Question

What rational equals 0.6 with a line over 6

Answer 1

The notation "0.6 with a line over 6" represents the repeating decimal \(0.6666\ldots\). This can also be expressed as \(0.6\overline{6}\).

To convert this repeating decimal into a fraction, let’s define:

\[
x = 0.6\overline{6}
\]

This can be rewritten as:

\[
x = 0.6666\ldots
\]

Now, to eliminate the repeating part, we can multiply by 10:

\[
10x = 6.6666\ldots
\]

Now we have two equations:

1. \(x = 0.6666\ldots\)
2. \(10x = 6.6666\ldots\)

Now we can subtract the first equation from the second:

\[
10x - x = 6.6666\ldots - 0.6666\ldots
\]

This simplifies to:

\[
9x = 6
\]

Now, solve for \(x\):

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

Which can be simplified to:

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

Therefore, the rational number that equals \(0.6\overline{6}\) is \(\frac{2}{3}\).