Question

Express $0.\bar{4}$ as a rational number simplest form?

Answer 1

Hint: To solve this kind of question we need to have the knowledge of recurring number and rational number. Rational numbers are the numbers which are expressed in p/q form where q is not equal to zero. To solve the question the first step is to consider the variable $x$ as the number given in the question. Multiply the number by $10$, and will further find the difference of the value of $x$ and $10x$, which results into a fraction.

Complete step-by-step solution:
The question asks us to express $0.\bar{4}$ as a rational number in the simplest form. Now the number $0.\bar{4}$ can be written as $0.44444$ which means the bar is used in the place of a repeatable number. On calculation of fraction the first step is to consider the value of $x$ as $0.\bar{4}$, which means:
$\Rightarrow x=0.\bar{4}$
$\Rightarrow x=0.444$
On multiplying the number of both the side with $10$, the decimal point shift by one place toward right we get:
$\Rightarrow 10x=4.\bar{4}$
On subtracting the $x$ from $10x$ and $0.\bar{4}$ from $4.\bar{4}$ , we get
$\Rightarrow 10x-x=4.\bar{4}-0.\bar{4}$
$\Rightarrow 9x=4$
On dividing both side of the number by $9$ we get:
$\Rightarrow \dfrac{9x}{9}=\dfrac{4}{9}$
Since there is no common factor of both the numerator and the denominator of the fraction, the fraction becomes $\dfrac{4}{9}$.
$\Rightarrow x=\dfrac{4}{9}$
$\therefore $ Express $0.\bar{4}$ as a rational number simplest form $\dfrac{4}{9}$.

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 $6.94$ is multiplied with $10$ the decimal point shifts towards right by one digit resulting in $69.4$.