Chapter 0.2, Problem 19E

To determine

The simplest form of $0.\overline{5}$ as fraction.

Answer to Problem 19E

The simplest of $0.\overline{5}$ as fraction is $\frac{5}{9}$ .

Explanation of Solution

Given information:

The repeating decimal $0.\overline{5}$ .

Formula used:

To convert a repeating decimal to a fraction where the repeating digits is single, multiply both sides by 10 and simplify it further.

Calculation:

Consider the providedrepeating decimal $0.\overline{5}$ .

Let us denote $0.\overline{5}$ it as $\text{x}=0.\overline{5}$ .

Recall that to convert a repeating decimal to a fraction where the repeating digits is single, multiply both sides by 10 and simplify it further.

So, multiply $\text{x}=0.\overline{5}$ by 10both sides, we get,

  $\begin{array}{c}\text{x}=0.\overline{5}\\ 10\times \text{x}=10\times 0.\overline{5}\\ 10\text{x}=5.\overline{5}\end{array}$

Now, subtract $\text{x}=0.\overline{5}$ from $\text{10x}=5.\overline{5}$ .

So, it will be written as,

  $\begin{array}{c}10\text{x}-\text{x}=5.\overline{5}-0.\overline{5}\\ 9\text{x}=5\\ \text{x}=\frac{5}{9}\end{array}$

Since, $\frac{5}{9}$ cannot be simplified further,

Hence, the simplified form of repeating decimal $0.\overline{5}$ is $\frac{5}{9}$ .