To turn \(0.(6)=0.6666...\) into a fraction you should do such steps:
1 step. Set up an equation by representing the repeating decimal with a variable. Using your example, you will let x represent the repeating decimal 0.(6), so you have x=0.666... .
2 step. Identify how many digits are in the repeating pattern, or n digits. Multiply both sides of the equation from Step 1 by \(10^n\) to create a new equation. Again, using your example, you see that the repeating pattern consists of just one digit: 6. Now multiply both sides of the equation by \(10^1 = 10\). Thus, you have \(10x = 10 \cdot 0.666...\) or \(10x = 6.666....\).
3 step. Subtract the equation in Step 1 from the equation in Step 2. Notice that when we subtract these equations, our repeating pattern drops off. Therefore, \(10x-x=6.666...-0.666...\\ 9x=6\).
4 step. You now have an equation that you can solve for x and simplify as much as possible, using x as a fraction: \(9x = 6\). If you divide both sides by 9, you get \(x=\dfrac{6}{9}\). When simplified, you get that \(x=\dfrac{2}{3}\).
Answer: \(0.(6)=0.666... =\dfrac{2}{3}\).
