Convert to a Simplified Fraction 0.6666666666666666666666666667

$0.\overline{6}$

Step 1

Create two equations where the only numbers to the right of the decimal place are the repeating part.

Step 2

Set up an equation with $0.\overline{6}$ .

$x = 0.\overline{6}$

Step 3

Multiply both sides of $x = 0.\overline{6}$ by $10$ to create the second equation.

$10 x = 6.\overline{6}$

Step 4

Subtract $x = 0.\overline{6}$ from $10 x = 6.\overline{6}$ to remove the repeating part.

$9 x = 6$

Step 5

Divide each term in $9 x = 6$ by $9$ and simplify.

Divide each term in $9 x = 6$ by $9$ .

$\frac{9 x}{9} = \frac{6}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{6}{9}$

Divide $x$ by $1$ .

$x = \frac{6}{9}$

Simplify the right side.

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 2}{3 \cdot 3}$

Cancel the common factor.

$x = \frac{3 \cdot 2}{3 \cdot 3}$

Rewrite the expression.

$x = \frac{2}{3}$

Step 6

Write as a mixed number by taking the whole number part along with the solution to the equation.

$0 \frac{2}{3}$

Step 7

Convert $0 \frac{2}{3}$ to an improper fraction.

A mixed number is an addition of its whole and fractional parts.

$0 + \frac{2}{3}$

Add $0$ and $\frac{2}{3}$ .

$\frac{2}{3}$