Question

Write each repeating decimal as a fraction in the simplest form.

$0.\overline{4}$

Short answer

The simplest form is $=\frac{4}{9}$.

Step-by-step solution

Step 1. Interpret the notation.

$0.\overline{4}$means 0.4444…..

Step 2. Assign a variable.

Let N = 0.4444…...

Step 3. Decide by what to multiply.

Since only one-digit repeats so we multiply each side by 10.

10N = 10(0.4444…..)

10N = 4.4444…..

Step 4. Solve for N.

Subtract N from 10N, so that the repeating part eliminates.

$$\begin{gathered} 10\mathrm{N}=4.4444\dots . \\ \frac{-\left(\mathrm{N}=0.4444\dots .\right)}{9\mathrm{N}=4} \end{gathered}$$

Then divide each side by 9 and simplify.

$\frac{9\mathrm{N}}{9}=\frac{4}{9}$

$\mathrm{N}=\frac{4}{9}$