Question

The value of $ 0.\overline {05} $ is equal to
(A) $ \dfrac{3}{{99}} $
(B) $ \dfrac{4}{{99}} $
(C) $ \dfrac{5}{{99}} $
(D) None of these

Answer 1

Hint: 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 the help of basic algebraic rules such as transposition. First we assume the recurring decimal expression to be a variable. Then, we multiply both sides of the equation by a power of ten so as to get the entire repeating is to the left of the decimal point. Then, we subtract the equations and obtain the value of the variable in the form of a fraction.

Complete step by step solution:
For converting the given repeating and recurring decimal expansion into fraction, let us assume $ x = 0.\overline {05} $ .
Writing the expanded form of the decimal expansion, we get
 $ x = 0.050505....... - - - - - (1) $
Since the repetition starts immediately after the decimal point and in pairs of two, so we should multiply the equation $ \left( 1 \right) $ by $ 100 $ 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 $ 100 $ , we get
 $ 100x = 100\left( {0.050505.....} \right) $
 $ \Rightarrow $ $ 100x = 5.050505....... - - - - - (2) $
Converting back to condensed form, we get
 $ \Rightarrow $ $ 100x = 5.\overline {05} $
Now subtracting equation $ \left( 1 \right) $ from equation \[\left( 2 \right)\], we get
 $ \left( {100x - x} \right) = \left( {5.050505....} \right) - \left( {0.050505....} \right) $
Simplifying with help of algebraic rules such as transposition, we get
 $ \Rightarrow $ $ \left( {100x - x} \right) = 5.0000.... $
 $ \Rightarrow $ $ 99x = 5 $
Simplifying with help of algebraic rules and dividing both sides of the above equation by $ 99 $ ,
 $ \Rightarrow $ $ x = \dfrac{5}{{99}} $
So, $ 0.\overline {05} $ can be represented as a fraction $ \dfrac{5}{{99}} $ .
So, the correct answer is “Option c”.

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. One must take care of the calculations so as to be sure of the final answer. We must cancel the common factors in numerator and denominator so that the fraction is in the simplest form.