Question

How do you find the repeating decimal $0.36$ with $36$ being repeated as 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 $2$ decimal places and then subtract the initial value and simplify to get the required value of $x$.

Complete step by step answer:
We have the number given to us as: $0.36$ ($36$ being repeated)
We will consider the number to be $x$ therefore, it can be written as:
$\Rightarrow x=0.36$ ($0.36$being repeated).
Now to remove the recurring decimal place, we will multiply both the sides of the equation by $100$.
On multiplying, we get:
$\Rightarrow 100x=36.36$
Now on subtracting the term $x$ from both the sides, we get:
$\Rightarrow 1000x-x=123.123-x$
Now on substituting the value of $x$ in the right-hand side, we get:
$\Rightarrow 100x-x=36.36-0.36$
Now on subtracting the value, we get:
 $\Rightarrow 99x=36$
Now on transferring the term $99$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{36}{99}$
Now on dividing the numerator and denominator by $3$, we get:
$\Rightarrow x=\dfrac{12}{33}$, which is the required solution.

Note:
It is to be remembered that whenever a value is added, subtracted, multiplied or divided on both the sides of the equation, the value of the equation does not change.
It is to be remembered that when a term which is in multiplication is transferred across the $=$ sign, it has to be written as division. Same rule applies for addition and subtraction.
In the above question, we have a smaller number in the numerator and a larger number in the denominator. These types of fractions are called proper fractions. There also exist improper fractions which are the opposite of it.