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

2 Answers
Jun 24, 2016

#0.789bar789 = 789/999#

Explanation:

This is written as #0.789bar789#

Let #x=0.789bar789# ...............................Equation (1)

Then #1000x = 789.789bar789# ............Equation (2)

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

So #1000x-x=789#

#=>999x=789#

Thus #x= 789/999#

Jun 24, 2016

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

Explanation:

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)#.