Use equation (1.8) to find the fraction that are equivalent to following repeating decimals.

11. 0.678571428571428571…

In this question it must convert the repeating decimal 0.678571428571428571… 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:

$\begin{array}{rcl}0.678571428571428571...& =& 0.25+0.428571+0.000000428571+...\\ & =& \frac{1}{4}+\frac{428571}{{10}^6}+\frac{428571}{{10}^{12}}+...\\ & =& \frac{1}{4}+\sum _{i=1}^{\infty }\frac{428571}{{10}^{6i}}\end{array}$

Implies that,

$\begin{array}{rcl}0.678571428571428571...& =& \frac{1}{4}+\sum _{i=1}^{\infty }\frac{428571}{{1000000}^i}\\ & =& \frac{1}{4}+428571\sum _{i=1}^{\infty }\frac{1}{{1000000}^i}\\ & & \end{array}$

Solve it further,

$$\begin{gathered} \begin{array}{rcl}0.678571428571428571...& =& \frac{1}{4}+428571\sum _{i=1}^{\infty }{\left(\frac{1}{1000000}\right)}^i\\ & =& \frac{1}{4}+428571\left[-1+\sum _{i=1}^{\infty }{\left(\frac{1}{1000000}\right)}^i \\ =\right]\\ & & \end{array} \end{gathered}$$

Further, solve the above expression,

$$\begin{gathered} 0.678571428571428571...\text{=}\frac{1}{4}+428571\left[-1+\frac{1}{1-\left(\frac{1}{1000000}\right)}\right] \\ \text{=}\frac{1}{2}+428571\left(-1+\frac{1000000}{999999}\right) \\ \text{=}\frac{1}{4}+428571\left(\frac{1}{999999}\right) \\ =\frac{19}{28} \end{gathered}$$

Hence, the required answer is $0.678571428571428571...=\frac{19}{28}$.