Question

Convert a recurring decimal to vulgar fraction.
$0.222...$ into a vulgar fraction.
A.$\dfrac{2}{9}$
B.$\dfrac{2}{99}$
C.$\dfrac{22}{99}$
D.$\dfrac{222}{999}$

Answer 1

Hint: We must know what is a vulgar fraction and when does a recurring decimal occur. We must do some manipulation so that a decimal with recurring decimal places is converted into a fraction. A method to do this is in the step – by -step solution below. For the solution, we will be making sure that the answer that the fraction formed must be in its simplest form.

Complete step-by-step answer:
We will discuss the vulgar fraction and recurring decimal.
A recurring decimal is generally seen when we are converting a fraction with the denominator which does not divide any power of $10$ , i.e. contains a factor other than $2$ or $5$ in their simplest form. This can be seen in day to day life, for example, $\dfrac{1}{3}$ , etc.
A vulgar fraction is nothing new, we can say it is just a simple fraction, with the condition that neither of the numerator or the denominator is zero.
Now, for converting a recurring decimal into a vulgar fraction, we need to do some manipulation with the decimal.
Given decimal is: $0.222...$
Let $0.222...$ be equal to $x$ .
$\Rightarrow 0.222...=x$ - (1)
Multiplying equation (1) with $10$ , we get:
$10\left( 0.222...=x \right)$
$\Rightarrow 2.222...=10x$ - (2)
Now, for the manipulation part, we need to somehow eliminate the recurring decimal part. If we subtract equation (1) from equation (2), we get:
$2.222...-0.222...=10x-x$
$\Rightarrow 2=9x$
Therefore, $x=\dfrac{2}{9}$ .
Now, initially, we assumed the recurring decimal to be equal to $x$ , and now we found the fractional representation of $x$ , and hence converted the recurring decimal into a vulgar fraction.
Therefore, $0.222..$ is equivalent to the vulgar fraction $\dfrac{2}{9}$ .
Hence, the option A. $\dfrac{2}{9}$ , is the correct option.

Note: We must be very careful while doing the manipulation as specifically in this question, all of the digits we are recurring, but there will be many questions in which all the digits are not recurring. All we need do is to eliminate the recurring part of the decimal and after that, conversion to fraction will be very easy.