Question

Write $0.\overline{475}$as a fraction. Describe the steps you take to write the fraction.

Short answer

The fraction number of $0.\overline{475}$ is $\frac{475}{999}$.

Step-by-step solution

Step 1. Given information

The decimal number is $0.\overline{475}$.

Step 2. Calculation 

Let $x=0.\overline{475}$

$x=0.475475475$ …… (1)

Clearly, 4 is the only digit which is repeating in equation (1)

Multiply equation (1) by 1000 on both sides,

$1000x=475.475475.....$ …… (2)

Subtract equation (1) from equation (2),

$\left(1000x=475.\mathrm{475475...}\right)-\left(x=0.\mathrm{475475...}\right)$

$\begin{array}{c}1000x-x=(475.\mathrm{475475.....})+(0.\mathrm{475475475.....})\\ 999x=475\\ x=\frac{475}{999}\end{array}$

Thus, the fraction number that can be formed is $\frac{475}{999}$.