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:

$x = 0 . ¯ ¯¯¯¯ ¯ 123$ $x = 0.bar123$

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

$1000 x = 123 . ¯ ¯¯¯¯ ¯ 123$ $1000x = 123.bar123$

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

$1000 x - x = 123 . ¯ ¯¯¯¯ ¯ 123 - 0 . ¯ ¯¯¯¯ ¯ 123$ $1000x - x = 123.bar123 - 0.bar123$

We can now solve for $x$ $x$ as follows:

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

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

$999 x = 123 + (0 . ¯ ¯¯¯¯ ¯ 123 - 0 . ¯ ¯¯¯¯ ¯ 123)$ $999x = 123 + (0.bar123 - 0.bar123)$

$999 x = 123 + 0$ $999x = 123 + 0$

$999 x = 123$ $999x = 123$

$\frac{999 x}{999} = \frac{123}{999}$ $(999x)/color(red)(999) = 123/color(red)(999)$

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

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

$x = 41/333$