To turn [tex] 3.(8)=3.888... [/tex] into a fraction you shouls 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 3.(8), so you have x=3.888... .
2 step. Identify how many digits are in the repeating pattern, or n digits. Multiply both sides of the equation from Step 1 by [tex] 10^n [/tex] to create a new equation. Again, using your example, you see that the repeating pattern consists of just one digit: 8. Now multiply both sides of the equation by [tex] 10^1 = 10 [/tex]. Thus, you have [tex] 10x = 10 \cdot 3.888... [/tex] or [tex] 10x = 38.888.... [/tex].
3 step. Subtract the equation in Step One from the equation in Step Two. Notice that when we subtract these equations, our repeating pattern drops off. Therefore, [tex] 10x-x=38.888...-3.888...\\ 9x=35 [/tex].
4 step. You now have an equation that you can solve for x and simplify as much as possible, using x as a fraction: [tex] 9x = 35 [/tex]. If you divide both sides by 9, you get [tex] x=\dfrac{35}{9} [/tex]. When simplified, you get that [tex] x=3\dfrac{8}{9} [/tex].
Answer: [tex] 3.(8)=3.888...=3\dfrac{8}{9} [/tex].
