Question

How do you convert $0.183$ ($183$ repeating) to a fraction?

Answer 1

Hint: We will consider the fraction to be a variable $x$ then since there is repetition, we will multiply the fraction to remove the $3$ decimal places and then subtract the initial value from the original value of the fraction such that we can eliminate the recurring terms in the decimal and then we will simplify the expression to get the required solution.

Complete step by step solution:
We have the number given to us as:
$\Rightarrow 0.183$ ($183$ repeating)
We will consider the number to be therefore, it can be written as:
$\Rightarrow x=0.183$ ($183$ repeating)
Now to remove the recurring decimal place and since there are $3$ digits which are recurring, we will multiply both the sides of the equation by $1000$. It is to be kept in mind that even after multiplying, the number is still recurring.
$\Rightarrow 1000x=183.183$ ($183$ repeating)
We can see that the number will keep on recurring even after multiplication therefore, to remove the recurring decimals, we will subtract the original value $x$ from both sides of the equation.
$\Rightarrow 1000x-x=183.183-x$
On simplifying the left-hand side of the equation, we get:
$\Rightarrow 999x=183.183-x$
On substituting the value of $x$ in the right-hand side of the equation, we get:
$\Rightarrow 999x=183.183-0.183$
On simplifying the right-hand side of the equation, we get:
$\Rightarrow 999x=183$
Note that in the above step we have removed the recurring decimal places. Now on transferring the term $999$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{183}{999}$
Now the above term can be written as:
 $\Rightarrow x=\dfrac{3\times 61}{3\times 333}$
On simplifying, we get:
$\Rightarrow x=\dfrac{61}{333}$, which is the required fraction.

Note: In this question we have been given a recurring number which can also be expressed using the bar notation in which a bar is put on the recurring decimal places therefore the number given in the question can be written as $x=0.\bar{183}$. These types of numbers are also called as irrational numbers.