How do you convert 0.789 (789 repeating) to a fraction?

Question

How do you convert 0.789  (789 repeating) to a fraction?

Eva Abramson · Accepted Answer

Do some algebra and reasoning to find #.bar(789)=263/333#.

The process for converting repeating decimals to fractions is confusing at first, but with practice it's pretty easy. You begin by setting #x# equal to #.789789...#: #x=.bar(789)# Then, multiply the equation by #1000#: #1000x=789.bar(789)# We do this so we can move one chunk of the repeating part to the left of the decimal point. This sets us up for the next, most important step: subtracting #x# from both sides. #1000x-x=789.bar(789)-x# On the left side of the equation, this is simply #999x#. On the right side, change #x# back to #.bar(789)#: #789.bar(789)-.bar(789)# And take a good look at this subtraction problem: #789.bar(789)# #ul(-color(white)(L).bar(789))# #?# The #.bar(789)# cancels! #789cancel(.bar(789))# #ul(-color(white)(L)cancel(.bar(789)))# #789# The right side of the equation becomes #789#, so we have: #999x=789# To solve for #x#, we divide #789# by #999# and simplify: #x=789/999=263/333# Therefore, #263/333=.bar(789)#.