Question

0.6 repeated + 9/10

Answer 1

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 \( 1 \frac{17}{30} \).