Chapter 0.2, Problem 21E

To determine

The simplest form of $0.\overline{13}$ as fraction.

Answer to Problem 21E

The simplest of $0.\overline{13}$ as fraction is $\frac{13}{99}$ .

Explanation of Solution

Given information:

The repeating decimal $0.\overline{13}$ .

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 $0.\overline{13}$ .

Let us denote $0.\overline{13}$ it as $\text{x}=0.\overline{13}$ .

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 $\text{x}=0.\overline{13}$ by 100 both sides, we get,

  $\begin{array}{c}\text{x}=0.\overline{13}\\ 100\times \text{x}=100\times 0.\overline{13}\\ 100\text{x}=13.\overline{13}\end{array}$

Now, subtract $\text{x}=0.\overline{13}$ from $\text{100x}=13.\overline{13}$ .

So, it will be written as,

  $\begin{array}{c}100\text{x}-\text{x}=13.\overline{13}-0.\overline{13}\\ 99\text{x}=13\\ \text{x}=\frac{13}{99}\end{array}$

Since, $\frac{13}{99}$ cannot be simplified further,

Hence, the simplified form of repeating decimal $0.\overline{13}$ is $\frac{13}{99}$ .