What is .123 (repeating) as a fraction?

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Answer 1 · 2018-04-12T12:46:05

See a solution process below:

First, we can write:

#x = 0.bar123#

Next, we can multiply each side by #1000# giving:

#1000x = 123.bar123#

Then we can subtract each side of the first equation from each side of the second equation giving:

#1000x - x = 123.bar123 - 0.bar123#

We can now solve for #x# as follows:

#1000x - 1x = (123 + 0.bar123) - 0.bar123#

#(1000 - 1)x = 123 + 0.bar123 - 0.bar123#

#999x = 123 + (0.bar123 - 0.bar123)#

#999x = 123 + 0#

#999x = 123#

#(999x)/color(red)(999) = 123/color(red)(999)#

#(color(red)(cancel(color(black)(999)))x)/cancel(color(red)(999)) = (3 xx 41)/color(red)(3 xx 333)#

#x = (color(red)(cancel(color(black)(3))) xx 41)/color(red)(color(black)(cancel(color(red)(3))) xx 333)#

#x = 41/333#