Question

6.6666666 in fraction form

Answer 1

To convert the repeating decimal \(6.6666666\ldots\) into a fraction, we can represent it as \(6 + 0.6666666\ldots\).

First, let's convert \(0.6666666\ldots\) to a fraction. We can let \(x = 0.6666666\ldots\).

Now, multiply both sides of the equation by 10:

\[
10x = 6.6666666\ldots
\]

Now we subtract the original equation from this new equation:

\[
10x - x = 6.6666666\ldots - 0.6666666\ldots
\]

This simplifies to:

\[
9x = 6
\]

Now, divide both sides by 9:

\[
x = \frac{6}{9} = \frac{2}{3}
\]

Now we substitute back into our original expression:

\[
6.6666666\ldots = 6 + 0.6666666\ldots = 6 + \frac{2}{3}
\]

To add these together, we can express \(6\) as a fraction:

\[
6 = \frac{18}{3}
\]

Now we add the two fractions:

\[
\frac{18}{3} + \frac{2}{3} = \frac{18 + 2}{3} = \frac{20}{3}
\]

Thus, the repeating decimal \(6.6666666\ldots\) can be expressed as the fraction \(\frac{20}{3}\).