Write $$ 0,2 \dot{6} $$ in the form $$ \frac{a}{b} $$ (where $$ a $$ and $$ b $$ are integers). Write your answer as a simplified fraction.

Question

UpStudy AI · Accepted Answer

Let $$ x = 0.2\dot{6} $$. Note that the non-repeating part is $$2$$ (one digit) and the repeating part is $$6$$ (one digit).

1. Multiply $$ x $$ by $$10$$ to shift the non-repeating part:
   $$
   10x = 2.\dot{6}
   $$

2. Multiply $$ x $$ by $$100$$ to shift one full cycle of the repeating part:
   $$
   100x = 26.\dot{6}
   $$

3. Subtract the first equation from the second:
   $$
   100x - 10x = 26.\dot{6} - 2.\dot{6}
   $$
   $$
   90x = 24
   $$

4. Solve for $$ x $$:
   $$
   x = \frac{24}{90} = \frac{4}{15}
   $$

Thus, $$ 0.2\dot{6} = \frac{4}{15} $$.
$$ 0.2\dot{6} = \frac{4}{15} $$

Write \( 0,2 \dot{6} \) in the form \( \frac{a}{b} \) (where \( a \) and \( b \) are integers). Write your answer as a simplified fraction.