Question

How do you convert \[0.\overline{18}\] \[\left( {18\;{\text{repeated}}} \right)\] to a fraction?

Answer 1

Hint: In this question we convert a decimal repeated number to a fraction. For converting decimal repeated numbers to fractions, there are two methods.
First is a simple calculation.
The second is by using the formula.
The formula is that decimal number multiplied with a repeated number (which is denoted below) minus non repeating part and it is divided by repeating number (which is denoted as below. And the formula is written as below.
\[ \Rightarrow \dfrac{{\left( {DN*F} \right) - NRP}}{D}\]
Where, \[DN\] is decimal number, \[F\] is \[10\] if one repeating number, \[100\]if two repeating number, \[1000\]if three repeating number etc, \[NRP\] is non repeating part of decimal number and \[D\] is\[9\] if one repeating number, \[99\] if two repeating number, \[999\] if three repeating number etc.

Complete step by step answer:
In this question, we have given a repeated decimal number and we need to convert it into fraction form.
We have given the decimal number is \[0.18\mathop {18}\limits^ - \]
Let’s assume that
\[ \Rightarrow x = 0.\mathop {18}\limits^ - ............\left( 1 \right)\]
Now we will multiply both sides with \[10\]in Equation \[1\] . Then,
\[ \Rightarrow 10x = 1.8\mathop {18}\limits^ - .............\left( 2 \right)\]
Again, we will multiply both sides with \[100\] in equation $1$. Then,
\[ \Rightarrow 100x = 18.\mathop {18}\limits^ - ...........\left( 3 \right)\]
Now, we will subtract the equation \[2\] from equation \[3\] as,
Then,
\[ \Rightarrow 100x - x = 18.\mathop {18}\limits^ - - .\mathop {18}\limits^ - \]
After simplification we will get,
\[ \Rightarrow 99x = 18\]
On further simplification of the equation we get,
\[\therefore x = \dfrac{{18}}{{99}}\]
Now, we will be divided numerator and denominator by \[9\] and get,
\[\therefore x = \dfrac{2}{{11}}\]

Therefore, the fraction is\[\dfrac{2}{{11}}\].

Note:
As we know that we can convert repeated decimal numbers to the fraction by using the formula.
By using the formula, we calculate the factor of the decimal repeated number.
The formula is
\[\dfrac{{\left( {DF*F} \right) - NRP}}{D}\]
\[
  DN = .\mathop {18}\limits^ - \\
  F = 100 \\
  NRP = 0 \\
  D = 99 \\
 \]
Put these values in the above formula.
Then,
\[ \Rightarrow \dfrac{{\left( {0.18 \times 100} \right) - 0}}{{99}} = \dfrac{{18}}{{99}} = \dfrac{2}{{11}}\]
Therefore, the fraction is \[\dfrac{2}{{11}}\].