Question

Write each repeating decimal as a fraction in simplest form.

$0.\overline{21}$

Short answer

The simplest form is$=\frac{21}{99}$

Step-by-step solution

Step 1. Interpret the notation.

$0.\overline{21}$means 0.212121…..

Step 2. Assign a variable.

Let N = 0. 212121…...

Step 3. Decide by what to multiply.

Since two-digits repeat so we multiply each side by 100.

100N = 100(0.212121…..)

100N = 21.212121…..

Step 4. Solve for N.

Subtract N from 100N, so that the repeating part eliminates.

$$\begin{gathered} 100N=21.212121\dots . \\ \frac{-\left(\mathrm{N}=0.212121\dots .\right)}{99\mathrm{N}=21} \end{gathered}$$

Then divide each side by 99 and simplify.

$\frac{99\mathrm{N}}{99}=\frac{21}{99}$

$\mathrm{N}=\frac{21}{99}$