Question

Convert $0.\overline {135}$ ($135$ being repeated) to a fraction?

Answer 1

Given question involves the concepts of rational numbers and repeating as well as recurring decimal expansions. We have to convert a repeating decimal expansion into a fraction. A bar on top of a decimal number means that the numbers are repeated after regular intervals. Such numbers can be represented as fractions with help of basic algebraic rules such as transposition.

Complete step by step solution:
For converting the given repeating and recurring decimal expansion into fraction, let us assume $x = 0.\overline {135}$.
Writing the expanded form of the decimal expansion, we get,
$x = 0.135135135..... - - - - (1)$
Since repetition of decimal expansion starts from the very first decimal place, we should multiply the complete decimal expansion by $1000$ so as to keep the repeating entity at the immediate right side of the decimal point so that we can subtract the two equations and get rid of the repeating entity.
So, multiplying both sides of equation $\left( 1 \right)$ with $1000$, we get,
$1000x = 1000\left( {0.135135135.....} \right)$
$\Rightarrow 1000x = 135.135135135..... - - - - - (2)$
Converting back to condensed form, we get
$\Rightarrow 1000x = 135.\overline {135}$
Hence, $1000x = 135.\overline {135}$
Now subtracting equation $\left( 1 \right)$ from equation $\left( 2 \right)$, we get,
$\left( {1000x - x} \right) = \left( {135.135135135....} \right) - \left( {0.135135135....} \right)$
Simplifying with help of algebraic rules such as transposition, we get
$\Rightarrow 999x = \left( {135} \right) - \left( 0 \right)$
$\Rightarrow 999x = \left( {135} \right)$
$\Rightarrow x = \dfrac{{135}}{{999}}$
On cancelling the common factors in both numerator and denominator, we get,
$\Rightarrow x = \dfrac{{15}}{{111}}$
On simplifying further, we get,
$\Rightarrow x = \dfrac{5}{{37}}$
So, $0.\overline {135}$ can be represented as fraction $x = \dfrac{5}{{37}}$.

Note: The method given above is the standard method to solve such types of questions with ease. Then, we have to decide by looking at the nature of repeating identity, what to multiply to keep the repeating entity at the immediate right side of the decimal point. Then, we can subtract the original equation from the new one and get the value of decimal expansion as a fraction. We can also verify the answer by converting back the fraction into decimal expansion.