Q19. - Algebra 1

Question

Write each repeating decimal as a fraction in simplest form.

$0.\overline{5}$

Step-by-Step Solution

Answer

The simplest form is $=\frac{5}{9}$.

Step 1. Interpret the notation.

 $0.\overline{5}$ means 0.5555…..

Step 2. Assign a variable.

Let N = 0.5555…... 

Step 3. Decide by what to multiply.

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

10N = 10(0.5555…..)

10N = 5.5555…..

Step 4. Solve for N.

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

$\frac{10\mathrm{N}=5.5555\dots .-\left(\mathrm{N}=0.5555\dots .\right)}{9\mathrm{N}=5}$

Then divide each side by 9 and simplify.

$\frac{9\mathrm{N}}{9}=\frac{5}{9}$

$\mathrm{N}=\frac{5}{9}$