te $$ 0,5 \dot{6} $$ 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 $$ x = 0.5\dot{6} $$, which means

$$
x = 0.56666\ldots
$$

Multiply by 10 to shift the decimal point:

$$
10x = 5.66666\ldots
$$

Subtract the original $$ x $$ from this expression:

$$
10x - x = 5.66666\ldots - 0.56666\ldots
$$

This gives

$$
9x = 5.1
$$

Express $$ 5.1 $$ as a fraction:

$$
5.1 = \frac{51}{10}
$$

Now solve for $$ x $$:

$$
x = \frac{\frac{51}{10}}{9} = \frac{51}{90}
$$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 3:

$$
x = \frac{51 \div 3}{90 \div 3} = \frac{17}{30}
$$

Thus, in simplified form, $$ 0.5\dot{6} $$ can be written as

$$
\frac{17}{30}
$$
$$ 0.5\dot{6} = \frac{17}{30} $$

te \( 0,5 \dot{6} \) in the form \( \frac{a}{b} \) (where \( a \) and \( b \) are integers). Write your answer as a simplified fraction.