How do you convert 0.8 (8 repeating) to a fraction?

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Answer 1 · 2016-06-07T16:25:04

$0 . ¯ 8 = \frac{8}{9}$ $0.bar8=8/9$

We have a repeating decimal, $0 . ¯ 8$ $0.bar8$

Let us put this equal to $x$ $x$ .

Then,

$x = 0 . ¯ 8$ $x=0.bar8$

Multiply both sides by $10$ $10$ :

$10 x = 8 . ¯ 8$ $10x=8.bar8$

We can write this $8 . ¯ 8$ $8.bar8$ as a sum of a whole number and a decimal number:

$10 x = 8 + 0 . ¯ 8$ $10x=color(red)(8)+color(blue)(0.bar8)$

Now, we know that $x = 0 . ¯ 8$ $x=0.bar8$

$10 x = 8 + x$ $10x=8+x$

$10 x - x = 8 + x - x$ $10xcolor(red)(-x)=8+xcolor(red)(-x)$

$9 x = 8$ $9x=8$

$\frac{9 x}{9} = \frac{8}{9}$ $(9x)/color(red)(9)=8/color(red)(9)$

$x = \frac{8}{9}$ $x=8/9$

Therefore,

$0 . ¯ 8 = \frac{8}{9}$ $0.bar8=8/9$