Question

What is 8.3 repeating as a fraction?

Answer 1

To turn
`8.(3)=8.333...` 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 8.(3), so you have x=8.333... .

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: 3. Now multiply both sides of the equation by
`10^1 = 10`. Thus, you have
`10x = 10 \cdot 8.333...` or
`10x = 83.333....`.

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=83.333...-8.333...\\ 9x=75`.

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 = 75`. If you divide both sides by 9, you get
`x=(75)/(9)`. When simplified, you get that
`x=(25)/(3)`.

Answer:
`8.(3)=8.333...=(25)/(3)=8(1)/(3)`.