To add \( 0.6 \) repeated (which is \( 0.666... \)) and \( \frac{9}{10} \), we first express \( 0.666... \) as a fraction.
Let \( x = 0.666... \). Then, multiplying both sides by 10:
\[ 10x = 6.666... \]
Subtracting the original \( x \) from this equation:
\[ 10x - x = 6.666... - 0.666... \] \[ 9x = 6 \] \[ x = \frac{6}{9} = \frac{2}{3} \]
So, \( 0.666... = \frac{2}{3} \).
Now, we need to add \( \frac{2}{3} \) and \( \frac{9}{10} \). To do this, we need a common denominator. The least common multiple of 3 and 10 is 30.
Converting both fractions:
\[ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \] \[ \frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30} \]
Now, add the two fractions:
\[ \frac{20}{30} + \frac{27}{30} = \frac{20 + 27}{30} = \frac{47}{30} \]
Thus, \( 0.6 \text{ repeated} + \frac{9}{10} = \frac{47}{30} \).
As a mixed number, this can also be written as:
\[ \frac{47}{30} = 1 \frac{17}{30} \]
So the final answer is \( \frac{47}{30} \) or \( 1 \frac{17}{30} \).