To express the repeating decimal \(0.08585858...\) (which can be denoted as \(0.\overline{0858}\)) as a fraction, we can set up an equation.
Let \(x = 0.08585858...\).
To eliminate the repeating part, we can multiply \(x\) by \(10000\) (since the repeating part has 4 digits):
\[ 10000x = 85.85858... \]
Now, we can set up another equation for \(x\):
\[ x = 0.08585858... \]
Next, we subtract the second equation from the first:
\[ 10000x - x = 85.85858... - 0.08585858... \]
This simplifies to:
\[ 9999x = 85.772727... \]
However, we want to focus on the stable repeating part which was not enough. Thus, we need to keep the original multiplication correct first.
Next, we recognize that within this procedure, we actually wanted the repeating part relative to \(10000\) starting from our initial assumption directly. Going right back preserves a clear transition.
From \(0.085\) multiplied directly to:
| We recall that;
0.0858 (repeats) = 8585... Above achieves a timed deduction just right with \(10000x = 85 + x\). Simplifying gives Reduction:
(9999x = 85 → x = 85/9999)
So, in this regard;
The decimal \( 0.\overline{0858}\) translates to the fraction \(\frac{85}{999} \).
Thus, the fraction representing \(0.08585858...\) is:
\[\frac{85}{999}.\]
And that describes the representation smoothly as requested.