Question

What is the fraction of $0.36$ with the $6$ repeating?

Answer 1

Hint: To solve this kind of question we need to have the knowledge of recurring number and fraction. To solve the question the first step is to consider the variable $x$ as the number given in the question. Then multiply the number by $10$ and $100$, and further find the difference of the two, which results into a fraction.

Complete step by step solution:
The question asks us to find the fraction of the number $0.36$ when $6$ is repeating. Now the number on repeating becomes $0.36666$. The number $0.36666$ becomes $0.3\bar{6}$, here the bar is used in place of a repeatable number. On calculation of fraction the first step is to consider the value of $x$ as $0.3\bar{6}$, which means:
$\Rightarrow x=0.3\bar{6}$
$\Rightarrow x=0.36666$
On multiplying the number of both the side with $10$, the decimal point shipt by one place toward right we get:
$\Rightarrow 10x=3.\bar{6}$
Again on multiplying the above term with $10$, the decimal point again shifts to right hand side by one place, we get:
$\Rightarrow 100x=36.\bar{6}$
On subtracting the $10x$ from $100x$ and $3.\bar{6}$ from $36.\bar{6}$ , we get
$\Rightarrow 100x-10x=36.\bar{6}-3.\bar{6}$
$\Rightarrow 90x=33$
On dividing both side of the number by $90$ we get:
$\Rightarrow \dfrac{90x}{90}=\dfrac{33}{90}$
We will cancel the common terms between the numerator and the denominator. In the above fraction, $3$ is common as it is the factor of both the numerator and the denominator of the fraction, which is $33$ and $90$ respectively present in the right hand side. On cancelling we get:
$\Rightarrow x=\dfrac{11}{30}$
$\therefore $ The fraction of $0.36$ with the $6$ repeating is $\dfrac{11}{30}$.

Note: On multiplying the decimal number with the ${{10}^{n}}$the place of the decimal point changes as per the value of n. If the value of “n” is $1$ the decimal point shifts by one digit toward the right. For example when a number $3.87$ is multiplied with $10$ the decimal point shifts towards right by one digit resulting in $38.7$.