.23bar023$

Notice that the repeating parts start in the 3rd decimal place for both $" "x" and "1000x$ . Consequently subtraction removes the repeating part.

$1000x-x=779.45$

$x(1000-1)=779.45$

$x=(779.45)/(999)$

But we do not wish to have a decimal numerator

$x=(779.45)/(999) xx 100/100" "=" "77945/99900$