Chapter 0.2, Problem 22E

To determine

The simplest form of $0.\overline{21}$ as fraction.

Answer to Problem 22E

The simplest of $0.\overline{21}$ as fraction is $\frac{7}{33}$ .

Explanation of Solution

Given information:

The repeating decimal $0.\overline{21}$ .

Formula used:

To convert a repeating decimal to a fraction where the repeating digits are double, multiply both sides by 100 and simplify it further.

Calculation:

Consider the providedrepeating decimal $0.\overline{21}$ .

Let us denote $0.\overline{21}$ it as $\text{x}=0.\overline{21}$ .

Recall that to convert a repeating decimal to a fraction where the repeating digits are double, multiply both sides by 100 and simplify it further.

So, multiply $\text{x}=0.\overline{21}$ by 100 both sides, we get,

  $\begin{array}{c}\text{x}=0.\overline{21}\\ 100\times \text{x}=100\times 0.\overline{21}\\ 100\text{x}=21.\overline{21}\end{array}$

Now, subtract $\text{x}=0.\overline{21}$ from $\text{100x}=21.\overline{21}$ .

So, it will be written as,

  $\begin{array}{c}100\text{x}-\text{x}=21.\overline{21}-0.\overline{21}\\ 99\text{x}=21\\ \text{x}=\frac{21}{99}\end{array}$

Simplify it further as,

  $\begin{array}{c}\text{x}=\frac{21}{99}\\ \text{x}=\frac{7}{33}\end{array}$

Hence, the simplified form of repeating decimal $0.\overline{21}$ is $\frac{7}{33}$ .