Question

What is the simplest fraction form of the sum 0.6¯ + 9/10?

Answer 1

To solve the problem, we need to convert \(0.6\overline{6}\) (which is the same as \(0.666...\)) into a fraction and then add it to \(\frac{9}{10}\).

### Step 1: Convert \(0.6\overline{6}\) to a fraction

Let \(x = 0.6\overline{6}\). We can represent this decimal as follows:

\[
x = 0.6666\ldots
\]

To eliminate the repeating part, multiply both sides by 10:

\[
10x = 6.6666\ldots
\]

Next, we can set up the following equation:

\[
10x = 6 + x
\]

Now, subtract \(x\) from both sides:

\[
10x - x = 6
\]

This simplifies to:

\[
9x = 6
\]

Now we divide both sides by 9:

\[
x = \frac{6}{9}
\]

We can simplify \(\frac{6}{9}\) by dividing numerator and denominator by their greatest common divisor, which is 3:

\[
x = \frac{2}{3}
\]

Thus, \(0.6\overline{6} = \frac{2}{3}\).

### Step 2: Add \(\frac{2}{3}\) to \(\frac{9}{10}\)

To add \(\frac{2}{3}\) and \(\frac{9}{10}\), we first need a common denominator. The least common multiple of the denominators 3 and 10 is 30.

We convert each fraction to have a denominator of 30:

\[
\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30}
\]

\[
\frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30}
\]

Now we can add the two fractions:

\[
\frac{20}{30} + \frac{27}{30} = \frac{20 + 27}{30} = \frac{47}{30}
\]

### Conclusion

Thus, the simplest fraction form of the sum \(0.6\overline{6} + \frac{9}{10}\) is

\[
\boxed{\frac{47}{30}}
\]