There are a couple of different approaches you can use for this. Here's one.
1. Determine how many digits repeat. (There is just one repeating digit.)
2. Call your number x. Multiply x by 10 to the power of the number of digits found in step 1.
[tex]x = 0.8\overline{3}\\10^{1}x=8.3\overline{3}[/tex]
3. Subtract the original number, then solve for x.
[tex]10x-x=9x=8.3\overline{3}-0.8\overline{3}=7.5\\\\x=\dfrac{7.5}{9}=\dfrac{5}{6}[/tex]
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If you recognize that 0.333... (repeating) is 1/3, then you know that 0.0333... (repeating) is 1/10×1/3 = 1/30. Add that to 0.8 = 4/5 and you get
... 4/5 + 1/30 = 24/30 + 1/30 = 25/30 = 5/6
