91Ó°ÊÓ

In this question it must convert the repeating decimal 0.61111 into fraction with the help of geometric series.

Repeating Decimal to Fraction:

A fraction is a number that represents a portion of a whole in mathematics. It is, in other words, a two-number ratio. In contrast, a decimal is a number with a decimal point between the total number and the fractional sections. There are two kinds of decimal numbers: terminating and non-terminating decimals, as well as repeating and non-repeating decimals. When dealing with many mathematical problems, converting decimal to fractional value is favoured because fractional values may be readily simplified.

The repeating decimal can be written as:

$\begin{array}{rcl}0.61111...& =& 0.6+0.01+0.001+...\\ & =& \frac{3}{\text{5}}+\frac{1}{100}+\frac{1}{1000}+...\\ & =& \frac{3}{\text{5}}+\underset{i=0}{\overset{∞}{∑}}\left(\frac{1}{100}\right)\frac{1}{{10}^i}\\ & & \text{=}\frac{3}{\text{5}}+\underset{i=0}{\overset{∞}{∑}}\left(\frac{1}{100}\right){\left(\frac{1}{10}\right)}^i\\ & & \end{array}$

Further, solve the above expression,

$\begin{array}{rcl}0.61111...& =& \frac{3}{5}+\frac{\frac{1}{100}}{1-\left(\frac{1}{10}\right)}\\ & =& \frac{11}{18}\\ & & \end{array}$

Hence, the required answer is $0.61111...=\frac{11}{18}$.